searchindex.js

Search.setIndex({envversion:42,terms:{kaibel:26,polynomi:[46,4,41,14,42,36,32,44,23,47],schlegel:[34,26],orthogon:[4,5,10,42],yellow:[34,9],four:[35,1,26,45,22,23],is_ultra_parallel:37,quadrant_fan:1,"0703456v2":42,skeleton:[42,17],exclude_point:35,whose:[0,1,42,35,26,46,14,15,41,10,30,45,36,24,40,22,23,5,47],typeerror:[0,5,26,42,19,10,4,22,47],twenty_four_cel:[4,15,34,26],cross_poly_4d:26,flow_polytop:[4,26],alf:46,swap:[35,15],under:[0,1,42,15,43,44,32],preprocess:14,some_el:30,testabl:23,palp_n:42,everi:[36,1,26,5],reduced_projective_vector:20,rise:0,cdd_type:25,a_fac:26,sage_object:[5,2,42,39,16,7,8,37,45,20,40,33,23,34,27],affect:[42,23,45],bindableclass:3,linear_part_project:10,p4_11169:41,cmd:[23,16],pconfig:[17,20],factori:[4,22],frozen:43,vector:[],math:[35,1,4,26,5,6,41,42,36,23],lrslib:26,point2d:[47,34],vindex_color:42,label_list:40,codim:[36,1,42,35,5],naiv:[0,15,26],ngen:[22,47,32],direct:[0,1,42,4,26,46,27,10,36,45,33,22,34,5],loop_over_parallelotope_point:15,doubt:6,n_bar:5,point_coordin:0,fan_2d_cyclically_ordered_rai:18,illumin:36,blue:[37,11,34,26,9],p_qq:15,box_max:[0,15,26],initial_form:23,contemp:42,nef:42,neg:[35,46,26,1,14,40,42,5],biject:[1,42],tohoku:36,interior_contain:[5,26,27],"new":[0,1,2,3,26,5,6,40,48,10,30,14,36,32,24,22,47,42,35],symmetr:[44,42,26],topolog:11,render_solid_3d:34,ck99:42,str_list:23,elimin:45,inner_loop_vari:15,"0x7f26f206d7d0":34,"3sec":4,zach:24,never:[11,19,17,47,10],here:[0,1,2,4,26,37,39,44,43,17,36,20,24,47,46,22,23,5,42],bounded_edg:[4,26],groebner_fan:23,projection_direct:[34,26],coord_index_list:34,path:4,bogdan:5,oriented_boundari:36,abstractli:3,interpret:[35,1,42,6,17,36,14,47],f_vector:[4,23,26],fg_pid:22,anymor:42,submodule_with_basi:22,precis:[1,26,14,15,10,29,4,20,23],loop:[15,26],subpolytop:[0,41],ray_matrix_normal_form:15,studi:[35,1],apart:35,isomorph:[],golden:4,parallelepip:4,fano_varieti:[1,45,40,36,22,5],render_solid:26,gosset:4,number_of_reduced_groebner_bas:23,linearli:[35,5,45,15,42,22],hypercub:[26,27,41,17,4,33,34],"096240cm":34,total:[35,15,23,17,43],label_color:[40,9],unit:[3,26,41,4,11,34],highli:16,plot:[],graphics3d:[35,42,37,26,40,17,9,4,22,11,34],describ:[5,26,46,6,42,22,23],would:[6,46,25,3,42],is_simpl:[4,16,26],maximal_con:23,kp2002:[26,43],call:[0,1,3,5,6,7,9,10,11,42,14,15,17,18,20,22,23,26,45,29,36,34,35,4,37,41,40,19,43,46,47],typo:36,recommend:[42,26],sturmfel:47,type:[5,42,3,26,46,6,40,41,27,8,10,9,30,36,20,33,34],hyperbolicmodel:[24,3],vertices_pc:[5,26,42],wall_label:40,relat:[35,1,4,5,40,42,36,20],notic:[36,22,4],warn:[36,5],berlin:[6,23,16],hold:[5,15],must:[0,5,3,10,42,14,15,17,22,23,26,27,4,34,35,36,37,41,40,19,44,45,47],base_zz:[31,41],springer:14,join:[36,34],hyperbolic_interfac:24,fukuda:[14,46,23],work:[0,1,2,35,26,5,14,15,7,13,9,36,19,33,46,22,23,34,42],backend_cdd:[46,21],prescrib:22,wors:5,introduc:22,root:[4,46],undirect:26,constant_term:[47,10,32],give:[0,5,48,4,46,7,42,9,36,45,33,22,11,34,23],"0cpu":42,face_v_indic:33,ambient_ray_indic:[36,5,40],brunskoch:5,want:[35,