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P`@7`$ @@7ѠF@ayTw@U%@UР?%TM@`$T @BB@DRR5#6cCdED3@c v'R@@R 3@),/gG%RJ`A:x`@@1ATRR :xm@a7!сz}vARҖR }RR h V @ROL@]"҆ RҖR IЀ3|AE`@@@T  5?esy`@7`RR 4RVR " `@7`W @ @@@@R$R ҪR(  @RRR@lv1VTWAA@W@1@T@!PBH@Z@?@Z@7Ѡ`+ @@@wZKA`~@q@\@Ҭ@ ]e @`] ( ^WAA@U`@1@T@!AH@f@?@f@7Ѡ P!A`\7@7р T@@@g`n5?l@7Ѡ`R @7  R@@7@Q@lT@1@TG@g@@? OT#@'TBB@#DRRt `6CcED3@c  v3Rg5@ R 30 RR % RR  @  @K@a@ @a7!a @ @A!GT  @`5 @҉ RR A @ @67@M(7S@ ) @c5 ? @i@@7@`b^C@LG!@  RR .`a A@"H@"b@?ր_ l @2@@R R n @@Rv R q@K  A@"H@Ba@?S@G!@@@RҶ R M@@R R @R@"R 7@ @ @1@T @1@T@7Ѡ@B{"@ 7B"b BB g p@#'@!@?TqTRCA @RG@Z@:@z@g:*!@ZkTB#@!@?`T*'@@!GH@"F@?D@`8@(@B(!C@? [@!@CH@H@?H ҳ`J @J@7р*@7Ѡ+@@@!/TV@.@U @1@T@1@T@@7@,@@7!!6`@7`F@7Ѡ'G@@g@@?`%T#@w@@@@?@$T@_A`@@T6@@%.200s() takes no arguments (%zd given)%.200s() takes exactly one argument (%zd given)Bad call flags in __Pyx_CyFunction_Call. METH_OLDARGS is no longer supported!%.200s() takes no keyword argumentsunbound method %.200S() needs an argument__pyx_capi__Unable to initialize pickling for %sBig-endian buffer not supported on little-endian compilerBuffer acquisition: Expected '{' after 'T'Cannot handle repeated arrays in format stringDoes not understand character buffer dtype format string ('%c')Expected a dimension of size %zu, got %dExpected a comma in format string, got '%c'Expected %d dimension(s), got %dUnexpected end of format string, expected ')'join() result is too long for a Python string'%.200s' object is not subscriptablecannot fit '%.200s' into an index-sized integermemviewslice is already initialized!scipy/spatial/_qhull.cpython-312-aarch64-linux-gnu.so.p/_qhull.c%s (%s:%d)Module '_qhull' has already been imported. Re-initialisation is not supported.%d.%dscipy.spatial._qhullcompiletime version %s of module '%.100s' does not match runtime version %s_cython_0_29_37Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingcython_runtime__builtins__int (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, PyObject *, int, int, int)_get_delaunay_infoint (int, double *, double const *, double *, double)_barycentric_insidevoid (int, double *, double const *, double *, int)_barycentric_coordinate_singlevoid (int, double *, double const *, double *)_barycentric_coordinatesvoid (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, double const *, double *)_lift_pointdouble (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, int, double *)_distplaneint (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, double const *, double)_is_point_fully_outsideint (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, double *, double const *, double, double)_find_simplex_bruteforceint (__pyx_t_5scipy_7spatial_6_qhull_DelaunayInfo_t *, double *, double const *, int *, double, double)_find_simplex_directed_find_simplexnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is not PyCapsule object_ARRAY_API is NULL pointermodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule compiled against API version 0x%x but this version of numpy is 0x%x . Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem .FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimenumpy.import_arraygetbuffer(obj, view, flags)init scipy.spatial._qhullCannot copy memoryview slice with indirect dimensions (axis %d)View.MemoryView.array_cwrapper__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)qhull error: insufficient memory to copy %d points qhull internal error (qh_determinate): only implemented for dimension >= 2 qh_detmaxoutside: MAXoutside %2.2g from qh.max_outside %2.2g, ONEmerge %2.2g, MINoutside %2.2g, DISTround %2.2g qhull internal error (qh_detsimplex): #points %d < dimension %d qh_detsimplex: det=%2.2g for point p%d, dim %d, nearzero? %d qh_distround: increase roundoff by random delta %2.2g for option 'R%2.2g' qh_distround: %2.2g, maxabs %2.2g, maxsumabs %2.2g, maxdistsum %2.2g qhull internal error (qh_facetarea_simplex): #points %d != dim %d -1 qh_facetarea_simplex: area=%2.2g for point p%d, toporient %d, nearzero? %d func_doc__doc__func_name__name____qualname____self__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations____module____reduce__base__reduce_cython____setstate_cython__Tstridessuboffsetsndimitemsizenbytesis_c_contigis_f_contigcopycopy_fortranmemview__getattr__furthest_sitecheck_activeget_pointsget_paraboloid_shift_scalevolume_areaget_simplex_facet_arrayget_hull_pointsget_hull_facetsget_voronoi_diagramget_extremes_2dcython_function_or_methodscipy.spatial._qhull._memoryviewsliceInternal class for passing memoryview slices to Pythonscipy.spatial._qhull.memoryviewscipy.spatial._qhull.Enumscipy.spatial._qhull.arrayscipy.spatial._qhull._Qhulldual_verticeshalfspacesridge_dictconvex_hullvertex_neighbor_verticesvertex_to_simplextransformqhull internal error (qh_projectpoints): newdim %d should be %d after projection qh_projectpoints: projected %d points from dim %d to dim %d %s %6.3g qhull internal error (qh_maxsimplex): qh.MAXwidth required for qh_maxsimplex. Used to estimate determinate input has same x coordinateqhull precision error (qh_maxsimplex for voronoi_center): %d points with the same x coordinate %4.4g qhull input error: input is less than %d-dimensional since all points have the same x coordinate %4.4g qh_maxsimplex: searching all points for %d-th initial vertex, better than mindet %4.4g, targetdet %4.4g qh_maxsimplex: searching all points ('Qs') for %d-th initial vertex, better than p%d det %4.4g, targetdet %4.4g, ratio %4.4g qh_maxsimplex: searching all points for %d-th initial vertex, better than p%d det %4.4g and mindet %4.4g, ratio %4.4g qh_maxsimplex: searching all points for %d-th initial vertex, better than p%d det %2.2g and mindet %4.4g, targetdet %4.4g qhull internal error (qh_maxsimplex): not enough points available qh_maxsimplex: selected point p%d for %d`th initial vertex, det=%4.4g, targetdet=%4.4g, mindet=%4.4g %s p%dqhull internal error (qh_voronoi_center): need at least %d points to construct a Voronoi center qh_voronoi_center: at infinity for qh_voronoi_center: det %2.2g factor %2.2g center:p%d dist %.2g, qh_scalelast: scale last coordinate from [%2.2g, %2.2g] to [%2.2g, %2.2g] qhull input error (qh_scalelast): can not scale last coordinate to [%4.4g, %4.4g]. Input is cocircular or cospherical. Use option 'Qz' to add a point at infinity. qhull input error (qh_scalelast): can not scale last coordinate to [%4.4g, %4.4g]. New bounds are too wide for compared to existing bounds [%4.4g, %4.4g] (width %4.4g) qhull input error: 'Qb%d' or 'QB%d' inverts paraboloid since high bound %.2g < low bound %.2g qhull input error: %d'th dimension's new bounds [%2.2g, %2.2g] too wide for existing bounds [%2.2g, %2.2g] qh_scalepoints: scaled %d'th coordinate [%2.2g, %2.2g] to [%.2g, %.2g] for %d points by %2.2g and shifted %2.2g qh_setdelaunay: project %d points to paraboloid for Delaunay triangulation qhull internal error (qh_projectinput): dimension after projection %d != hull_dim %d qhull error: insufficient memory to project %d points qh_projectinput: updating lower and upper_bound qhull internal error (qh_projectinput): HALFspace defined without qh.feasible_point qh_projectinput: projected points to paraboloid for Delaunay qh_sethalfspace: halfspace at offset %6.2g to point: %6.2gqhull input error: feasible point is not clearly inside halfspace feasible point: %6.16g halfspace: at offset: and distance: qh_distplane: from p%d to f%d qh_findgooddist: p%d is %2.2g above good facet f%d qh_findgooddist: no good facet for p%d above f%d qh_furthestvertex: all vertices of f%d are in f%d. Returning 0.0 for max and mindist qh_furthestvertex: v%d dist %2.2g is furthest (mindist %2.2g) of f%d above f%d qh_maxouter: max distance from facet to outer plane is %4.4g, qh.max_outside is %4.4g qh_findbesthorizon: test neighbors of f%d bestdist %2.2g f%d ischeckmax? %d noupper? %d minsearch %2.2g is_5x? %d searchdist %2.2g qh_findbesthorizon: p%d, newbest? %d, bestfacet f%d, bestdist %2.2g, numfacet %d, coplanarfacets %d, numdist %d qhull topology error (qh_findbestnew): merging has formed and deleted a cone of new facets. Can not continue. qhull internal error (qh_findbestnew): no new facets for point p%d qh_findbestnew: point p%d facet f%d. Stop? %d if dist > %2.2g, Last qh_addpoint p%d, qh.visit_id %d, vertex_visit %d, Last merge #%d qh_findbestnew: bestfacet f%d bestdist %2.2g for p%d f%d bestoutside? %d qh_findbest: point p%d starting at f%d isnewfacets? %d, unless %d exit if > %2.2g, testhorizon? %d, noupper? %d, Last qh_addpoint p%d, Last merge #%d, max_outside %2.2g qh_findbest: neighbors of f%d, bestdist %2.2g f%d qh_sharpnewfacets: %d qh_findbestlower: all neighbors of facet %d are flipped or upper Delaunay. Search all facets qh_findbestlower: f%d dist %2.2g for f%d p%d qh_gausselim: 0 pivot at column %d. (%2.2g < %2.2g) Matrix:zero pivot for Gaussian eliminationqh_gausselem: resultqh_getangle: %4.4g qhull internal error (qh_getcenter): not defined for %d points qh_getcentrum: for f%d, %d vertices dist= %2.2g computing area of each facet and volume of the convex hull qh_getarea: computing area for each facet and its volume to qh.interior_point (dist*area/dim) qh_facetarea: f%d area %2.2g qh_sethyperplane_gauss: nearly singular or axis parallel hyperplane during p%d. qh_sethyperplane_gauss: singular or axis parallel hyperplane at normalization during p%d. qh_setfacetplane: facet f%d created. Last point added to hull was p%d. Last merge was #%d. Current summary is: qh_sethyperplane_det: degenerate norm during p%d, use qh_sethyperplane_gauss instead. qh_setfacetplane: flipped orientation due to nearzero gauss and interior_point test. During p%d qh_setfacetplane: ====== vertex p%d(v%d) increases max_outside to %2.2g for new facet f%d last p%d DISTANTqh_setfacetplane: f%d offset %2.2g normal: %2.2g qhull internal error (qh_option): option (%d chars) has more than %d chars. May overflow temporary buffer. Option '%s' %sqh_option: option would overflow qh.qhull_options. Truncated '%s' qhull internal error (qh_initflags): qh.NOerrexit was not cleared before calling qh_initflags(). It should be cleared after setjmp(). Exit qhull. delaunayincidencemathematicanormalsoffFilesummaryvoronoiqhull input warning: no maximum cosine angle given for option 'An'. A1.0 is coplanar Angle-premerge-Angle-postmergeqhull input warning: no centrum radius given for option 'Cn' Centrum-premerge-Centrum-postmergeqhull option error: expecting a positive number for maximum roundoff 'En'. Got '%s' qhull option warning: no maximum roundoff given for option 'En' Distance-roundoffqhull option error: expecting 'Hn,n,n,...' for feasible point of halfspace intersection. Got '%s' qhull error: insufficient memory for 'Hn,n,n' Halfspace-aboutHalfspaceqhull option warning: missing random perturbation for option 'Rn' Random-perturbqhull option warning: missing visible distance for option 'Vn' Visibleqhull option warning: missing coplanar distance for option 'Un' U-coplanarqhull option error: expecting a positive number for outside width 'Wn'. Got '%s' qhull option warning: missing outside width for option 'Wn' W-outsideFareaFArea-totalFcoplanarsFCentrumsFd-cdd-inFD-cdd-outFFacets-xridgeFinnerFIDsFmergesFMapleFneighborsFNeighbors-vertexFouterFOptionsFpoint-intersectFPoint-nearestFQhullFsummaryFSizeFtrianglesFVertex-averageFxtremesqhull option warning: unknown 'F' output option 'F%c', skip to next space Gall-pointsGcentrumsGintersectionsGinnerGno-planesGouterGpointsGridgesGtransparentGverticesqhull option warning: missing dimension for option 'GDn' qhull option warning: can only drop one dimension. Previous 'GD%d' ignored GDrop-dimqhull option warning: unknown 'G' geomview option 'G%c', skip to next space Pdrop-facets-dim-lessPDrop-facets-dim-morePgood-facetsPGood-facet-neighborsPoutput-forcedPprecision-ignoreqhull option warning: missing facet count for keep area option 'PAn' PArea-keepqhull option warning: missing facet area for option 'PFn' PFacet-area-keepqhull option warning: missing merge count for option 'PMn' PMerge-keepqhull option warning: unknown 'P' print option 'P%c', skip to next space Qallow-shortQbBound-unit-boxQbbound-lastqh_initflags: project dimension %d Qb-project-dimQbound-dim-lowQBound-dim-highQcoplanar-keepQfurthest-outsideQgood-facets-onlyQinterior-keepQmax-outside-onlyQrandom-outsideQsearch-initial-simplexQtriangulateQTestPointsqhull option warning: missing number of test points for option 'QTn' QupperDelaunayQvertex-neighbors-convexQxact-mergeQz-infinity-pointQ0-no-premergeQ1-angle-mergeQ10-no-narrowQ11-trinormals QtriangulateQ12-allow-wideQ14-merge-pinched-verticesQ15-check-duplicatesqhull option warning: unknown 'Q' qhull option 'Q1%c', skip to next space Q2-no-merge-independentQ3-no-merge-verticesqhull option warning: can not follow '1', '2', or '3' with a digit. 'Q%c%c' skipped Q4-avoid-old-into-newQ5-no-check-outerQ6-no-concave-mergeQ7-no-breadth-firstQ8-no-near-insideQ9-pick-furthestqhull option warning: good point already defined for option 'QGn'. Ignored qhull option warning: missing good point id for option 'QGn'. Ignored QGood-if-dont-see-pointQGood-if-see-pointQJoggleqhull option warning: missing random seed for option 'QRn' QRotate-idQRandom-seedqhull option warning: good vertex already defined for option 'QVn'. Ignored qhull option warning: no good point id given for option 'QVn'. Ignored QV-good-facets-not-pointQV-good-facets-pointQwarn-allowqhull option warning: unknown 'Q' qhull option 'Q%c', skip to next space Tannotate-outputTcheck-frequentlyTflushTstatisticsTverifyTz-stdoutqhull option warning: output file undefined(stdout). Option 'Tz' ignored. qhull option warning: missing point id for cone for trace option 'TCn' TCone-stopqhull option warning: missing frequency count for trace option 'TFn' TFacet-logqhull option error: cannot open 'TI' file "%s" TInput-fileqhull option warning: qh.fout was not set by caller of qh_initflags. Cannot use option 'TO' to redirect output. Ignoring option 'TO' qhull option error: cannot open file "%s" for writing as option 'TO'. It is already in use or read-only TOutput-fileqhull option warning: missing count of added points for trace option 'TAn' TA-stop-addTrace-pointqhull option warning: negative point id for trace option 'TPn'. Expecting 'TP-1' for tracing after qh_buildhull and qh_postmerge qhull option warning: missing point id or -1 for trace option 'TPn' qhull option warning: missing merge id for trace option 'TMn' Trace-mergeqhull option warning: missing rerun count for trace option 'TRn' TRerunqhull option warning: missing furthest point id for trace option 'TVn' TV-stop-before-pointTV-stop-after-pointqhull option warning: missing max width for trace option 'TWn' TWide-traceqhull option warning: unknown 'T' trace option 'T%c', skip to next space qhull option warning: unknown option '%c'(%x) qhull option warning: missing space after option '%c'(%x), reserved for sub-options, ignoring '%c' options to next space qhull option warning: 'TCn' (stopCone) ignored when used with 'QJn' (joggle) qhull option warning: additional output formats ('Fc',etc.) are not compatible with Geomview ('G'). Use option 'Po' to override qhull option error: see previous warnings, use 'Qw' to override: '%s' (last offset %d) qh_initflags: option flags initialized qh_initqhull_globals: for %s | %s qhull input error (qh_initqhull_globals): expecting between 1 and %d points. Got %d %d-d points _pre-mergeqhull option warning: 'Q0-no-premerge' ignored due to exact merge ('Qx') or pre-merge ('C-n' or 'A-n') qhull option warning: joggle ('QJ') produces simplicial output (i.e., triangles in 2-D). Unless merging is requested, option 'Qt' has no effect Qbbound-last-qj_zero-centrumqhull internal error (qh_initqhull_globals): if qh.VORONOI is set, qh.DELAUNAY must be set. Qhull constructs the Delaunay triangulation in order to compute the Voronoi diagram qhull option error: can not use Delaunay('d') or Voronoi('v') with halfspace intersection('H') qhull option error: use upper-Delaunay('Qu') or infinity-point('Qz') with Delaunay('d') or Voronoi('v') qhull option error: can not use infinity-point('Qz') with upper-Delaunay('Qu') qhull option error: can not use merge-pinched-vertices ('Q14') with good-facets-only ('Qg') qh_initqhull_globals: disable qh.MERGEpinched for 2-d. It has no effectqhull option warning: option 'Qbb' (scale-last-coordinate) is normally used with 'd' or 'v' qhull option error: test vertex neighbors('Qv') needs a merge option qhull error: dimension %d must be > 1 qh_initqhull_globals: initialize globals. input_dim %d, numpoints %d, malloc? %d, projected %d to hull_dim %d Q3-no-merge-vertices-dim-highqh_initqhull_globals: trace last of TR%d runs at level %d QRotate-randomqhull configuration error (qh_RANDOMmax in user_r.h): random integer %d > qh_RANDOMmax (%.8g) qhull configuration warning (qh_RANDOMmax in user_r.h): average of 1000 random integers (%.2g) is much different than expected (%.2g). Is qh_RANDOMmax (%.2g) wrong? qhull input error: not enough points(%d) to construct initial simplex (need %d) qh_initqhull_outputflags: %s Pgoodqhull option error: transparent Delaunay('Gt') needs 3-d Delaunay('d') w/o 'GDn' qhull option error: option 'Fp' is only used for halfspace intersection('Hn,n,n'). qhull option error: option 'Ft' is not available for Voronoi vertices ('v') or halfspace intersection ('H') qhull option error: option 'FC' is not available for Voronoi vertices('v') FvoronoiFverticesqhull option warning: 'QJ' (joggle) will usually prevent coincident input sites for options 'Fc' and 'FP' qhull option error: Mathematica and Maple output is only available for 2-d and 3-d convex hulls and 2-d Delaunay triangulations qhull option error: Geomview output is only available for 2-d, 3-d and 4-d qhull option error: no output specified for Geomview qhull option error: Geomview output for Voronoi diagrams only for 2-d qhull option warning: coplanars, vertices, and centrums output not available for 4-d output(ignored). Could use 'GDn' instead. qhull output warning: ignoring coplanar points, option 'Qc' was not set for the first run of qhull. Qcoplanarqhull option warning: drop dimension 'GD%d' is only available for 3-d/4-d Geomview qh_order_vertexneighbors: order neighbors of v%d for 3-d qhull internal error (qh_order_vertexneighbors): no neighbor of v%d for f%d p%d(v%d)%d %d %dVECT 1 2 1 2 1 # f%d %8.4g %8.4g %8.4g %8.4g %8.4g %8.4g { OFF %d 1 1 # f%d 0 %8.4g %d %8.4g %8.4g %8.4g 1.0 } VECT 1 2 1 2 1 # p%d VECT 1 1 1 1 1 %8.4g %8.4g %8.4g 1 {appearance {-normal -edge normscale 0} {INST geom { define centrum CQUAD # f%d -0.3 -0.3 0.0001 0 0 1 1 0.3 -0.3 0.0001 0 0 1 1 0.3 0.3 0.0001 0 0 1 1 -0.3 0.3 0.0001 0 0 1 1 } transform { {INST geom { : centrum } transform { # f%d %8.4g %8.4g %8.4g 0 1 }}} VECT 1 %d 1 %d 1 OFF 3 1 1 # intersect f%d f%d %8.4g %8.4g %8.4g # %8.4g %8.4g %8.4g %8.4g # p%d(coplanar facets) projected p%d 3 0 1 2 %8.4g %8.4g %8.4g 1.0 qhull internal error (qh_detvnorm): too few points(%d) to compute separating plane qh_detvnorm: Voronoi vertex or midpointqh_detvnorm: points %d %d midpoint dist %2.2g qh_detvnorm: points %d %d angle %2.2g nearzero %d qh_detvnorm: points %d %d Voronoi vertex %d dist %2.2g %d %d %d qh_markkeep: only keep %d largest and/or %d most merged facets and/or min area %.2g p%d: %8.4g - r%d tested nonconvex mergevertex mergevertex2 simplicialtop simplicialbot vertices: between f%d and f%d - ridges (tentative ids): r%d - ridges: - horizon ridge to visible facet - all ridges: NULLvertex - p%d(v%d): %5.2g delridge newfacet seen seen2 neighbors: f%d3 %d %d %d %8.4g %8.4g %8.4g 1 # f%d f%d 3 %d %d %d %8.4g %8.4g %8.4g 1 #r%d f%d f%d qhull warning: no facets printed qhull internal error (qh_printend): number of ridges %d != number printed %d and at end %d } )); qhull input error: filename expected, none found. qhull input error: missing quote after filename -- %s %d %d Vertices and facets: qhull warning: output for ridges and intersections not implemented in 2-d qhull warning: output for outer/inner planes and centrums not implemented in 4-d qhull warning: output for vertices not implemented in 4-d qhull warning: 'Gnh' generates no output in 4-d {appearance {linewidth 3} LIST # %s | %s {appearance {+edge -evert linewidth 2} LIST # %s | %s LIST # %s | %s 4VECT %d %d 1 VECT %d %d 1 # 1 point per line 1 # 1 color for all 0 1 1 1 # color of points 4OFF %d %d 1 {appearance {-edge -normal normscale 0} { INST geom {define vsphere OFF 18 32 48 0 0 1 1 0 0 0 1 0 -1 0 0 0 -1 0 0 0 -1 0.707107 0 0.707107 0 -0.707107 0.707107 0.707107 -0.707107 0 -0.707107 0 0.707107 -0.707107 -0.707107 0 0 0.707107 0.707107 -0.707107 0.707107 0 0.707107 0.707107 0 0.707107 0 -0.707107 0 0.707107 -0.707107 -0.707107 0 -0.707107 0 -0.707107 -0.707107 3 0 6 11 3 0 7 6 3 0 9 7 3 0 11 9 3 1 6 8 3 1 8 14 3 1 13 6 3 1 14 13 3 2 11 13 3 2 12 11 3 2 13 15 3 2 15 12 3 3 9 12 3 3 10 9 3 3 12 16 3 3 16 10 3 4 7 10 3 4 8 7 3 4 10 17 3 4 17 8 3 5 14 17 3 5 15 14 3 5 16 15 3 5 17 16 3 6 13 11 3 7 8 6 3 9 10 7 3 11 12 9 3 14 8 17 3 15 13 14 3 16 12 15 3 17 10 16 } transforms { TLIST }}} %8.4g 0 0 0 # v%d 0 %8.4g 0 0 0 0 %8.4g 0 1 qhull warning: input sites of Delaunay regions (option 'i'). Use option 'p' or 'o' for Voronoi centers. Disable warning with option 'Pp' %s | %s begin %d %d real qhull warning: output is the Delaunay triangulation PLOT(CURVES( PLOT3D(POLYGONS( { %d %d %d %d %s | %s begin %d %d real qhull internal error (qh_printbegin): can not use this format for dimension %d MERGEridge DUPLICATEridge NULLfacet - f%d - flags: top bottom simplicial tricoplanar upperDelaunay visible notG isarea coplanarhorizon mergehorizon cycledone keepcentrum mergeridge1 mergeridge2 newmerge flipped notfurthest degenerate redundant - area: %2.2g - replacement: f%d - shares same visible/horizon as f%d - owner of normal & centrum is facet f%d - was horizon to f%d - merges: %dmax - merges: %d - normal: - offset: %10.7g - center: - maxoutside: %10.7g - outside set(furthest p%d): - outside set: - outside set: %d points. Furthest - furthest distance= %2.2g - coplanar set(furthest p%d): - coplanar set: - coplanar set: %d points. furthest distance= %2.2g - vertices: - neighboring facets: MERGEridge DUPLICATEridge[[%16.8f, %16.8f], [%16.8f, %16.8f]] Line[{{%16.8f, %16.8f}, {%16.8f, %16.8f}}] ][%16.8f, %16.8f, %16.8f]}]{%16.8f, %16.8f, %16.8f}0 OFF 3 1 1 # ridge between f%d f%d # ridge between f%d f%d 3 0 1 2 %8.4g %8.4g %8.4g OFF 3 1 1 # f%d # r%d between f%d f%d no normal for facet f%d ,, [Polygon[{qhull input error (qh_printafacet): option 'Fp' needs qh->feasible_point %d %d %d %6.16g qhull internal error (qh_addpoint): NULL facet. Need to call qh_findbestfacet first qh_findhorizon: find horizon for point p%d facet f%d visibleempty horizonqhull topology error (qh_findhorizon): empty horizon for p%d. It was above all facets. qhull internal error (qh_findhorizon): does not work for tricoplanar facets. Use option 'Q11' qhull internal error (qh_findhorizon): visible facet f%d does not have neighbors coplanar horizonqh_findhorizon: point p%d is coplanar to horizon f%d, dist=%2.7g < qh->MINvisible(%2.7g) qh_findhorizon: %d horizon facets(good %d), %d visible(good %d), %d coplanar qhull internal error (qh_facetintersect): f%d or f%d not in other's neighbors ERRONEOUSqh_facetintersect: f%d skip %d matches f%d skip %d qh_makenew_simplicial: create facet f%d top %d from v%d and horizon f%d skip %d top %d and visible f%d skip %d, flip? %d qh_makenewfacets: created %d new facets f%d..f%d from point p%d to horizon qh_getpinchedmerges: try to merge pinched vertices for dupridges in new facets with apex p%d(v%d) max dupdist %2.2g qhull internal error (qh_getpinchedmerges): expecting MRGdupridge from qh_mark_dupridges. Got merge f%d f%d type %d qhull internal error (qh_findbest_pinchedvertex): expecting merge of adjacent, simplicial new facets. f%d or f%d is not simplicial qhull internal error (qh_findbest_pinchedvertex): expecting subridge of qh.hull_dim-2 vertices for the intersection of new facets f%d and f%d minus their apex. Got %d vertices qh_neighbor_vertices_facet: found %d vertex neighbors for v%d in f%d (simplicial? %d) qh_neighbor_vertices: %d non-subridge, vertex neighbors for v%d qhull internal error (qh_findbest_pinchedvertex): did not find best vertex for subridge of dupridge between f%d and f%d, while processing p%d qh_findbest_pinchedvertex: best pinched p%d(v%d) and vertex p%d(v%d) are closest (%2.2g) for duplicate subridge between f%d and f%d qh_getpinchedmerges: dupridge (MRGdupridge) of coplanar horizon would produce a wide merge (%.0fx) due to pinched vertices v%d and v%d (dist %2.2g) for f%d and f%d. qh_mergecycle_all will merge one or both facets qhull precision warning (qh_getpinchedmerges): pinched vertices v%d and v%d (dist %2.2g, %.0fx) would produce a wide merge for f%d and f%d. Will merge dupridge instead qh_getpinchedmerges: will make the apex a coplanar point. apex p%d(v%d) is the nearest vertex to v%d on dupridge. Dist %2.2g qh_getpinchedmerges: will merge new facets to resolve dupridge between f%d and f%d with pinched v%d and v%d qh_getpinchedmerges: will merge pinched v%d into v%d to resolve dupridge between f%d and f%d qh_buildcone: created %d newfacets for p%d(v%d) new facet balance %2.2g qhull internal error (qh_addpoint): infinite loop (%d retries) of merging pinched vertices due to dupridge for point p%d, facet f%d, and %d vertices qh_premerge: premerge centrum %2.2g angle %4.4g for apex p%d newfacet_list f%d qh_forcedmerges: merge dupridges qhull internal error (qh_forcedmerges): qh_settemppop (size %d) is not qh->facet_mergeset (size %d) qhull internal error (qh_forcedmerges): f%d and f%d had a dupridge but as f%d and f%d they are no longer neighbors qh_check_dupridge: dupridge between f%d and f%d (vertex dist %2.2g), dist %2.2g, reverse dist %2.2g, ratio %2.2g while processing p%d qhull topology error (qh_check_dupridge): wide merge (%.1fx wider) due to dupridge between f%d and f%d (vertex dist %2.2g), merge dist %2.2g, while processing p%d - Allow error with option 'Q12' - Experimental option merge-pinched-vertices ('Q14') may avoid this error. It merges nearly adjacent vertices. - A bounding box for the input sites may alleviate this error. qh_forcedmerges: dupridge f%d is degenerate with fewer than %d neighbors qh_forcedmerges: merged %d facets, %d flipped facets, and %d degenredundant facets across dupridges qh_addpoint: added p%d to convex hull with point balance %2.2g qh_buildhull: start build hull qhull internal error (qh_buildhull): visible or new facet f%d in facet list qhull internal error (qh_buildhull): new vertex f%d in vertex list qh_buildhull: stop point or cone P%d in initial hull qh_buildhull: stop after adding %d vertices qhull internal error (qh_nextfurthest): null facet or infinite loop detected for qh.facet_next f%d facet_tail f%d qh_furthestout: p%d is furthest outside point of f%d qhull internal error (qh_nextfurthest): num_outside %d is too low by at least %d, or a random real %g >= 1.0 qh_outcoplanar: move outsideset to coplanarset for qh->NARROWhull qhull internal error (qh_buildhull): %d outside points were never processed. qh_buildhull: completed the hull construction qhull input error: %d attempts to construct a convex hull with joggled input. Increase joggle above 'QJ%2.2g' or modify qh_JOGGLE... parameters in user_r.h _runqhull error: insufficient memory to joggle %d points qh_detjoggle: joggle=%2.2g maxwidth=%2.2g qhull input error (qh_joggleinput): the current joggle for 'QJn', %.2g, is too large for the width of the input. If possible, recompile Qhull with higher-precision reals. _joggle-seedqh_joggleinput: joggle input by %4.4g with seed %d qh_copynonconvex: moved nonconvex flag from r%d to r%d between f%d and f%d qh_findbest_ridgevertex: best pinched p%d(v%d) and vertex p%d(v%d) are closest (%2.2g) for duplicated ridge r%d (same vertices) between f%d and f%d qhull internal error: cannot call qh_findbestneighor for f%d while qh.CENTERtype is qh_ASvoronoi qhull internal error (qh_findbestneighbor): no neighbors for f%d qh_findbestneighbor: f%d is best neighbor for f%d testcentrum? %d nonconvex? %d dist %2.2g min %2.2g max %2.2g qh_vertexridges: found %d ridges for v%d qh_mergevertex_del: deleted v%d when merging f%d into f%d qh_remove_extravertices: test non-simplicial f%d for extra vertices qh_remove_extravertices: v%d deleted because it's lost all ridges qh_remove_extravertices: v%d removed from f%d because it's lost all ridges At %02d:%02d:%02d & %2.5g CPU secs, qhull has created %d facets and merged %d. The current hull contains %d facets and %d vertices. Last point was p%d qh_buildtracing: start trace T%d for point TP%d above facet f%d At %02d:%02d:%02d & %2.5g CPU secs, qhull has created %d facets and merged %d. The current hull contains %d facets and %d vertices. There are %d outside points. Next is point p%d(v%d), %2.2g above f%d. qh_addpoint: add p%d(v%d) %2.2g above f%d to hull of %d facets, %d merges, %d outside at %4.4g CPU secs. Previous p%d(v%d) delta %4.4g CPU, %d facets, %d merges, %d hyperplanes, %d distplanes, %d retries qhull internal error: qh_checklists failed on reset of qh.visit_id %u qhull internal error: qh_checklists failed on reset of qh.vertex_visit %u qh_joggle_restart: qhull restart because of %s qhull internal error (qh_partitionpoint): cannot partition p%d of f%d into visible facet f%d nearly incident point (narrow hull)qh_partitionpoint: point p%d is outside facet f%d newfacet? %d, newoutside? %d (or narrowhull) nearly incident pointqh_partitionpoint: point p%d is coplanar to facet f%d (dropped) qh_partitionpoint: point p%d is inside all facets, closest to f%d dist %2.2g qh_partitioncoplanar: partition coplanar point p%d starting with f%d dist? %2.2g, allnew? %d, gh.repart_facetid f%d qh_partitioncoplanar: point p%d is more than near-inside facet f%d dist %2.2g allnew? %d qh_partitioncoplanar: point p%d is inside facet f%d dist %2.2g allnew? %d qhull internal error (qh_partitioncoplanar): cannot partition coplanar p%d of f%d into visible facet f%d qh_partitioncoplanar: repartition coplanar point p%d from f%d as an outside point above corner facet f%d dist %2.2g with angle %2.2g Qhull internal error (qh_partitioncoplanar): infinite loop due to recursive call to qh_partitionpoint. Repartition point p%d from f%d as a outside point dist %2.2g nearest vertices %2.2g Qhull topology error (qh_partitioncoplanar): can not repartition coplanar point p%d from f%d as outside point above f%d. It previously failed to form a cone of facets, dist %2.2g, nearest vertices %2.2g Qhull precision warning: repartition coplanar point p%d from f%d as an outside point above twisted facet f%d dist %2.2g nearest vertices %2.2g Qhull precision warning: repartition coplanar point p%d from f%d as an outside point above hidden facet f%d dist %2.2g nearest vertices %2.2g qh_partitioncoplanar: == p%d from f%d increases qh.max_outside to %2.2g of f%d last p%d qh_partitioncoplanar: point p%d is coplanar with facet f%d (or inside) dist %2.2g and at-infinity 'good'similar and redundantredundantsimilarcoplanar and interiorinteriorcoplanarApproximateTotalqhull internal error: qh_checklists failed at qh_printsummary Early exit due to 'TAn', 'TVn', 'TCn', 'TRn', or precision error with 'QJn'. Furthest-site Voronoi vertices by the convex hull of %d points in %d-d: Voronoi diagram by the convex hull of %d points in %d-d: Number of Voronoi regions%s: %d Total number of deleted points due to merging: %d Number of nearly incident points: %d Total number of nearly incident points: %d Number of%s Voronoi vertices: %d Number of%s non-simplicial Voronoi vertices: %d Furthest-site Delaunay triangulation by the convex hull of %d points in %d-d: Delaunay triangulation by the convex hull of %d points in %d-d: Number of input sites%s: %d Number of%s Delaunay regions: %d Number of%s non-simplicial Delaunay regions: %d Halfspace intersection by the convex hull of %d points in %d-d: Number of halfspaces: %d Number of non-redundant halfspaces: %d Number of %s halfspaces: %d Number of intersection points: %d Number of 'good' intersection points: %d Number of%s non-simplicial intersection points: %d Convex hull of %d points in %d-d: Number of vertices: %d Number of %s points: %d Number of facets: %d Number of 'good' facets: %d Number of%s non-simplicial facets: %d Number of triangulated facets: %d Statistics for: %s | %s QR%d Number of points processed: %d Number of hyperplanes created: %d Number of facets in hull: %d Number of distance tests for qhull: %d Number of distance tests for merging: %d Number of distance tests for checking: %d Number of merged facets: %d Number of merged pinched vertices: %d CPU seconds to compute hull (after input): %2.4g Percentage of runs with precision errors: %4.1f After %d retries, input joggled by: %2.2g Input joggled by: %2.2g %s facet area: %2.8g %s volume: %2.8g Maximum distance of point above facet: %2.2g Maximum distance of vertex below facet: %2.2gqhull error (qh_memalloc): negative request size (%d). Did int overflow due to high-D? qh_mem %p n %8d alloc quick: %d bytes (tot %d cnt %d) qhull error (qh_memalloc): insufficient memory to allocate short memory buffer (%d bytes) qhull internal error (qh_memalloc): short totbuffer %d != totshort+totfree... %d qh_mem %p n %8d alloc short: %d bytes (tot %d cnt %d) qhull internal error (qh_memalloc): qhmem has not been initialized. qhull error (qh_memalloc): insufficient memory to allocate %d bytes qh_mem %p n %8d alloc long: %d bytes (tot %d cnt %d) qhull internal error (qh_memcheck): qh is 0. It does not point to a qhT qhull internal error (qh_memcheck): either qh->qhmem is overwritten or qh->qhmem is not initialized. Call qh_meminit or qh_new_qhull before calling qh_mem routines. ferr 0x%x, IsTracing %d, ALIGNmask 0x%x qh_memcheck: check size of freelists on qh->qhmem qh_memcheck: A segmentation fault indicates an overwrite of qh->qhmem qhull internal error (qh_memcheck): totfree %d not equal to freelist total %d qh_memcheck: total size of freelists totfree is the same as qh->qhmem.totfree qh_mem %p n %8d free short: %d bytes (tot %d cnt %d) qh_mem %p n %8d free long: %d bytes (tot %d cnt %d) qh_delridge_merge: delete ridge r%d between f%d and f%d qh_delridge_merge: drop merge of v%d into v%d (dist %2.2g r%d r%d) due to deleted, duplicated ridge r%d qh_find_newvertex: find new vertex for v%d from v%d r%d qh_find_newvertex: vertices not in ridges for v%d qh_find_newvertex: found v%d for old v%d from %d vertices and %d ridges. qh_find_newvertex: no vertex for renaming v%d (all duplicated ridges) during p%d qhull internal error (qh_memsize): qh_memsize called after qh_memsetup qh_memsize: quick memory of %d bytes qhull warning (qh_memsize): free list table has room for only %d sizes memory statistics: %7d quick allocations %7d short allocations %7d long allocations %7d short frees %7d long frees %7d bytes of short memory in use %7d bytes of short memory in freelists %7d bytes of dropped short memory %7d bytes of unused short memory (estimated) %7d bytes of long memory allocated (max, except for input) %7d bytes of long memory in use (in %d pieces) %7d bytes of short memory buffers (minus links) %7d bytes per short memory buffer (initially %d bytes) %7d calls to qh_setlarger %7.2g average copy size freelists(bytes->count): %d->%ddupridgedegenqh_appendmergeset: f%d is already redundant (%d) or f%d is already redundant (%d). Ignore merge f%d and f%d type %d qh_appendmergeset: f%d is already degenerate. Ignore merge f%d type %d (MRGdegen) qhull internal error (qh_appendmergeset): expecting temp set defined for qh.facet_mergeset (0x%x) and qh.degen_mergeset (0x%x). Got NULL qhull internal error (qh_appendmergeset): except for MRGdupridge, cannot merge a non-flipped facet f%d into flipped f%d, mergetype %d, dist %4.4g qh_appendmergeset: dupridge will merge a non-flipped facet f%d into flipped f%d, dist %4.4g qhull internal error (qh_appendmergeset): facet f%d or f%d is already a mirrored facet (i.e., 'redundant') qhull internal error (qh_appendmergeset): mirrored facets f%d and f%d do not have the same vertices qh_appendmergeset: append merge f%d and f%d type %d (%s) to qh.degen_mergeset (size %d) qh_appendmergeset: append merge f%d and f%d type %d (%s) dist %2.2g angle %4.4g to qh.facet_mergeset (size %d) qh_degen_redundant_facet: test facet f%d for degen/redundant qh_degen_redundant_facet: f%d is flipped, will merge later qhull internal error (qh_degen_redundant_facet): facet f%d has deleted neighbor f%d (qh.visible_list) qh_degen_redundant_facet: f%d is contained in f%d. merge qh_degen_redundant_facet: f%d is degenerate. qh_test_appendmerge: coplanar angle %4.4g between f%d and f%d qh_test_centrum_merge: concave f%d to coplanar f%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_centrum_merge: concave f%d to f%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_centrum_merge: coplanar f%d to f%d dist %4.4g, reverse dist %4.4g angle %4.4g qh_test_nonsimplicial_merge: concave centrum for f%d or f%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_nonsimplicial_merge: twisted concave f%d v%d to f%d v%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_nonsimplicial_merge: concave coplanar f%d v%d to f%d v%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_nonsimplicial_merge: concave f%d v%d to f%d v%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_test_nonsimplicial_merge: coplanar f%d v%d to f%d v%d, dist %4.4g and reverse dist %4.4g, angle %4.4g during p%d qh_getmergeset: started. qh_getmergeset: %d merges found qh_test_degen_neighbors: test for degenerate neighbors of f%d qhull internal error (qh_test_degen_neighbors): facet f%d has deleted neighbor f%d (qh.visible_list) qh_test_degen_neighbors: f%d is degenerate with %d neighbors. Neighbor of f%d. qhull internal error (qh_appendvertexmerge): expecting temp set defined for qh.vertex_mergeset (0x%x). Got NULL qhull internal error (qh_appendvertexmerge): expecting two distinct ridges for MRGvertices. Got r%d r%d qh_appendvertexmerge: append merge v%d into v%d r%d r%d dist %2.2g type %d (%s) qhull internal error (qh_checkdelfacet): cannot delete f%d. It is referenced by merge f%d f%d mergetype %d qhull internal error (qh_checkdelridge): expecting empty qh.vertex_mergeset in order to avoid calling qh_delridge_merge. Got %d merges qhull internal error (qh_checkdelridge): unexpected 'nonconvex' flag for ridge r%d in newfacet f%d. Otherwise need to call qh_delridge_merge qhull internal error (qh_checkdelridge): unexpected 'nonconvex' flag for ridge r%d in visible facet f%d. Otherwise need to call qh_delridge_merge not concave, flipped, or dupridgeclearly convexqh_checkzero: skip convexity check until first pre-merge qh_checkzero: testall %d, facets are %s qh_checkzero: qh_premerge is needed. New facet f%d or its horizon f%d is non-simplicial, flipped, dupridge, or mergehorizon qh_checkzero: facet f%d and f%d are not clearly convex. v%d dist %.2g qh_checkzero: facet f%d and horizon f%d are not clearly convex. v%d dist %.2g qhull internal error (qh_freemergesets): expecting mergesets. Got a NULL mergeset, qh.facet_mergeset (0x%x), qh.degen_mergeset (0x%x), qh.vertex_mergeset (0x%x) qhull internal error (qh_freemergesets): expecting empty mergesets. Got qh.facet_mergeset (%d merges), qh.degen_mergeset (%d merges), qh.vertex_mergeset (%d merges) qh_getmergeset_initial: %d merges found qhull internal error (qh_initmergesets): expecting NULL mergesets. Got qh.facet_mergeset (0x%x), qh.degen_mergeset (0x%x), qh.vertex_mergeset (0x%x) qh_maybe_duplicateridges: will merge v%d into v%d (dist %2.2g) due to opposite oriented ridges r%d/r%d for f%d and f%d qh_maybe_duplicateridges: will merge v%d into v%d (dist %2.2g) due to duplicate ridges with the same vertices r%d/r%d in merged facet f%d qh_test_redundant_neighbors: test neighbors of f%d vertex_visit %d qh_test_redundant_neighbors: f%d is degenerate with %d neighbors. qhull internal error (qh_test_redundant_neighbors): facet f%d has deleted neighbor f%d (qh.visible_list) qh_test_redundant_neighbors: f%d is contained in f%d. merge qh_renamevertex: rename v%d to v%d in %d ridges with old f%d and neighbor f%d qh_renamevertex: renaming v%d to v%d in several facets for qh_redundant_vertex or MRGsubridge qhull internal error (qh_renameridgevertex): oldvertex v%d not found in r%d. Cannot rename to v%d qh_renameridgevertex: ridge r%d deleted. It contained both v%d and v%d qh_renameridgevertex: swapped the top and bottom of ridge r%d qh_renamevertex: ignore duplicate check for r%d. top f%d (size %d) will be degenerate after rename v%d to v%d qh_renamevertex: ignore duplicate check for r%d. bottom f%d (size %d) will be degenerate after rename v%d to v%d qh_maybe_duplicateridge: will merge v%d into v%d (dist %2.2g) due to duplicate ridges r%d/r%d with the same vertices. mergevertex set qh_renamevertex: rename vertices in non-simplicial neighbor f%d of v%d qh_maydropneighbor: test f%d for no ridges to a neighbor qhull internal error (qh_maydropneighbor): not valid for simplicial f%d while adding furthest p%d qhull internal error (qh_maydropneighbor): facet f%d has deleted neighbor f%d (qh.visible_list) qh_maydropneighbor: facets f%d and f%d are no longer neighbors while adding furthest p%d qhull internal error (qh_maydropneighbor): not valid for simplicial neighbor f%d of f%d while adding furthest p%d qh_maydropneighbors: f%d is degenerate. qh_renamevertex: renaming v%d to v%d in oldfacet f%d for qh_rename_sharedvertex qh_renamevertex: renaming pinched v%d to v%d between f%d and f%d noneMERGEDqh_tracemerge: trace facet and vertex after merge of f%d into f%d type %d (%s), furthest p%d TRACEqh_tracemerge: trace vertex deleted at furthest p%d At %d:%d:%d & %2.5g CPU secs, qhull has merged %d facets with max_outside %2.2g, min_vertex %2.2g. The hull contains %d facets and %d vertices. qh_willdelete: move f%d to visible list, set its replacement as f%d, and clear f.neighbors and f.ridges qhull internal error (qh_willdelete): expecting qh.visible_list at before qh.newfacet_list f%d. Got NULL makeridges: appended r%d to ridges for f%d. Next is ridges for neighbor f%d qh_mark_dupridges: identify dupridges in facetlist f%d, allmerges? %d qh_mark_dupridges): dupridge due to duplicate vertices for subridges f%d and f%d qh_mark_dupridges: found %d duplicated ridges (MRGdupridge) for qh_getpinchedmerges qh_mark_dupridges: found %d duplicated ridges (MRGdupridge) for qh_premerge. Prepare facets for merging qh_makeridges: make ridges for f%d qh_mark_dupridges: restore missing neighbors and ridges due to qh_MERGEridge qhull topological error (qh_mark_dupridges): multiple dupridges for f%d and f%d, including reverse qhull internal error (qh_mergefacet): merge f%d into f%d for mergetype %d (%s) does not work for tricoplanar facets. Use option 'Q11' qh_mergefacet: ========= trace wide merge #%d(%2.2g) for f%d into f%d for mergetype %d (%s), last point was p%d qh_mergefacet: ========= trace merge #%d for f%d into f%d for mergetype %d (%s), furthest is p%d qh_mergefacet: #%d merge f%d into f%d for merge for mergetype %d (%s), mindist= %2.2g, maxdist= %2.2g, max_outside %2.2g qhull precision error (qh_mergefacet): wide merge for facet f%d into f%d for mergetype %d (%s). maxdist %2.2g (%.1fx) mindist %2.2g (%.1fx) vertexdist %2.2g Allow with 'Q12' (allow-wide) qhull precision error (qh_mergefacet): wide merge for pinched facet f%d into f%d for mergetype %d (%s). maxdist %2.2g (%.fx) mindist %2.2g (%.1fx) vertexdist %2.2g Allow with 'Q12' (allow-wide) qhull internal error (qh_mergefacet): either f%d and f%d are the same or one is a visible facet, mergetype %d (%s) qhull topology error: Only %d facets remain. The input is too degenerate or the convexity constraints are too strong. Option 'Qx' may avoid this problem. MERGINGqh_mergesimplex: merge apex v%d of f%d into facet f%d qh_mergesimplex: merge opposite v%d of f%d into facet f%d qh_mergesimplex: update vertex neighbors of f%d qh_mergesimplex: merge ridges and neighbors of f%d into f%d qhull topology error (qh_mergesimplex): f%d is a dupridge of f%d, cannot merge f%d into f%d qh_mergesimplex: move r%d with f%d to f%d, new neighbor? %d, maybe horizon? %d qh_mergesimplex: merged simplex f%d v%d into facet f%d qh_mergefacet2d: merged v%d and neighbor f%d of f%d into f%d qh_mergeneighbors: merge neighbors of f%d and f%d qhull internal error (qh_mergevertices): facets did not share a ridge qh_mergeridges: merge ridges of f%d into f%d qh_mergevertex_neighbors: merge vertex neighborset for f%d into f%d qh_mergevertex_neighbors: of f%d into f%d at furthest p%d f0= %p qh_mergefacet: end of wide tracing qh_mergefacet: check f%d and f%d for redundant and degenerate neighbors qh_merge_degenredundant: merge %d degenerate, redundant, and mirror facets qhull internal error (qh_merge_degenredunant): f%d is redundant but visible f%d has no replacement qh_merge_degenredundant: merge redundant f%d into f%d (arg f%d) qh_merge_degenredundant: facet f%d has no neighbors. Deleted qh_merge_degenredundant: deleted v%d because f%d has no neighbors qh_merge_degenredundant: facet f%d has %d neighbors, merge into f%d dist %2.2g qh_flippedmerges: begin qhull internal error (qh_flippedmerges): facet_mergeset (%d merges) not at top of tempstack (%d merges) qh_flippedmerges: merge flipped f%d into f%d dist %2.2g during p%d qh_flippedmerges: merged %d flipped and %d degenredundant facets into a good neighbor qh_reducevertices: reduce extra vertices, shared vertices, and redundant vertices qhull internal error (qh_rename_sharedvertex): v%d's neighbors not in f%d qh_rename_sharedvertex: p%d(v%d) is shared by f%d(%d ridges) and f%d qh_redundant_vertex: check if v%d from a deleted ridge can be renamed qh_neighbor_intersections: %d vertices in neighbor intersection of v%d qh_reducevertices: renamed %d shared vertices and %d redundant vertices. Degen? %d qh_all_merges: starting to merge %d facet and %d degenerate merges for new facets f%d, othermerge? %d qh_all_merges: drop merge of f%d (del? %d) into f%d (del? %d) mergetype %d, dist %4.4g, angle %4.4g. One or both facets is deleted qh_all_merges: drop merge of f%d (tested? %d) into f%d (tested? %d) mergetype %d, dist %2.2g, angle %4.4g. Merge independent sets of coplanar merges qh_all_merges: merge f%d and f%d type %d dist %2.2g angle %4.4g qh_merge_twisted: merge #%d for twisted f%d and f%d qhull precision error (qh_merge_twisted): twisted facet f%d does not contain pinched vertices. Too wide to merge into neighbor. mindist %2.2g maxdist %2.2g vertexdist %2.2g maxpinched %2.2g neighbor f%d mindist %2.2g maxdist %2.2g qhull precision error (qh_merge_twisted): twisted facet f%d with pinched vertices. Could merge vertices, but too wide to merge into neighbor. mindist %2.2g maxdist %2.2g vertexdist %2.2g neighbor f%d mindist %2.2g maxdist %2.2g qhull internal error (qh_merge_nonconvex): expecting mergetype MRGcoplanar..MRGconcavecoplanar. Got merge f%d and f%d type %d qh_merge_nonconvex: merge #%d for f%d and f%d type %d qh_merge_nonconvex: avoid merging old facet f%d dist %2.2g. Use f%d dist %2.2g instead qh_test_vneighbors: testing vertex neighbors for convexity qh_test_vneighbors: found %d non-convex, vertex neighbors qh_all_merges: skip qh_reducevertices due to post-merging, no qh.VERTEXneighbors (%d), or hull_dim %d ==2 or >%d qh_all_merges: merged %d coplanar %d concave %d concavecoplanar %d twisted facets and %d degen or redundant facets. reason %s with 'C%.2g' and 'A%.2g' qh_postmerge: postmerge. test vneighbors? %d qh_all_vertexmerges: starting to merge %d vertex merges for apex p%d facet f%d qhull internal error (qh_merge_pinchedvertices): qh.visible_list (f%d), newfacet_list (f%d), or newvertex_list (v%d) not empty qh_merge_pinchedvertices: merge one of %d pinched vertices before adding apex p%d. Try to resolve duplicate ridges in newfacets qh_remove_mergetype: remove merge f%d f%d v%d v%d r%d r%d dist %2.2g type %dqh_merge_pinchedvertices: merge %d pinched vertices from dupridges in merged facets, apex p%d qh_rename_adjacentvertex: merge #%d rename v%d (%d neighbors) to v%d (%d neighbors) dist %2.2g qh_rename_adjacentvertex: ignore rename. Either v%d (%d) or v%d (%d) is deleted qhull internal error (qh_rename_adjacentvertex): expecting neighbor facets for v%d and v%d. Got %d and %d neighbors resp. qh_rename_adjacentvertex: simplicial f%d contains old v%d and new v%d. Will be marked degenerate by qh_renamevertex qh_replacefacetvertex: replace v%d with v%d in f%d qhull internal error (qh_replacefacetvertex): f%d is not simplicial qhull internal error (qh_replacefacetvertex): f%d already contains new v%d qh_rename_adjacentvertex: partition old p%d(v%d) as a coplanar point for furthest f%d dist %2.2g. Maybe repartition later (QH0031) qh_merge_pinchedvertices: merge degenerate f%d into an adjacent facet qh_merge_pinchedvertices: merge f%d into f%d mergeType %d qhull internal error (qh_next_vertexmerge): expecting two vertices for vertex merge. Got v%d v%d and optional f%d qh_next_vertexmerge: drop merge of v%d (del? %d) into v%d (del? %d) due to deleted vertex of r%d and r%d qh_drop_mergevertex: unset mergevertex for r%d and r%d due to dropped vertex merge v%d to v%d. Sets mergevertex2 qhull topology error (qh_next_vertexmerge): no nearly adjacent vertices to resolve opposite oriented ridges r%d and r%d in f%d and f%d. Nearest v%d and v%d dist %2.2g (%.1fx) qhull topology error (qh_next_vertexmerge): no nearly adjacent vertices to resolve duplicate ridges r%d and r%d. Nearest v%d and v%d dist %2.2g (%.1fx) qhull topology error (qh_next_vertexmerge): no nearly adjacent vertices to resolve dupridge. Nearest v%d and v%d dist %2.2g (%.1fx) qhull internal error (qh_replacefacetvertex): f%d does not contain old v%d anglecoplanarconcaveconcavecoplanartwistedflipsubridgemirrorcoplanarhorizonqh_triangulate_link: relink neighbors f%d and f%d of null facet f%d qh_triangulate_link: relink neighbors f%d and f%d of mirrored facets f%d and f%d qhull internal error (qh_triangulate_link): neighbors f%d and f%d do not match for null facet or mirrored facets f%d and f%d qhull precision error: point p%d is outside facet f%d, distance= %6.8g maxoutside= %6.8g nearest vertices %2.2g qh_checkconvex: check that facets are not-flipped and for qh.ZEROcentrum that simplicial vertices are below their neighbor (dist<0.0) qh_checkconvex: check that facets are not-flipped and that simplicial vertices are convex by qh.DISTround ('En', 'Rn') qh_checkconvex: check that facets are not-flipped and that their centrums are convex by qh.DISTround ('En', 'Rn') flipped facetqhull precision error: f%d is flipped (interior point is outside) non-convex initial simplexqhull precision error: initial simplex is not convex, since p%d(v%d) is %6.4g above opposite f%d qhull precision error: initial simplex is not convex, since p%d(v%d) is within roundoff of opposite facet f%d (dist %6.4g) concave ridgeqhull precision error: f%d is concave to f%d, since p%d(v%d) is %6.4g above f%d coplanar ridgeqhull precision error: f%d is clearly not convex to f%d, since p%d(v%d) is %6.4g above or coplanar with f%d with qh.ZEROcentrum qhull precision error: f%d is coplanar to f%d, since p%d(v%d) is within %6.4g of f%d, during p%d qh_checkconvex: starting with f%d, also check that centrums of non-simplicial ridges are below their neighbors (dist<0.0) qhull warning: recomputing centrums for convexity test. This may lead to false, precision errors. qhull precision error: f%d is concave to f%d. Centrum of f%d is %6.4g above f%d qhull precision error: f%d is coplanar or concave to f%d. Centrum of f%d is %6.4g above f%d newfacetqh.facet_qh_checklists: check qh.%s_list f%d and qh.vertex_list v%d qh_checklists: check %slist f%d and qh.vertex_list v%d qhull internal error (qh_checklists): either qh.facet_tail f%d is NULL, or its id is not 0, or its next is not NULL qhull internal error (qh_checklists): f%d already in facetlist causing an infinite loop ... f%d > f%d ... > f%d > f%d. Truncate facetlist at f%d qhull internal error (qh_checklists): unknown or overwritten facet f%d, either id >= qh.facet_id (%d) or f.visitid %u > qh.visit_id %u. Facetlist terminated at previous facet f%d qhull internal error (qh_checklists): expecting f%d.previous == f%d. Got f%d qhull internal error (qh_checklists): qh.visible_list f%d is after qh.newfacet_list f%d. It should be at, before, or NULL qhull internal error (qh_checklists): qh.facet_next f%d for qh_addpoint is not on qh.facet_list f%d qhull internal error (qh_checklists): qh.newfacet_list f%d is not on qh.facet_list f%d qhull internal error (qh_checklists): qh.visible_list f%d is not on qh.facet_list f%d qhull internal error (qh_checklists): either qh.vertex_tail v%d is NULL, or its id is not 0, or its next is not NULL qhull internal error (qh_checklists): v%d already in vertexlist causing an infinite loop ... v%d > v%d ... > v%d > v%d. Truncate vertexlist at v%d qhull internal error (qh_checklists): unknown or overwritten vertex v%d, either id >= qh.vertex_id (%d) or v.visitid %u > qh.visit_id %u. vertexlist terminated at previous vertex v%d qhull internal error (qh_checklists): expecting v%d.previous == v%d. Got v%d qhull internal error (qh_checklists): new vertex list v%d is not on vertex list qhull internal error (qh_checkvertex): unknown point id %p qhull internal error (qh_checkvertex): unknown vertex id v%d >= qh.vertex_id (%d) qhull internal error (qh_checkvertex): expecting v%d.visitid <= qh.vertex_visit (%d). Got visitid %d qhull internal error (qh_checkvertex): neighbor f%d does not contain v%d qh_clearcenters: switched to center type %d qhull internal error (qh_delvertex): vertex v%d was deleted but it was not partitioned as a coplanar point qh_findfacet_all: p%d, noupper? %d, f%d, dist %2.2g, isoutside %d, totpart %d qh_findgood: f%d is closest(%2.2g) to thresholds qh_findgood: found %d good facets with %d good horizon and qh.GOODclosest f%d qhull warning: good vertex p%d does not match last good facet f%d. Ignored. qhull warning: point p%d is not a vertex('QV%d'). qhull warning: point p%d is a vertex for every facet('QV-%d'). qh_findgood_all: f%d is closest(%2.2g) to split thresholds qh_findgood_all: undo selection of qh.GOODclosest f%d since it would fail qh_inthresholds in qh_skipfacet qh_findgood_all: %d good facets remain out of %d facets qhull internal error (qh_nearvertex): qh.VERTEXneighbors and facet->center required for tricoplanar facets qhull internal error (qh_nearvertex): did not find bestvertex for f%d p%d qh_nearvertex: v%d dist %2.2g for f%d p%d qhull error (qh_newhashtable): negative request (%d) or size (%d). Did int overflow due to high-D? qhull error: 2^32 or more vertices. vertexT.id field overflows. Vertices would not be sorted correctly. qh_newvertex: vertex p%d(v%d) created qhull internal error (qh_facet3vertex): only %d vertices for simplicial facet f%d qhull internal error (qh_facet3vertex): ridges for facet %d don't match up. got at least %d qhull internal warning (point_add): unknown point %p id %d qhull internal error (point_add): point p%d is out of bounds(%d) qh_check_maxout: check and update qh.min_vertex %2.2g and qh.max_outside %2.2g qh_check_maxout: determine actual minvertex Qhull precision warning: in post-processing (qh_check_maxout) p%d(v%d) is %2.2g below f%d nearest vertices %2.2g qh_check_maxout: p%d(v%d) is %.2g from f%d nearest vertices %2.2g qh_check_maxout: determine actual maxoutside Qhull precision warning: in post-processing (qh_check_maxout) p%d for f%d is %2.2g above twisted facet f%d nearest vertices %2.2g Qhull precision warning: in post-processing (qh_check_maxout) p%d for f%d is %2.2g above hidden facet f%d nearest vertices %2.2g qh_check_maxout: p%d is %.2g above f%d Qhull precision warning (qh_check_maxout): f%d.maxoutside (%4.4g) is greater than computed qh.max_outside (%2.2g) + qh.DISTround (%2.2g). It should be less than or equal qh_check_maxout: p%d(v%d) is qh.min_vertex %2.2g below facet f%d. Point p%d for f%d is qh.max_outside %2.2g above f%d. %d points are outside of not-good facets Qhull precision error (qh_check_maxout): large increase in qh.max_outside during post-processing dist %2.2g (%.1fx). See warning QH0032/QH0033. Allow with 'Q12' (allow-wide) and 'Pp' Qhull precision error (qh_check_maxout): a facet merge, vertex merge, vertex, or coplanar point produced a wide facet %2.2g (%.1fx). Trace with option 'TWn' to identify the merge. Allow with 'Q12' (allow-wide) Qhull precision error (qh_check_maxout): large increase in qh.min_vertex during post-processing dist %2.2g (%.1fx). See warning QH7083. Allow with 'Q12' (allow-wide) and 'Pp' Qhull precision error (qh_check_maxout): a facet or vertex merge produced a wide facet: v%d below f%d distance %2.2g (%.1fx). Trace with option 'TWn' to identify the merge. Allow with 'Q12' (allow-wide) qh_prependfacet: prepend f%d before f%d qh_furthestnext: made f%d next facet(dist %.2g) qh_printlists: max_outside %2.2g all facets: qh.visible_list f%d, newfacet_list f%d, facet_next f%d for qh_addpoint qh.newvertex_list v%d all vertices:qh_resetlists: reset newvertex_list v%d, newfacet_list f%d, visible_list f%d, facet_list f%d next f%d vertex_list v%d -- NEWfacets? %d, NEWtentative? %d, stats? %d qh_checkfacet: check f%d newmerge? %d qhull internal error (qh_checkfacet): unknown facet id f%d >= qh.facet_id (%d) qhull internal error (qh_checkfacet): expecting f%d.visitid <= qh.visit_id (%d). Got visitid %d qhull internal error (qh_checkfacet): facet f%d is on qh.visible_list qhull internal error (qh_checkfacet): redundant facet f%d not on qh.visible_list qhull internal error (qh_checkfacet): degenerate facet f%d is not on qh.visible_list and qh.degen_mergeset is empty qhull internal error (qh_checkfacet): facet f%d does not have a normal qhull internal error (qh_checkfacet): f%d is 'dupridge' but it is not a newfacet on qh.newfacet_list f%d qhull internal error (qh_checkfacet): f%d is 'newmerge' but it is not a newfacet on qh.newfacet_list f%d. Missing call to qh_reducevertices vertices for fridges for foutsideset for fcoplanarset for fneighbors for fqhull internal error (qh_checkfacet): deleted vertex v%d in f%d qhull internal error (qh_checkfacet): vertices of f%d are not in descending id order at v%d qhull internal error (qh_checkfacet): for simplicial facet f%d, #vertices %d + #neighbors %d != 2*qh->hull_dim qhull internal error (qh_checkfacet): for facet f%d, #vertices %d or #neighbors %d < qh->hull_dim qhull internal error (qh_checkfacet): for facet f%d, #ridges %d < #neighbors %d or(3-d) > #vertices %d or(2-d) not all 2 qhull internal error (qh_checkfacet): facet f%d still has a MERGEridge or DUPLICATEridge neighbor qhull internal error (qh_checkfacet): facet f%d has deleted neighbor f%d (qh.visible_list) qhull internal error (qh_checkfacet): facet f%d has neighbor f%d, but f%d does not have neighbor f%d qhull internal error (qh_checkfacet): facet f%d has a duplicate neighbor f%d vertices for rqhull internal error (qh_checkfacet): facet f%d has a duplicate ridge r%d qhull internal error (qh_checkfacet): ridge between f%d and f%d has %d vertices qhull internal error (qh_checkfacet): for facet f%d, neighbor f%d of ridge r%d not in facet qhull internal error (qh_checkfacet): ridge r%d is nonconvex (%d), mergevertex (%d) or not tested (%d) for facet f%d, neighbor f%d qhull internal error (qh_checkfacet): facet f%d does not have a ridge for neighbor f%d qhull internal error (qh_checkfacet): vertex v%d in r%d not in f%d intersect f%d qhull topology error (qh_checkfacet): vertex v%d in f%d intersect f%d but not in a ridge. Last point was p%d qh_checkfacet: vertex v%d in f%d intersect f%d but not in a ridge. Repaired by qh_remove_extravertices in qh_reducevertices qhull internal error (qh_checkfacet): facet f%d skip %d and neighbor f%d skip %d do not match qhull internal error (qh_checkfacet): ridges r%d and r%d (f%d) have the same vertices qh_checkpolygon: check all facets from f%d, qh.NEWtentative? %d qhull internal error: qh_checklists failed in qh_checkpolygon qhull internal error (qh_checkpolygon): f%d is 'newfacet' but it is not on qh.newfacet_list f%d or visible_list f%d qhull internal error (qh_checkpolygon): f%d is on qh.newfacet_list f%d but it is not 'newfacet' qhull internal error (qh_checkpolygon): f%d is 'visible' but it is not on qh.visible_list f%d qhull internal error (qh_checkpolygon): f%d is on qh.visible_list f%d but it is not 'visible' qhull internal error (qh_checkpolygon): f%d has outside points before qh.facet_next f%d qhull internal error (qh_checkpolygon): expecting empty f.neighbors and f.ridges for visible facet f%d. Got %d neighbors and %d ridges qhull internal error (qh_checkpolygon): v%d is 'newfacet' but it is not on new vertex list v%d qhull internal error (qh_checkpolygon): unknown point %p for vertex v%d first_point %p qhull internal error (qh_checkpolygon): actual number of facets is %d, cumulative facet count is %d - %d visible facets qhull internal error (qh_checkpolygon): missing vertex neighbors for v%d neighbors for vqhull internal error (qh_checkpolygon): vertex neighbors inconsistent (tot_vneighbors %d != tot_facetvertices %d). Maybe duplicate or missing vertex qhull internal error (qh_checkpolygon): facet f%d occurs twice in neighbors of vertex v%d qhull internal error (qh_checkpolygon): facet f%d is a neighbor of vertex v%d but v%d is not a vertex of f%d qhull internal error (qh_checkpolygon): v%d is a vertex of facet f%d but f%d is not a neighbor of v%d qhull internal error (qh_checkpolygon): actual number of vertices is %d, cumulative vertex count is %d qhull internal error (qh_checkpolygon): #vertices %d != #facets %d qhull warning: #vertices %d + #facets %d - #edges %d != 2. A vertex appears twice in a edge list. May occur during merging. qh_vertexneighbors: determining neighboring facets for each vertex qh_appendfacet: append f%d to facet_list qh_appendvertex: append v%d to qh.newvertex_list and set v.newfacet qh_attachnewfacets: delete interior ridges qh_attachnewfacets: attach horizon facets to new facets qhull internal error (qh_attachnewfacets): could not find visible facet for horizon f%d of newfacet f%d qh_attachnewfacets: clear f.ridges and f.neighbors for visible facets, may become invalid before qh_deletevisible qh_checkflipped: facet f%d flipped, allerror? %d, distance= %6.12g during p%d qhull %s Statistics: %s | %s qhull input error: either QGn or QVn point is > p%d qh_maxmin: dim min max width nearzero min-point max-point %3d % 14.8e % 14.8e % 14.8e %4.4e p%-9d p%-d last coordinate scaled to (%4.4g, %4.4g), width %4.4e for option 'Qbb' qh_maxmin: found the max and min points (by dim):_max-widthError-roundoffqh_detroundoff: increase qh.ANGLEround by option 'R%2.2g' Angle-premerge-with-randomAngle-postmerge-with-randomCentrum-premerge-with-randomCentrum-postmerge-with-random_one-mergeqhull option error: the joggle for 'QJn', %.2g, is below roundoff for distance computations, %.2g Visible-distanceU-max-coplanarWidth-outside_wide-facetqhull input warning: minimum visibility V%.2g is greater than minimum outside W%.2g. Flipped facets are likely. qh_initbuild: create sentinels for qh.facet_tail and qh.vertex_tail qh_createsimplex: created simplex qh_initialhull: qh.interior_pointqh_initialhull: ignore f%d flipped. Test qh.interior_point (p-2) for clearly flipped qh_initialhull: initial orientation incorrect, qh.interior_point is %2.2g from f%d. Reversing orientation of all facets qh_initialhull: initial orientation incorrect, qh.interior_point is %2.2g from f%d. Either axis-parallel facet or coplanar firstfacet f%d. Force outside orientation of all facets initial Delaunay cocircular or cosphericalQhull precision error: initial Delaunay input sites are cocircular or cospherical. Option 'Qs' searches all points. Use option 'QJ' to joggle the input, otherwise cannot compute the upper Delaunay triangulation or upper Voronoi diagram of cocircular/cospherical points. Qhull precision error: initial Delaunay input sites are cocircular or cospherical. Use option 'Qz' for the Delaunay triangulation or Voronoi diagram of cocircular/cospherical points; it adds a point "at infinity". Alternatively use option 'QJ' to joggle the input. Use option 'Qs' to search all points for the initial simplex. input sites with last coordinate projected to a paraboloid initial simplex is flatQhull precision error: Initial simplex is flat (facet %d is coplanar with the interior point) _narrow-hullqhull precision warning: The initial hull is narrow. Is the input lower dimensional (e.g., a square in 3-d instead of a cube)? Cosine of the minimum angle is %.16f. If so, Qhull may produce a wide facet. Options 'Qs' (search all points), 'Qbb' (scale last coordinate), or 'QbB' (scale to unit box) may remove this warning. See 'Limitations' in qh-impre.htm. Use 'Pp' to skip this warning. qh_initialhull: simplex constructed qh_partitionall: partition all points into outside sets Trace level T%d, IStracing %d, point TP%d, merge TM%d, dist TW%2.2g, qh.tracefacet_id %d, traceridge_id %d, tracevertex_id %d, last qh.RERUN %d, %s | %s Options selected for Qhull %s: %s qhull input error: 'Qg QVn' (only good vertex) does not work with merging. Use 'QJ' to joggle the input or 'Q0' to turn off merging. qhull input error: 'Qg' (ONLYgood) needs a good threshold('Pd0D0'), a good point(QGn or QG-n), or a good vertex with 'QJ' or 'Q0' (QVn). qhull input error: point for QV%d is inside initial simplex. It can not be made a vertex. qh_initbuild: initial hull created and points partitioned _near-insideqhull precision error: facet f%d is flipped, distance= %6.12g A flipped facet occurs when its distance to the interior point is greater than or equal to %2.2g, the maximum roundoff error. qh_delfacet: delete f%d qh_deletevisible: delete %d visible facets and %d vertices qhull internal error (qh_deletevisible): qh->num_visible %d is not number of visible facets %d qhull internal error: negative hashsize %d passed to qh_gethash [poly_r.c] qhull internal error (qh_infiniteloop): potential infinite loop detected. If visible, f.replace. If newfacet, f.samecycle qh_makenewplanes: make new hyperplanes for facets on qh.newfacet_list f%d qhull internal error (qh_makenew_nonsimplicial): simplicial f%d sharing two ridges with f%d qh_makenew_nonsimplicial: created facet f%d from v%d and r%d of horizon f%d qh_triangulate: triangulate non-simplicial facets qh_triangulate: delete null facets from facetlist f%d. A null facet has the same first (apex) and second vertices qh_triangulate_facet: triangulate facet f%d qhull internal error (qh_triangulate): ridges still defined for f%d qh_triangulate_null: delete null facet f%d qh_triangulate: delete %d or more mirrored facets. Mirrored facets have the same vertices due to a null facet qh_triangulate_mirror: delete mirrored facets f%d and f%d and link their neighbors qh_triangulate: update neighbor lists for vertices from v%d qh_triangulate: identify degenerate tricoplanar facets from f%d qh_triangulate: delete visible facets -- non-simplicial, null, and mirrored facets qh_triangulate: delete f%d. All tricoplanar facets degenerate for non-simplicial facet qhull internal error (qh_triangulate): tricoplanar facet f%d not owned by its visible, non-simplicial facet f%d qh_triangulate: all tricoplanar facets degenerate for last non-simplicial facet f%d qh_triangulate: and replace facet->f.triowner with tricoplanar facets that own center, normal, etc. QH%.4d qhull internal error (userprintf_r.c): fp and qh not defined for qh_fprintf '%s' qhull internal error (userprintf_r.c): fp is 0. Wrong qh_fprintf was called. [QH%.4d]qh_newfacet: created facet f%d qhull warning: more than 2^32 ridges. Qhull results are OK. Since the ridge ID wraps around to 0, two ridges may have the same identifier. qh_newridge: created ridge r%d qh_removefacet: removed f%d from facet_list, newfacet_list, and visible_list qh_removevertex: remove v%d from qh.vertex_list %s set is null %s set=%p maxsize=%d size=%d elems= %p%7.2g *0 cnt*%7d%7.3g %s qhull warning (qh_stddev): expecting num > 0. Got num %d, tot %4.4g, tot2 %4.4g. Returning 0.0 %s qhull invoked by: %s | %s %s with options: %s precision constants: %6.2g max. abs. coordinate in the (transformed) input ('Qbd:n') %6.2g max. roundoff error for distance computation ('En') %6.2g max. roundoff error for angle computations %6.2g min. distance for outside points ('Wn') %6.2g min. distance for visible facets ('Vn') %6.2g max. distance for coplanar facets ('Un') %6.2g max. facet width for recomputing centrum and area %6.2g max. distance for near-inside points %6.2g max. cosine for pre-merge angle %6.2g radius of pre-merge centrum %6.2g max. cosine for post-merge angle %6.2g radius of post-merge centrum %6.2g max. distance for merging two simplicial facets %6.2g max. roundoff error for arithmetic operations %6.2g min. denominator for division zero diagonal for Gauss: %6.2e A Qhull error has occurred. Qhull should have corrected the above precision error. Please send the input and all of the output to qhull_bug@qhull.org Precision problems were detected during construction of the convex hull. This occurs because convex hull algorithms assume that calculations are exact, but floating-point arithmetic has roundoff errors. To correct for precision problems, do not use 'Q0'. By default, Qhull selects 'C-0' or 'Qx' and merges non-convex facets. With option 'QJ', Qhull joggles the input to prevent precision problems. See "Imprecision in Qhull" (qh-impre.htm). If you use 'Q0', the output may include coplanar ridges, concave ridges, and flipped facets. In 4-d and higher, Qhull may produce a ridge with four neighbors or two facets with the same vertices. Qhull reports these events when they occur. It stops when a concave ridge, flipped facet, or duplicate facet occurs. When computing the Delaunay triangulation of coordinates > 1.0, - use 'Qbb' to scale the last coordinate to [0,m] (max previous coordinate) When computing the Delaunay triangulation: - use 'Qz' to add a point at-infinity. This reduces precision problems. If you need triangular output: - use option 'Qt' to triangulate the output - use option 'QJ' to joggle the input points and remove precision errors - use option 'Ft'. It triangulates non-simplicial facets with added points. If you must use 'Q0', try one or more of the following options. They can not guarantee an output. - use 'QbB' to scale the input to a cube. - use 'Po' to produce output and prevent partitioning for flipped facets - use 'V0' to set min. distance to visible facet as 0 instead of roundoff - use 'En' to specify a maximum roundoff error less than %2.2g. - options 'Qf', 'Qbb', and 'QR0' may also help To guarantee simplicial output: - use option 'Qt' to triangulate the output - use option 'QJ' to joggle the input points and remove precision errors - use option 'Ft' to triangulate the output by adding points - use exact arithmetic (see "Imprecision in Qhull", qh-impre.htm) qhull error while handling previous error in qh_errexit. Exit program _maxoutside While executing: %s | %s Last point added to hull was p%d. Qhull has finished constructing the hull. Qhull has started post-merging. size in bytes: merge %d ridge %d vertex %d facet %d normal %d ridge vertices %d facet vertices or neighbors %d qhull internal error (qh_produce_output2): temporary sets not empty(%d) qhull internal error (qh_produce_output): temporary sets not empty(%d) At error exit: at error exit A Qhull internal error has occurred. Please send the input and output to qhull_bug@qhull.org. If you can duplicate the error with logging ('T4z'), please include the log file. The input to qhull appears to be less than %d dimensional, or a computation has overflowed. Qhull could not construct a clearly convex simplex from points: The center point is coplanar with a facet, or a vertex is coplanar with a neighboring facet. The maximum round off error for computing distances is %2.2g. The center point, facets and distances to the center point are as follows: center point distance= %4.2g These points are the dual of the given halfspaces. They indicate that the intersection is degenerate. These points either have a maximum or minimum x-coordinate, or they maximize the determinant for k coordinates. Trial points are first selected from points that maximize a coordinate. Because of the high dimension, the min x-coordinate and max-coordinate points are used if the determinant is non-zero. Option 'Qs' will do a better, though much slower, job. Instead of 'Qs', you can change the points by randomly rotating the input with 'QR0'. The min and max coordinates for each dimension are: %d: %8.4g %8.4g difference= %4.4g If the input should be full dimensional, you have several options that may determine an initial simplex: - use 'QJ' to joggle the input and make it full dimensional - use 'QbB' to scale the points to the unit cube - use 'QR0' to randomly rotate the input for different maximum points - use 'Qs' to search all points for the initial simplex - use 'En' to specify a maximum roundoff error less than %2.2g. - trace execution with 'T3' to see the determinant for each point. If the input is lower dimensional: - use 'QJ' to joggle the input and make it full dimensional - use 'Qbk:0Bk:0' to delete coordinate k from the input. You should pick the coordinate with the least range. The hull will have the correct topology. - determine the flat containing the points, rotate the points into a coordinate plane, and delete the other coordinates. - add one or more points to make the input full dimensional. A Qhull topology error has occurred. Qhull did not recover from facet merges and vertex merges. This usually occurs when the input is nearly degenerate and substantial merging has occurred. See http://www.qhull.org/html/qh-impre.htm#limit A wide merge error has occurred. Qhull has produced a wide facet due to facet merges and vertex merges. This usually occurs when the input is nearly degenerate and substantial merging has occurred. See http://www.qhull.org/html/qh-impre.htm#limit qhull internal error (qh_errexit): either error while reporting error QH%d, or qh.NOerrexit not cleared after setjmp(). Exit program with error status %d qhull internal error (qh_setaddnth): nth %d is out-of-bounds for set: qhull internal error (qh_setcheck): actual size %d of %s%d is greater than max size %d qhull internal error (qh_setcheck): %s%d(size %d max %d) is not null terminated. qhull internal error (qh_setdelnth): nth %d is out-of-bounds for set: qh_update_vertexneighbors_cone: update v.neighbors for qh.newvertex_list (v%d) and qh.newfacet_list (f%d) qh_update_vertexneighbors_cone: deleted %d visible vertexneighbors of v%d qh_update_vertexneighbors_cone: delete interior vertices, if any, for qh.visible_list (f%d) qh_update_vertexneighbors_cone: will delete interior vertex p%d(v%d) of visible f%d qh_update_vertexneighbors_cone: delete interior vertices for qh.visible_list (f%d) qhull internal error (qh_setnew_delnthsorted): nth %d is out-of-bounds for set: qhull internal error (qh_setreplace): elem %p not found in set qhull internal error (qh_setsize): current set size %d is greater than maximum size %d set: qh_settemp: temp set %p of %d elements, depth %d qhull internal error (qh_settemppop): pop from empty temporary stack qh_settemppop: depth %d temp set %p of %d elements qhull error (qh_settemppush): can not push a NULL temp qh_settemppush: depth %d temp set %p of %d elements qhull internal error (qh_settruncate): size %d out of bounds for set: qh_update_vertexneighbors: update v.neighbors for qh.newvertex_list (v%d) and qh.newfacet_list (f%d) qh_update_vertexneighbors: delete %d of %d vertex neighbors for v%d. Removes to-be-deleted, visible facets qh_update_vertexneighbors: delete interior vertices for qh.visible_list (f%d) qh_update_vertexneighbors: delete interior vertex p%d(v%d) of visible f%d qh_update_vertexneighbors: delete old vertices for qh.visible_list (f%d) qh_update_vertexneighbors: will delete interior vertex p%d(v%d) of visible f%d qhull internal error (qh_settempfree): set %p(size %d) was not last temporary allocated(depth %d, set %p, size %d) %s VERTEX: %s RIDGE: %s FACET: %s OTHER FACET: ERRONEOUS and NEIGHBORING FACETS to output qh_matchnewfacets: match neighbors for new facets. qhull internal error (qh_matchnewfacets): expecting simplicial facets on qh.newfacet_list f%d for qh_matchneighbors, qh_matchneighbor, and qh_matchdupridge. Got non-simplicial f%d qh_matchneighbor: newfacet f%d skip %d hash %d hashcount %d two new facets with the same verticesqhull topology warning (qh_matchneighbor): will merge vertices to undo new facets -- f%d and f%d have the same vertices (skip %d, skip %d) and same horizon ridges to f%d and f%d qh_matchneighbor: f%d skip %d matched with new f%d skip %d a ridge with more than two neighborsqhull topology error: facets f%d, f%d and f%d meet at a ridge with more than 2 neighbors. Can not continue due to no qh.PREmerge and no 'Qx' (MERGEexact) qhull topology error (qh_matchneighbor): matchfacet f%d is in f%d neighbors but not vice versa. Can not continue. qh_matchneighbor: new f%d skip %d duplicates ridge for f%d skip %d matching f%d ismatch %d at hash %d qh_matchneighbor: no match for f%d skip %d at hash %d ridge with multiple neighborsqh_matchdupridge: find dupridge matches for f%d skip %d hash %d hashcount %d qh_matchdupridge: duplicate ridge due to duplicate facets (f%d skip %d and f%d skip %d) previously reported as QH7084. Maximize dupdist to force vertex merge qhull topology error (qh_matchdupridge): missing qh_DUPLICATEridge at f%d skip %d for new f%d skip %d hash %d ismatch %d. Set by qh_matchneighbor qh_matchdupridge: allow tricoplanar dupridge for new f%d skip %d and f%d skip %d qh_matchdupridge: make good forced merge of dupridge f%d skip %d into f%d skip %d, keep new f%d skip %d and f%d skip %d, dist %4.4g qh_matchdupridge: make forced merge of dupridge for new f%d skip %d and f%d skip %d, maxdist %4.4g in qh_forcedmerges qh_matchdupridge: try good dupridge flipped f%d skip %d into new f%d skip %d at dist %2.2g otherdist %2.2g qh_matchdupridge: try good dupridge flipped new f%d skip %d into f%d skip %d at dist %2.2g otherdist %2.2g qh_matchdupridge: try good dupridge f%d skip %d into new f%d skip %d at dist %2.2g otherdist %2.2g qh_matchdupridge: try good dupridge new f%d skip %d into f%d skip %d at dist %2.2g otherdist %2.2g qh_matchdupridge: try furthest dupridge f%d skip %d new f%d skip %d at dist %2.2g qhull internal error (qh_matchdupridge): no MERGEridge match for dupridge new f%d skip %d at hash %d..%d qhull internal error (qh_matchdupridge): no maximum or good match for dupridge new f%d skip %d at hash %d..%d qh_matchdupridge: keep dupridge f%d skip %d and f%d skip %d, dist %4.4g qhull internal error (qh_matchnewfacets): %d neighbors did not match up hash %d f%d neighbors:qh_matchnewfacets: maxdupdist %2.2g, new facets %d, unused hash entries %d, hashsize %d qhull internal error: qh_checklists failed on qh_collectstatistics qhull error: filename is more than %d characters, %s qh_mergecycle_all: merge new facets into coplanar horizon facets. Bulk merge a cycle of facets with the same horizon facet qhull internal error (qh_mergecycle_all): f%d without normal qh_mergecycle: merge #%d for facets from cycle f%d into coplanar horizon f%d qh_mergecycle: ========= trace merge %d of samecycle %d into trace f%d, furthest is p%d same cycle:MERGING CYCLEqhull internal error (qh_mergecycle): does not work for tricoplanar facets. Use option 'Q11' qh_mergecycle_neighbors: delete shared neighbors from newfacet qh_mergecycle_neighbors: update neighbors qh_mergecycle_neighbors: deleted %d neighbors and added %d qh_mergecycle_ridges: delete shared ridges from newfacet qh_mergecycle_ridges: add ridges to newfacet qhull internal error (qh_mergecycle_ridges): bad ridge r%d qh_mergecycle_ridges: found %d old ridges and %d new ones qh_mergecycle_vneighbors: update vertex neighbors for newfacet qh_basevertices: found %d vertices qh_mergecycle_vneighbors: deleted v%d when merging cycle f%d into f%d qh_mergecycle_vneighbors: merged vertices from cycle f%d into f%d qh_mergecycle_facets: make newfacet new and samecycle deleted qh_mergecycle_facets: merged facets from cycle f%d into f%d qh_mergecycle: end of trace facet qh_mergecycle_all: merged %d same cycles or facets into coplanar horizons and %d degenredundant facets qh_partitionvisible: partition outside and coplanar points of visible and merged facets f%d into new facets f%d qhull topology error (qh_partitionvisible): all new facets deleted as degenerate facets. Can not continue. qh_partitionvisible: partition %d deleted vertices as coplanar? %d points into new facets f%d qhull internal error (qh_partitionvisible): all new facets deleted or none defined. Can not partition deleted v%d. qh_partitionvisible: partitioned %d points from outsidesets, %d points from coplanarsets, and %d deleted vertices good qhull qhullqhull error (qh_new_qhull): start qhull_cmd argument with "qhull " or set to "qhull" precision statisticsave. distance of a new vertex to a facetmax. distance of a new vertex to a facetmax. distance of an output vertex to a facetmin. distance of an output vertex to a facetmin. denominator in hyperplane computationprecision problems (corrected unless 'Q0' or an error)coplanar half ridges in outputconcave half ridges in outputflipped facetscoplanar horizon facets for new verticescoplanar points during partitioningdegenerate hyperplanes recomputed with gaussian eliminationnearly singular or axis-parallel hyperplaneszero divisors during back substitutezero divisors during gaussian eliminationdupridges with multiple neighborsdupridges with flip facet into good neighbordupridges with flip facet into good flip neighborsummary informationnumber of vertices in outputnumber of facets in outputnumber of non-simplicial facets in outputsimplicial facets that were non-simplicialaverage number of ridges per facetmaximum number of ridgesaverage number of neighbors per facetmaximum number of neighborsaverage number of vertices per facetmaximum number of verticesaverage number of neighbors per vertexcpu seconds for qhull after inputvertices created altogetherfacets created altogetherridges created altogetherfacets before post mergeaverage merges per facet (at most 511) maximum merges for a facet (at most 511)average cosine (angle) of facet normals for all ridges maximum cosine of facet normals (flatest) across a ridge minimum cosine of facet normals (sharpest) across a ridgetotal area of facets maximum facet area minimum facet areabuild hull statisticspoints processedretries due to precision problems max. random jogglemax. vertices at any one timeave. visible facets per iteration ave. visible facets without an horizon neighbor ave. facets deleted per iteration maximumave. visible vertices per iterationave. horizon facets per iterationave. new or merged facets per iteration maximum (includes initial simplex)average new facet balance standard deviationaverage partition balance countsearches of all points for initial simplexdeterminants for facet area determinants not computed because vertex too lowdeterminants for initial hull or voronoi verticespoints ignored (!above max_outside)points ignored (pinched apex)points ignored (!above a good facet)points ignored (didn't create a good new facet)good facets founddistance tests for facet visibilitydistance tests to report minimum vertexpoints checked for facets' outer planes ave. distance tests per checkresets of visit_id resets of vertex_visit max visit_id/2 max vertex_visit/2partitioning statistics (see previous for outer planes)total vertices deleted maximum vertices deleted per iterationcalls to findbest ave. facets tested max. facets tested ave. coplanar searchcalls to findbestnew ave. clearly better calls due to qh_sharpnewfacetscalls to findhorizon new bestfacets during qh_findbesthorizonangle tests for repartitioned coplanar points repartitioned coplanar points above a corner facet repartitioned coplanar points above a hidden facet repartitioned coplanar points above a twisted facetinside points near inside points kept with a facet inside points that were coplanar with a facetcalls to findbestlower with search of vertex neighbors with rare search of all facets facets per search of all facetsdifference in max_outside at final checkdistance tests for initial partitionpartitions of a pointdistance tests for partitioningdistance tests for checking flipped facetsdistance tests for checking convexitydistance tests for checking good pointdistance tests for outputdistance tests for statisticstotal number of distance testspartitions of coplanar points or deleted vertices distance tests for these partitionsdistance tests for computing furtheststatistics for matching ridgestotal lookups for matching ridges of new facetsaverage number of tests to match a ridgetotal lookups of subridges (duplicates and boundary)average number of tests per subridgeduplicated ridges in same merge cycleduplicated ridges with flipped facetsstatistics for determining mergesangles computed for ridge convexitybest merges used centrum instead of verticesdistance tests for best mergedistance tests for centrum convexitydistance tests for vertex convexitydistance tests for checking simplicial convexitycoplanar angles in getmergesetcoplanar centrums or vertices in getmergesetconcave ridges in getmergesetconcave-coplanar ridges in getmergesettwisted ridges in getmergesetstatistics for mergingmerge iterationsave. initial non-convex ridges per iteration maximum ave. additional non-convex ridges per iteration maximum additional in one passinitial non-convex ridges for post merging additional non-convex ridgesmax distance of vertex or coplanar point above facet (w/roundoff)max distance of vertex below facet (or roundoff)centrums frozen due to a wide mergecentrums frozen due to extra verticestotal number of facets or cycles of facets mergedmerged a simplexsimplices merged into coplanar horizoncycles of facets merged into coplanar horizon ave. facets per cycle max. facetsnew facets merged into coplanar horizonnew facets merged into horizonnew facets mergedhorizon facets merged into new facetsvertices deleted by mergingvertices deleted by merging into coplanar horizonvertices deleted by degenerate facetmerges due to flipped facets in duplicated ridgemerges due to redundant neighbors detected by qh_test_nonsimplicial_merge instead of qh_test_redundant_neighborsnon-convex vertex neighborsmerges due to angle coplanar facets average merge distance maximum merge distancemerges due to coplanar facetsmerges due to concave facetsmerges due to concave-coplanar facetscoplanar/concave merges due to avoiding old mergemerges due to degenerate facetsmerges due to removing flipped facetsmerges due to dupridgesmerges due to twisted facetsstatistics for vertex mergesmerge pinched vertices for a duplicate ridgemerge pinched vertices for a dupridgerenamed vertices shared by two facetsrenamed vertices in a pinched facetrenamed vertices shared by multiple facetsrename failures due to duplicated ridges found new vertex in ridgedeleted ridges due to renamed verticesdropped neighbors due to renamed verticesmerge degenerate facets due to dropped neighbors facets deleted because of no neighborsvertices removed from facets due to no ridges deletedretry qh_addpoint after merge pinched vertex tot. merge pinched vertex due to dupridge max. merge pinched vertex for a qh_addpointvertex intersections for locating redundant verticesintersections failed to find a redundant vertexintersections found redundant vertices ave. number found per vertex max. found for a vertex ave. number of ridges per tested vertex max. number of ridges per tested vertexmemory usage statistics (in bytes)for facets and their normals, neighbor and vertex setsfor vertices and their neighbor setsfor input points, outside and coplanar sets, and qhTfor ridges and their vertex setsVoronoi ridge statisticsnon-simplicial Voronoi vertices for all ridges ave. distance to ridge max. distance to ridgebounded ridges ave. distance of midpoint to ridge max. distance of midpoint to ridgebounded ridges with ok normal ave. angle to ridge max. angle to ridgebounded ridges with near-zero normalTriangulation statistics ('Qt')non-simplicial facets triangulated ave. new facets created (may be deleted) max. new facets creatednull new facets deleted (duplicated vertex)mirrored pairs of new facets deleted (same vertices)degenerate new facets in output (same ridge)run-idqh_new_qhull: initialize Qhull qh_new_qhull: build new Qhull for %d %d-d points with %s qhull input error: halfspace intersection needs a feasible point. Either prepend the input with 1 point or use 'Hn,n,n'. See manual. qhull input warning: more coordinates for 'H%s' than dimension %d qh_sethalfspace_all: compute dual for halfspace intersection qhull error: insufficient memory to compute dual of %d halfspaces The halfspace was at index %d qhull error (qh_meminit): insufficient memory qh_meminitbuffers: memory initialized with alignment %d qhull error (qh_memsetup): largest mem size %d is >= buffer size %d or initial buffer size %d qhull error (qh_memsetup): insufficient memory qhull option warning: no dimension given for Print option 'P%c' at: %s. Ignored qhull option warning: dimension %d for Print option 'P%c' is >= %d. Ignored qhull option warning: value %2.4g for Print option 'P%c' is > +1 or < -1. Ignored qhull option warning: no dimension given for Qhull option 'Q%c' qhull option warning: dimension %d for Qhull option 'Q%c' is >= %d. Ignored qh_rotatepoints: rotate points byqh_qhull: all facets are clearly convex and no coplanar points. Post-merging and check of maxout not needed. First post-mergeFor post-mergingFor testing vertex neighborsqh_qhull: finished qh_buildhull and qh_postmerge, start tracing (TP-1) Testing all coplanar points. qhull internal error (qh_qhull): temporary sets not empty(%d) at end of Qhull Qhull: algorithm completed qh_check_points: check all points below %2.2g of all facet planes qhull input warning: merging without checking outer planes('Q5' or 'Po'). Verify may report that a point is outside of a facet. qh_check_bestdist: check points below nearest facet. Facet_list f%d qh_check_bestdist: check that all points are within %2.2g of best facet qhull output completed. Verifying that %d points are below %2.2g of the nearest %sfacet. qhull precision error (qh_check_bestdist): point p%d is outside facet f%d, distance= %6.8g maxoutside= %6.8g %d points were well inside the hull. If the hull contains a lens-shaped component, these points were not verified. Use options 'Qci Tv' to verify all points. qhull precision error (qh_check_bestdist): a coplanar point is %6.2g from convex hull. The maximum value is qh.outside_err (%6.2g) qh_check_bestdist: max distance outside %2.2g qhull input warning: exact merge ('Qx'). Verify may report that a point is outside of a facet. See qh-optq.htm#Qx qhull input warning: no outer plane check ('Q5') or no processing of near-inside points ('Q8'). Verify may report that a point is outside of a facet. Output completed. Verifying that all points are below outer planes of all %sfacets. Will make %2.0f distance computations. Output completed. Verifying that all points are below %2.2g of all %sfacets. Will make %2.0f distance computations. qhull warning (qh_check_points): missing normal for facet f%d qhull precision error (qh_check_points): %d additional points outside facet f%d, maxdist= %6.8g qhull precision error (qh_check_points): a coplanar point is %6.2g from convex hull. The maximum value(qh.outside_err) is %6.2g qh_check_points: max distance outside %2.2g qh_freebuild: free global sets qh_freebuild: free temporary sets (qh_settempfree_all) qh_freebuild: free memory from qh_inithull and qh_buildhull qh_freebuild: delete the previously-seen ridges of f%d qh_freeqhull: free global memory qh_freebuffers: freeing up global memory buffers qh_freebuffers: finished qh_freeqhull: clear qhT except for qh.qhmem and qh.qhstat qh_markvoronoi: isLower %d numcenters %d {appearance {+edge -face} OFF %d %d 1 # Voronoi centers and cells 0 # infinity not used %d %d %d 1 # %d f%d # p%d(v%d) # p%d is coplanar or isolated Buffer has wrong number of dimensions (expected %d, got %d)buffer dtypeItem size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer exposes suboffsets but no stridesC-contiguous buffer is not indirect in dimension %dBuffer and memoryview are not contiguous in the same dimension.Buffer is not indirectly contiguous in dimension %d.Buffer is not indirectly accessible in dimension %d.Buffer not compatible with direct access in dimension %d.Item size of buffer (%zd byte%s) does not match size of '%s' (%zd byte%s)calling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptionco_argcountco_posonlyargcountco_kwonlyargcountco_nlocalsco_stacksizeco_flagsco_codeco_constsco_namesco_varnamesco_freevarsco_cellvarsco_linetablereplace%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectqh_normalize: norm=%2.2g too small during p%d qh_eachvoronoi: Voronoi ridge of %d vertices between sites %d and %d qhull internal error (qh_detvridge3): neighbors of vertex p%d are not connected at facet %d qhull internal error (qh_setzero): index %d or size %d out of bounds for set: qhull warning: CDD format is not available for centrums, halfspace intersections, and OFF file format. qhull internal error (qh_printextremes_2d): loop in facet list. facet %d nextfacet %d %s | %s 0 2 10 %d %d %d %d %d %d %d %d %d %d 2 %6.16g %6.16g 1 %d qhull internal error (qh_printvdiagram): unknown print format %d. printfacetlist: vertices printfacetlist: facetlist printfacetlist: end qh_backnormal: zero diagonal at column %d. zero diagonal on back substitutionqh_matchvertices: matched by skip %d(v%d) and skip %d(v%d) same? %d %s() got multiple values for keyword argument '%U'%.200s() keywords must be strings%s() got an unexpected keyword argument '%U'qhull internal error (qh_setdelnthsorted): nth %d is out-of-bounds for set: 1Nscipy.spatial._qhull.Delaunay.convex_hull Wrappers for Qhull triangulation, plus some additional N-D geometry utilities .. versionadded:: 0.9 zipzeroszxi_shapexi has different dimensionality than triangulationxix_shapexwrong dimensionality in xiworkvolume_areavolumevertices_vertices verticesvertex_to_simplex_vertex_to_simplexvertex_neighbor_vertices_vertex_neighbor_verticesvupdate_updateunpackuniqueunable to allocate shape and strides.unable to allocate array data. tsearch(tri, xi) Find simplices containing the given points. This function does the same thing as `Delaunay.find_simplex`. Parameters ---------- tri : DelaunayInfo Delaunay triangulation xi : ndarray of double, shape (..., ndim) Points to locate Returns ------- i : ndarray of int, same shape as `xi` Indices of simplices containing each point. Points outside the triangulation get the value -1. See Also -------- Delaunay.find_simplex Notes ----- .. versionadded:: 0.9 Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.spatial import Delaunay, delaunay_plot_2d, tsearch >>> rng = np.random.default_rng() The Delaunay triangulation of a set of random points: >>> pts = rng.random((20, 2)) >>> tri = Delaunay(pts) >>> _ = delaunay_plot_2d(tri) Find the simplices containing a given set of points: >>> loc = rng.uniform(0.2, 0.8, (5, 2)) >>> s = tsearch(tri, loc) >>> plt.triplot(pts[:, 0], pts[:, 1], tri.simplices[s], 'b-', mask=s==-1) >>> plt.scatter(loc[:, 0], loc[:, 1], c='r', marker='x') >>> plt.show() tsearch (line 2145)tsearchtriangulatetritransform_transformtolisttol__test__sumstructstringsourcestopstepstartsrcsplitsizesimplicesshape__setstate_cython____setstate__setsself._qh cannot be converted to a Python object for picklingselfscipy.spatial._qhullridge_verticesridge_pointsridge_dict_ridge_dictrestartresizereshaperequired_optionsregionsrefcheck__reduce_ex____reduce_cython____reduce__rcond_limitrcondrange__qualname__qhull_optionsqhull: did not free _qhullqhullqhull __pyx_vtable____pyx_unpickle_Enum__pyx_type__pyx_state__pyx_result__pyx_getbuffer__pyx_checksum__pyx_PickleErrorproperty__prepare___pointspointspoint_regionplane_distance pieces)pickleparaboloid_shiftparaboloid_scalepackout_outorderoptionsoption_setoptionobjnumpy.core.umath failed to importnumpy.core.multiarray failed to importnumpynsimplexnrhsnpointsnpnon-simplical facet encountered: no default __reduce__ due to non-trivial __cinit__nineq__new__neighborsndim__name__namenmsize__module__mode_optionmodemin_boundmin__metaclass__memviewmemory allocation failedmax_boundmaxmap__main__mamlift_pointsldbldalatin1kjoinjiworkivertexitemsize <= 0 for cython.arrayitemsizeisnanisimplexisMaskedArrayipivinvalid size for new points arrayintpintersectionsinterior_pointintc__init__infoincremental mode not enabled or already closedincremental__import__idihalfspacesgot differing extents in dimension %d (got %d and %d)good__getstate__get_voronoi_diagramget_simplex_facet_arrayget_pointsget_paraboloid_shift_scaleget_hull_pointsget_hull_facetsget_extremes_2d_get_barycentric_transformsgetfurthest_site__func__fortranformatflagsfinfofind_simplexfillerrorequationseps_broadepsenumerateencodeemptydual_volumedual_verticesdual_pointsdual_facetsdual_equationsdual_areadtype_is_objectdtypedstdoubledot__doc____dict____del__d_copy_docstrcopycoplanarconvex_hullconcatenateclosecline_in_tracebackclear__class__check_activec bytes (/build/scipy-CRQwve/scipy-1.11.4/scipy/spatial/_qhull.pyxbruteforcebaseaxisastypeascontiguousarrayasarrayasanyarrayarrayarrarea are incompatible with incremental modeanyanormallocate_buffer__all___add_pointsadd_pointsadd_halfspaces Voronoi.ridge_dict Voronoi(points, furthest_site=False, incremental=False, qhull_options=None) Voronoi diagrams in N dimensions. .. versionadded:: 0.12.0 Parameters ---------- points : ndarray of floats, shape (npoints, ndim) Coordinates of points to construct a Voronoi diagram from furthest_site : bool, optional Whether to compute a furthest-site Voronoi diagram. Default: False incremental : bool, optional Allow adding new points incrementally. This takes up some additional resources. qhull_options : str, optional Additional options to pass to Qhull. See Qhull manual for details. (Default: "Qbb Qc Qz Qx" for ndim > 4 and "Qbb Qc Qz" otherwise. Incremental mode omits "Qz".) Attributes ---------- points : ndarray of double, shape (npoints, ndim) Coordinates of input points. vertices : ndarray of double, shape (nvertices, ndim) Coordinates of the Voronoi vertices. ridge_points : ndarray of ints, shape ``(nridges, 2)`` Indices of the points between which each Voronoi ridge lies. ridge_vertices : list of list of ints, shape ``(nridges, *)`` Indices of the Voronoi vertices forming each Voronoi ridge. regions : list of list of ints, shape ``(nregions, *)`` Indices of the Voronoi vertices forming each Voronoi region. -1 indicates vertex outside the Voronoi diagram. When qhull option "Qz" was specified, an empty sublist represents the Voronoi region for a point at infinity that was added internally. point_region : array of ints, shape (npoints) Index of the Voronoi region for each input point. If qhull option "Qc" was not specified, the list will contain -1 for points that are not associated with a Voronoi region. If qhull option "Qz" was specified, there will be one less element than the number of regions because an extra point at infinity is added internally to facilitate computation. furthest_site True if this was a furthest site triangulation and False if not. .. versionadded:: 1.4.0 Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. ValueError Raised if an incompatible array is given as input. Notes ----- The Voronoi diagram is computed using the `Qhull library `__. Examples -------- Voronoi diagram for a set of point: >>> import numpy as np >>> points = np.array([[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], ... [2, 0], [2, 1], [2, 2]]) >>> from scipy.spatial import Voronoi, voronoi_plot_2d >>> vor = Voronoi(points) Plot it: >>> import matplotlib.pyplot as plt >>> fig = voronoi_plot_2d(vor) >>> plt.show() The Voronoi vertices: >>> vor.vertices array([[0.5, 0.5], [0.5, 1.5], [1.5, 0.5], [1.5, 1.5]]) There is a single finite Voronoi region, and four finite Voronoi ridges: >>> vor.regions [[], [-1, 0], [-1, 1], [1, -1, 0], [3, -1, 2], [-1, 3], [-1, 2], [0, 1, 3, 2], [2, -1, 0], [3, -1, 1]] >>> vor.ridge_vertices [[-1, 0], [-1, 0], [-1, 1], [-1, 1], [0, 1], [-1, 3], [-1, 2], [2, 3], [-1, 3], [-1, 2], [1, 3], [0, 2]] The ridges are perpendicular between lines drawn between the following input points: >>> vor.ridge_points array([[0, 3], [0, 1], [2, 5], [2, 1], [1, 4], [7, 8], [7, 6], [7, 4], [8, 5], [6, 3], [4, 5], [4, 3]], dtype=int32) Voronoi.pointsVoronoi.add_pointsVoronoi._updateVoronoi.__init__VoronoiView.MemoryViewValueErrorUnable to convert item to objectTypeErrorTinvs Takes care of basic dealings with the Qhull objects TRuntimeErrorQzQx QxQuQtQhull options Qhull internal error: loop in facet listQhull instance is closed_QhullUser.close_QhullUser._update_QhullUser._add_points_QhullUser.__init___QhullUser.__del___QhullUserQhullError_QhullQcQbkQbb Qc Qz Q12Qbb Qc QzQbbQbBQJQG_option_presentQGQBkQ11Points cannot contain NaNPickleErrorOut of bounds on buffer access (axis %d)ONo points givenNeed at least 2-D dataMemoryErrorInvalid shape in axis %d: %d.Invalid mode, expected 'c' or 'fortran', got %sInput points cannot be a masked arrayInput interior point cannot be a masked arrayInput halfspaces cannot be a masked arrayIndirect dimensions not supportedIndexErrorIncompatible checksums (0x%x vs (0xb068931, 0x82a3537, 0x6ae9995) = (name))ImportErrorHalfspaceIntersection.halfspaces HalfspaceIntersection(halfspaces, interior_point, incremental=False, qhull_options=None) Halfspace intersections in N dimensions. .. versionadded:: 0.19.0 Parameters ---------- halfspaces : ndarray of floats, shape (nineq, ndim+1) Stacked Inequalities of the form Ax + b <= 0 in format [A; b] interior_point : ndarray of floats, shape (ndim,) Point clearly inside the region defined by halfspaces. Also called a feasible point, it can be obtained by linear programming. incremental : bool, optional Allow adding new halfspaces incrementally. This takes up some additional resources. qhull_options : str, optional Additional options to pass to Qhull. See Qhull manual for details. (Default: "Qx" for ndim > 4 and "" otherwise) Option "H" is always enabled. Attributes ---------- halfspaces : ndarray of double, shape (nineq, ndim+1) Input halfspaces. interior_point :ndarray of floats, shape (ndim,) Input interior point. intersections : ndarray of double, shape (ninter, ndim) Intersections of all halfspaces. dual_points : ndarray of double, shape (nineq, ndim) Dual points of the input halfspaces. dual_facets : list of lists of ints Indices of points forming the (non necessarily simplicial) facets of the dual convex hull. dual_vertices : ndarray of ints, shape (nvertices,) Indices of halfspaces forming the vertices of the dual convex hull. For 2-D convex hulls, the vertices are in counterclockwise order. For other dimensions, they are in input order. dual_equations : ndarray of double, shape (nfacet, ndim+1) [normal, offset] forming the hyperplane equation of the dual facet (see `Qhull documentation `__ for more). dual_area : float Area of the dual convex hull dual_volume : float Volume of the dual convex hull Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. ValueError Raised if an incompatible array is given as input. Notes ----- The intersections are computed using the `Qhull library `__. This reproduces the "qhalf" functionality of Qhull. Examples -------- Halfspace intersection of planes forming some polygon >>> from scipy.spatial import HalfspaceIntersection >>> import numpy as np >>> halfspaces = np.array([[-1, 0., 0.], ... [0., -1., 0.], ... [2., 1., -4.], ... [-0.5, 1., -2.]]) >>> feasible_point = np.array([0.5, 0.5]) >>> hs = HalfspaceIntersection(halfspaces, feasible_point) Plot halfspaces as filled regions and intersection points: >>> import matplotlib.pyplot as plt >>> fig = plt.figure() >>> ax = fig.add_subplot(1, 1, 1, aspect='equal') >>> xlim, ylim = (-1, 3), (-1, 3) >>> ax.set_xlim(xlim) >>> ax.set_ylim(ylim) >>> x = np.linspace(-1, 3, 100) >>> symbols = ['-', '+', 'x', '*'] >>> signs = [0, 0, -1, -1] >>> fmt = {"color": None, "edgecolor": "b", "alpha": 0.5} >>> for h, sym, sign in zip(halfspaces, symbols, signs): ... hlist = h.tolist() ... fmt["hatch"] = sym ... if h[1]== 0: ... ax.axvline(-h[2]/h[0], label='{}x+{}y+{}=0'.format(*hlist)) ... xi = np.linspace(xlim[sign], -h[2]/h[0], 100) ... ax.fill_between(xi, ylim[0], ylim[1], **fmt) ... else: ... ax.plot(x, (-h[2]-h[0]*x)/h[1], label='{}x+{}y+{}=0'.format(*hlist)) ... ax.fill_between(x, (-h[2]-h[0]*x)/h[1], ylim[sign], **fmt) >>> x, y = zip(*hs.intersections) >>> ax.plot(x, y, 'o', markersize=8) By default, qhull does not provide with a way to compute an interior point. This can easily be computed using linear programming. Considering halfspaces of the form :math:`Ax + b \leq 0`, solving the linear program: .. math:: max \: y s.t. Ax + y ||A_i|| \leq -b With :math:`A_i` being the rows of A, i.e. the normals to each plane. Will yield a point x that is furthest inside the convex polyhedron. To be precise, it is the center of the largest hypersphere of radius y inscribed in the polyhedron. This point is called the Chebyshev center of the polyhedron (see [1]_ 4.3.1, pp148-149). The equations outputted by Qhull are always normalized. >>> from scipy.optimize import linprog >>> from matplotlib.patches import Circle >>> norm_vector = np.reshape(np.linalg.norm(halfspaces[:, :-1], axis=1), ... (halfspaces.shape[0], 1)) >>> c = np.zeros((halfspaces.shape[1],)) >>> c[-1] = -1 >>> A = np.hstack((halfspaces[:, :-1], norm_vector)) >>> b = - halfspaces[:, -1:] >>> res = linprog(c, A_ub=A, b_ub=b, bounds=(None, None)) >>> x = res.x[:-1] >>> y = res.x[-1] >>> circle = Circle(x, radius=y, alpha=0.3) >>> ax.add_patch(circle) >>> plt.legend(bbox_to_anchor=(1.6, 1.0)) >>> plt.show() References ---------- .. [Qhull] http://www.qhull.org/ .. [1] S. Boyd, L. Vandenberghe, Convex Optimization, available at http://stanford.edu/~boyd/cvxbook/ HalfspaceIntersection.dual_verticesHalfspaceIntersection.add_halfspacesHalfspaceIntersection._updateHalfspaceIntersection.__init__HalfspaceIntersectionHFeasible point must be a (ndim-1,) arrayFailed to allocate memory in setlist.init()Empty shape tuple for cython.arrayEllipsisDelaunay.vertex_to_simplexDelaunay.vertex_neighbor_verticesDelaunay.transform Delaunay(points, furthest_site=False, incremental=False, qhull_options=None) Delaunay tessellation in N dimensions. .. versionadded:: 0.9 Parameters ---------- points : ndarray of floats, shape (npoints, ndim) Coordinates of points to triangulate furthest_site : bool, optional Whether to compute a furthest-site Delaunay triangulation. Default: False .. versionadded:: 0.12.0 incremental : bool, optional Allow adding new points incrementally. This takes up some additional resources. qhull_options : str, optional Additional options to pass to Qhull. See Qhull manual for details. Option "Qt" is always enabled. Default:"Qbb Qc Qz Qx Q12" for ndim > 4 and "Qbb Qc Qz Q12" otherwise. Incremental mode omits "Qz". .. versionadded:: 0.12.0 Attributes ---------- points : ndarray of double, shape (npoints, ndim) Coordinates of input points. simplices : ndarray of ints, shape (nsimplex, ndim+1) Indices of the points forming the simplices in the triangulation. For 2-D, the points are oriented counterclockwise. neighbors : ndarray of ints, shape (nsimplex, ndim+1) Indices of neighbor simplices for each simplex. The kth neighbor is opposite to the kth vertex. For simplices at the boundary, -1 denotes no neighbor. equations : ndarray of double, shape (nsimplex, ndim+2) [normal, offset] forming the hyperplane equation of the facet on the paraboloid (see `Qhull documentation `__ for more). paraboloid_scale, paraboloid_shift : float Scale and shift for the extra paraboloid dimension (see `Qhull documentation `__ for more). transform : ndarray of double, shape (nsimplex, ndim+1, ndim) Affine transform from ``x`` to the barycentric coordinates ``c``. This is defined by:: T c = x - r At vertex ``j``, ``c_j = 1`` and the other coordinates zero. For simplex ``i``, ``transform[i,:ndim,:ndim]`` contains inverse of the matrix ``T``, and ``transform[i,ndim,:]`` contains the vector ``r``. If the simplex is degenerate or nearly degenerate, its barycentric transform contains NaNs. vertex_to_simplex : ndarray of int, shape (npoints,) Lookup array, from a vertex, to some simplex which it is a part of. If qhull option "Qc" was not specified, the list will contain -1 for points that are not vertices of the tessellation. convex_hull : ndarray of int, shape (nfaces, ndim) Vertices of facets forming the convex hull of the point set. The array contains the indices of the points belonging to the (N-1)-dimensional facets that form the convex hull of the triangulation. .. note:: Computing convex hulls via the Delaunay triangulation is inefficient and subject to increased numerical instability. Use `ConvexHull` instead. coplanar : ndarray of int, shape (ncoplanar, 3) Indices of coplanar points and the corresponding indices of the nearest facet and the nearest vertex. Coplanar points are input points which were *not* included in the triangulation due to numerical precision issues. If option "Qc" is not specified, this list is not computed. .. versionadded:: 0.12.0 vertex_neighbor_vertices : tuple of two ndarrays of int; (indptr, indices) Neighboring vertices of vertices. The indices of neighboring vertices of vertex `k` are ``indices[indptr[k]:indptr[k+1]]``. furthest_site True if this was a furthest site triangulation and False if not. .. versionadded:: 1.4.0 Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. ValueError Raised if an incompatible array is given as input. Notes ----- The tessellation is computed using the Qhull library `Qhull library `__. .. note:: Unless you pass in the Qhull option "QJ", Qhull does not guarantee that each input point appears as a vertex in the Delaunay triangulation. Omitted points are listed in the `coplanar` attribute. Examples -------- Triangulation of a set of points: >>> import numpy as np >>> points = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1]]) >>> from scipy.spatial import Delaunay >>> tri = Delaunay(points) We can plot it: >>> import matplotlib.pyplot as plt >>> plt.triplot(points[:,0], points[:,1], tri.simplices) >>> plt.plot(points[:,0], points[:,1], 'o') >>> plt.show() Point indices and coordinates for the two triangles forming the triangulation: >>> tri.simplices array([[2, 3, 0], # may vary [3, 1, 0]], dtype=int32) Note that depending on how rounding errors go, the simplices may be in a different order than above. >>> points[tri.simplices] array([[[ 1. , 0. ], # may vary [ 1. , 1. ], [ 0. , 0. ]], [[ 1. , 1. ], [ 0. , 1.1], [ 0. , 0. ]]]) Triangle 0 is the only neighbor of triangle 1, and it's opposite to vertex 1 of triangle 1: >>> tri.neighbors[1] array([-1, 0, -1], dtype=int32) >>> points[tri.simplices[1,1]] array([ 0. , 1.1]) We can find out which triangle points are in: >>> p = np.array([(0.1, 0.2), (1.5, 0.5), (0.5, 1.05)]) >>> tri.find_simplex(p) array([ 1, -1, 1], dtype=int32) The returned integers in the array are the indices of the simplex the corresponding point is in. If -1 is returned, the point is in no simplex. Be aware that the shortcut in the following example only works correctly for valid points as invalid points result in -1 which is itself a valid index for the last simplex in the list. >>> p_valids = np.array([(0.1, 0.2), (0.5, 1.05)]) >>> tri.simplices[tri.find_simplex(p_valids)] array([[3, 1, 0], # may vary [3, 1, 0]], dtype=int32) We can also compute barycentric coordinates in triangle 1 for these points: >>> b = tri.transform[1,:2].dot(np.transpose(p - tri.transform[1,2])) >>> np.c_[np.transpose(b), 1 - b.sum(axis=0)] array([[ 0.1 , 0.09090909, 0.80909091], [ 1.5 , -0.90909091, 0.40909091], [ 0.5 , 0.5 , 0. ]]) The coordinates for the first point are all positive, meaning it is indeed inside the triangle. The third point is on an edge, hence its null third coordinate. Delaunay.pointsDelaunay.plane_distanceDelaunay.lift_pointsDelaunay.find_simplexDelaunay.convex_hullDelaunay.add_pointsDelaunay._updateDelaunay.__init__DelaunayConvexHull.vertices ConvexHull(points, incremental=False, qhull_options=None) Convex hulls in N dimensions. .. versionadded:: 0.12.0 Parameters ---------- points : ndarray of floats, shape (npoints, ndim) Coordinates of points to construct a convex hull from incremental : bool, optional Allow adding new points incrementally. This takes up some additional resources. qhull_options : str, optional Additional options to pass to Qhull. See Qhull manual for details. (Default: "Qx" for ndim > 4 and "" otherwise) Option "Qt" is always enabled. Attributes ---------- points : ndarray of double, shape (npoints, ndim) Coordinates of input points. vertices : ndarray of ints, shape (nvertices,) Indices of points forming the vertices of the convex hull. For 2-D convex hulls, the vertices are in counterclockwise order. For other dimensions, they are in input order. simplices : ndarray of ints, shape (nfacet, ndim) Indices of points forming the simplical facets of the convex hull. neighbors : ndarray of ints, shape (nfacet, ndim) Indices of neighbor facets for each facet. The kth neighbor is opposite to the kth vertex. -1 denotes no neighbor. equations : ndarray of double, shape (nfacet, ndim+1) [normal, offset] forming the hyperplane equation of the facet (see `Qhull documentation `__ for more). coplanar : ndarray of int, shape (ncoplanar, 3) Indices of coplanar points and the corresponding indices of the nearest facets and nearest vertex indices. Coplanar points are input points which were *not* included in the triangulation due to numerical precision issues. If option "Qc" is not specified, this list is not computed. good : ndarray of bool or None A one-dimensional Boolean array indicating which facets are good. Used with options that compute good facets, e.g. QGn and QG-n. Good facets are defined as those that are visible (n) or invisible (-n) from point n, where n is the nth point in 'points'. The 'good' attribute may be used as an index into 'simplices' to return the good (visible) facets: simplices[good]. A facet is visible from the outside of the hull only, and neither coplanarity nor degeneracy count as cases of visibility. If a "QGn" or "QG-n" option is not specified, None is returned. .. versionadded:: 1.3.0 area : float Surface area of the convex hull when input dimension > 2. When input `points` are 2-dimensional, this is the perimeter of the convex hull. .. versionadded:: 0.17.0 volume : float Volume of the convex hull when input dimension > 2. When input `points` are 2-dimensional, this is the area of the convex hull. .. versionadded:: 0.17.0 Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. ValueError Raised if an incompatible array is given as input. Notes ----- The convex hull is computed using the `Qhull library `__. Examples -------- Convex hull of a random set of points: >>> from scipy.spatial import ConvexHull, convex_hull_plot_2d >>> import numpy as np >>> rng = np.random.default_rng() >>> points = rng.random((30, 2)) # 30 random points in 2-D >>> hull = ConvexHull(points) Plot it: >>> import matplotlib.pyplot as plt >>> plt.plot(points[:,0], points[:,1], 'o') >>> for simplex in hull.simplices: ... plt.plot(points[simplex, 0], points[simplex, 1], 'k-') We could also have directly used the vertices of the hull, which for 2-D are guaranteed to be in counterclockwise order: >>> plt.plot(points[hull.vertices,0], points[hull.vertices,1], 'r--', lw=2) >>> plt.plot(points[hull.vertices[0],0], points[hull.vertices[0],1], 'ro') >>> plt.show() Facets visible from a point: Create a square and add a point above the square. >>> generators = np.array([[0.2, 0.2], ... [0.2, 0.4], ... [0.4, 0.4], ... [0.4, 0.2], ... [0.3, 0.6]]) Call ConvexHull with the QG option. QG4 means compute the portions of the hull not including point 4, indicating the facets that are visible from point 4. >>> hull = ConvexHull(points=generators, ... qhull_options='QG4') The "good" array indicates which facets are visible from point 4. >>> print(hull.simplices) [[1 0] [1 2] [3 0] [3 2]] >>> print(hull.good) [False True False False] Now plot it, highlighting the visible facets. >>> fig = plt.figure() >>> ax = fig.add_subplot(1,1,1) >>> for visible_facet in hull.simplices[hull.good]: ... ax.plot(hull.points[visible_facet, 0], ... hull.points[visible_facet, 1], ... color='violet', ... lw=6) >>> convex_hull_plot_2d(hull, ax=ax)
# may vary >>> plt.show() References ---------- .. 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See qhull/io.c:qh_printextremes_2d Return the voronoi diagram currently in Qhull. Returns ------- voronoi_vertices : array of double, shape (nvoronoi_vertices, ndim) Coordinates of the Voronoi vertices ridge_points : array of double, shape (nridges, 2) Voronoi ridges, as indices to the points array. ridge_vertices : list of lists, shape (nridges, *) Voronoi vertices for each Voronoi ridge, as indices to the Voronoi vertices array. Infinity is indicated by index ``-1``. regions : list of lists, shape (nregion, *) Voronoi vertices of all regions. point_region : array of int, shape (npoint,) Index of the Voronoi region for each input point. Returns the facets contained in the current Qhull. This function does not assume that the hull is simplicial, meaning that facets will have different number of vertices. It is thus less efficient but more general than get_simplex_facet_array. Returns ------- facets: list of lists of ints The indices of the vertices forming each facet. Returns all points currently contained in Qhull. It is equivalent to retrieving the input in most cases, except in halfspace mode, where the points are in fact the points of the dual hull. Returns ------- points: array of double, shape (nrpoints, ndim) The array of points contained in Qhull. Return array of simplical facets currently in Qhull. Returns ------- facets : array of int, shape (nfacets, ndim+1) Indices of coordinates of vertices forming the simplical facets neighbors : array of int, shape (nfacets, ndim) Indices of neighboring facets. The kth neighbor is opposite the kth vertex, and the first neighbor is the horizon facet for the first vertex. Facets extending to infinity are denoted with index -1. equations : array of double, shape (nfacets, ndim+2) Uninitialize this instance add_halfspaces(halfspaces, restart=False) Process a set of additional new halfspaces. Parameters ---------- halfspaces : ndarray New halfspaces to add. The dimensionality should match that of the initial halfspaces. restart : bool, optional Whether to restart processing from scratch, rather than adding halfspaces incrementally. Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. See Also -------- close Notes ----- You need to specify ``incremental=True`` when constructing the object to be able to add halfspaces incrementally. Incremental addition of halfspaces is also not possible after `close` has been called. tsearch(tri, xi) Find simplices containing the given points. This function does the same thing as `Delaunay.find_simplex`. Parameters ---------- tri : DelaunayInfo Delaunay triangulation xi : ndarray of double, shape (..., ndim) Points to locate Returns ------- i : ndarray of int, same shape as `xi` Indices of simplices containing each point. Points outside the triangulation get the value -1. See Also -------- Delaunay.find_simplex Notes ----- .. versionadded:: 0.9 Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.spatial import Delaunay, delaunay_plot_2d, tsearch >>> rng = np.random.default_rng() The Delaunay triangulation of a set of random points: >>> pts = rng.random((20, 2)) >>> tri = Delaunay(pts) >>> _ = delaunay_plot_2d(tri) Find the simplices containing a given set of points: >>> loc = rng.uniform(0.2, 0.8, (5, 2)) >>> s = tsearch(tri, loc) >>> plt.triplot(pts[:, 0], pts[:, 1], tri.simplices[s], 'b-', mask=s==-1) >>> plt.scatter(loc[:, 0], loc[:, 1], c='r', marker='x') >>> plt.show() lift_points(self, x) Lift points to the Qhull paraboloid. plane_distance(self, xi) Compute hyperplane distances to the point `xi` from all simplices. find_simplex(self, xi, bruteforce=False, tol=None) Find the simplices containing the given points. Parameters ---------- tri : DelaunayInfo Delaunay triangulation xi : ndarray of double, shape (..., ndim) Points to locate bruteforce : bool, optional Whether to only perform a brute-force search tol : float, optional Tolerance allowed in the inside-triangle check. Default is ``100*eps``. Returns ------- i : ndarray of int, same shape as `xi` Indices of simplices containing each point. Points outside the triangulation get the value -1. Notes ----- This uses an algorithm adapted from Qhull's ``qh_findbestfacet``, which makes use of the connection between a convex hull and a Delaunay triangulation. After finding the simplex closest to the point in N+1 dimensions, the algorithm falls back to directed search in N dimensions. Vertices of facets forming the convex hull of the point set. :type: *ndarray of int, shape (nfaces, ndim)* The array contains the indices of the points belonging to the (N-1)-dimensional facets that form the convex hull of the triangulation. .. note:: Computing convex hulls via the Delaunay triangulation is inefficient and subject to increased numerical instability. Use `ConvexHull` instead. Neighboring vertices of vertices. Tuple of two ndarrays of int: (indptr, indices). The indices of neighboring vertices of vertex `k` are ``indices[indptr[k]:indptr[k+1]]``. Lookup array, from a vertex, to some simplex which it is a part of. :type: *ndarray of int, shape (npoints,)* Affine transform from ``x`` to the barycentric coordinates ``c``. :type: *ndarray of double, shape (nsimplex, ndim+1, ndim)* This is defined by:: T c = x - r At vertex ``j``, ``c_j = 1`` and the other coordinates zero. For simplex ``i``, ``transform[i,:ndim,:ndim]`` contains inverse of the matrix ``T``, and ``transform[i,ndim,:]`` contains the vector ``r``. add_points(points, restart=False) Process a set of additional new points. Parameters ---------- points : ndarray New points to add. The dimensionality should match that of the initial points. restart : bool, optional Whether to restart processing from scratch, rather than adding points incrementally. Raises ------ QhullError Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled. See Also -------- close Notes ----- You need to specify ``incremental=True`` when constructing the object to be able to add points incrementally. Incremental addition of points is also not possible after `close` has been called. close() Finish incremental processing. Call this to free resources taken up by Qhull, when using the incremental mode. After calling this, adding more points is no longer possible. Compute barycentric affine coordinate transformations for given simplices. Returns ------- Tinvs : array, shape (nsimplex, ndim+1, ndim) Barycentric transforms for each simplex. Tinvs[i,:ndim,:ndim] contains inverse of the matrix ``T``, and Tinvs[i,ndim,:] contains the vector ``r_n`` (see below). Notes ----- Barycentric transform from ``x`` to ``c`` is defined by:: T c = x - r_n where the ``r_1, ..., r_n`` are the vertices of the simplex. The matrix ``T`` is defined by the condition:: T e_j = r_j - r_n where ``e_j`` is the unit axis vector, e.g, ``e_2 = [0,1,0,0,...]`` This implies that ``T_ij = (r_j - r_n)_i``. For the barycentric transforms, we need to compute the inverse matrix ``T^-1`` and store the vectors ``r_n`` for each vertex. These are stacked into the `Tinvs` returned.  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