5,4,26,43,15,7,10,17,36,40,22],keep:[],unsign:35,jensen:23,"331036cm":34,facet_of:[36,5],end:[35,1,37,25,26,45,19,4,3,44],thing:[35,1,42,36,23,5],point_opt:34,hom:[0,1,22],homogeneity_spac:23,classifi:[14,5,19,36],ilio:41,how:[0,46,3,26,42,6,15,48,8,17,36,20,22,34],contains_origin:0,ring_to_gfan_format:23,disappear:6,whitney_numb:47,answer:[46,26],verifi:[14,36,7],vertices_matrix:[35,26],updat:[6,11,37],point_idx_list:35,recogn:34,facet_color:[42,34],irration:41,poly_dmax:42,after:[30,1,42,26,40],diagram:[],befor:[35,5,26,42,6,15,36,40,22,47],wrong:[6,1,22,36,46],"_compute_cone_lattic":43,parallel:[37,26,6,40,2,10,47],demonstr:1,kreuzer:[42,41],attempt:[36,5,35,26,18],third:26,classmethod:35,poly_spam:26,exclud:[35,26,27,40,29,47],doubly_indexed_whitney_numb:47,maintain:23,green:[47,34,26,9],enter:[4,9],exclus:22,lambda:[6,4,37,42],order:[35,5,42,4,26,18,6,15,27,29,43,36,14,20,40,46,33,23,34,47],oper:[5,4,26,45,36,22],fetter:4,fukudaprodon:14,hyperbolicgeodesickm:37,over:[],verts_for_norm:23,orang:[37,34],becaus:[6,1,10,5],cohen:6,ray_in_l:5,incid:[33,43,26,27],"poincar\u00e9":[24,37,3],attracting_fixed_point:19,v_i:42,v_j:42,fit:15,fix:[0,1,37,3,26,5,6,27,17,30,45,19,36,32,46,11,35],sigma_p:1,better:[36,5,47],uoa:41,complex:[36,11,3,17,19],render_line_1d:34,easier:[0,35,3,26,43],them:[0,5,2,42,46,6,40,27,10,36,45,32,22,47],"0sec":42,thei:[46,37,26,1,15,42,36,40,22,47,5],rectangl:3,"break":[35,5],all_point:42,wedg:[5,34],radius_squar:26,choic:[0,1,4,46,36,45,22,5],face_v:33,fan_isomorph:18,fgp_module_class:22,face_h:33,arrow:[40,34],each:[0,1,48,5,6,7,9,42,14,15,16,17,18,22,23,26,29,4,32,33,34,35,36,41,40,43,44,46,47],debug:[14,46,7,42,15],side:[47,27],pc_nostar:35,mean:[1,37,26,5,14,15,41,10,4,40,22,11,47],nbegin:25,symmetri:[44,46,23,26,42],show_point:42,buchberg:23,clear_cach:48,loop_polyhedra_with_recycl:26,varchenko:47,gave:6,coordinates_of:[34,26],"0x7f8fc40c1758":[],extract:5,kuai:47,unbound:[],symmetricgroup:[4,47],bayreuth:35,rewrit:[35,46],p_solid:[34,26],adapt:16,plot3d:[42,34,26],rank_funct:47,vertex_s:42,got:40,normal_con:17,alltogeth:26,kernel_fan:1,coord_indices_of:34,linear:[],written:[42,26,43],polyhedra_field:30,"244762cm":34,situat:[36,1,40],infin:[1,37,3,26,46,19,44,11],free:[0,1,42,26,5,7,8,17,30,18,36,32,22],standard:[35,5,4,42,14,36,22,23],show_pindic:42,polyhedralfan:23,ertex:[],lineal:23,loop_polyhedra:26,reconfigur:16,supercon:5,angl:[40,37,34],hom21:1,traceback:[0,1,3,5,6,11,42,14,15,10,18,20,22,23,26,45,29,4,34,35,36,37,41,40,19,47],filter:42,leila:24,ish:[47,48],iso:29,unabl:[4,42],subtl:[1,5],confus:44,sageobject:[5,2,42,39,16,7,8,37,45,20,40,33,23,34,27],polyhedron:[],get_geodes:[37,3],rang:[5,3,4,26,41,6,40,42,9,36,18,23,47],"939161cm":34,render:[44,46,23,34,26],tachyon:34,independ:[35,5,42,26,27,15,17,45,33,22],rank:[0,5,42,48,26,39,14,7,8,17,30,18,36,32,22,47],necess:5,restrict:[0,1,26,42,14,15,35,4,20,23,47],simplicial_polytop:26,instruct:[35,26],alreadi:[36,5,45,26,41],wrapper:0,inequalitii:7,thick:[44,40,37,34],agre:[36,42],primari:24,nefpartit:42,cartesian:[36,5,26,45],iaggelonica:5,region_containing_point:47,sometim:5,parentnewtonpolygon:44,poset_el:26,wigglesworth:24,too:[1,4,5,15,36,22,34],tol:4,similarli:[46,11],isometry_group_is_project:3,john:1,isometr:[4,3],consol:34,circl:3,is_conform:3,tool:[33,23,42],grini:42,hypersimplex:4,somewhat:20,conserv:4,technic:[35,10],schlegel_edg:26,"840780cm":34,simplicial_complex:17,keyword:[5,26,46,40,43,8,17,36,22,34],princeton:5,provid:[1,4,26,44,5,14,40,43,16,42,36,20,24,22,23,47],tree:4,second:[1,37,26,42,43,36,44,47,5],project:[0,1,42,3,4,26,41,19,10,36,20,23,34,47,35],matter:[36,26],strict_quoti:5,bdry_point_test:3,half3d:26,jmol:34,latticeeuclideangroupel:[29,39],rad:34,wheelgraph:48,mind:[36,40],affine_algebraic_patch:8,rai:[46,1,7,36,13,14,15,17,18,21,23,25,26,27,30,33,34,5,45,41,40,31,47],raw:36,seed:[16,20],hrep2vrep:7,seen:4,seem:36,incompat:[1,5],minu:26,recreat:17,unbounded_region:47,latter:[35,5,26,42,6,7,10,20],ray_list:[36,5],thoma:[23,26],even:[0,1,46,42,36,45,22,5],part_of:42,though:[6,1,41,22],face_h_indic:33,object:[],bss:[0,35,26],ofil:42,monomi:[36,23],regular:[35,1,4,26,45,17,36,20],is_v:27,jean:24,sub:[0,46,41,29,22],filter_polytop:42,"_pivot":7,don:[36,5,42,40],doc:23,flow:4,doe:[35,46,4,26,1,14,40,42,9,30,36,22,11,5,23],yang:47,ray_zord:40,parentmethod:24,toric_lattic:22,left:[6,45,23,27],sum:[35,5,4,26,44,46,41,42,36,45,22,23,47],dot:[34,26],quadraticfield:[14,4],triang:[35,17],"746680cm":34,opposit:[44,40,26],random:[0,5,3,26,42,6,10,36,14,45,32],sage:[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],radiu:[40,26],syntax:[22,47,10],polygon_opt:34,vertex_digraph:26,involv:[1,22,43,46],lineality_dim:23,vectorspacemorph:26,egyptian_pyramid:26,fixed_point_set:19,column_matrix:[5,45,42,15],sugar:[47,32],inequalit:[14,27],tropic:23,oct:26,klein:[19,24,11,3,37],googlegroup:23,likewis:5,stop:[45,26],ceil:41,infti:3,right_kernel:2,report:36,reconstruct:20,is_fibr:1,square_from_ieq:46,integervector:4,emb:[36,1,5],simplex_indic:26,equilateral_triangl:13,associahedron:[4,16],patch:[36,1,35],twice:[36,45],bad:41,nrai:[36,5],palp_modifi:42,rhoad:48,one_dimensional_con:5,riemannian:3,get_idx:26,polymak:[23,16],is_con:5,mandatori:[36,5,22],result:[35,46,4,1,15,7,42,40,36,45,39,22,34,5],respons:3,fail:[1,42,5],themselv:[0,5,19],is_rank:47,best:[15,26],order_polytop:47,awar:1,said:5,onto:[0,1,10],databas:[],wikipedia:[4,26,41],figur:[6,46],common_perpendicula:37,prepare_inner_loop:15,awai:1,approach:[42,3],inequalities_list:26,attribut:[36,5,40],is_nefpartit:42,extend:[1,43,30,36,23,47],bipyramid:26,qiaoyu:47,extens:[5,26,46,43,36,33,22,47],recent:[0,1,3,5,6,11,42,14,15,10,18,20,22,23,26,45,29,4,34,35,36,37,41,40,19,47],no_decomposit:41,toler:46,star_gener:[36,1],expos:40,c24:26,hashabl:[5,43,42],cox:[1,42],howev:[0,1,42,3,26,5,6,40,19,35,36,22],vertex_graph:[36,18,26],r_cone:14,facet:[0,5,42,35,4,26,27,15,16,17,36,45,23,34],bigraph:48,linear_subspac:5,uni:[35,5],com:23,monograph:42,compromis:36,projection_dir:26,kwd:[5,13,4,26,41,40,43,8,17,9,10,36,28,21,31,22,23,34,38,47],had:[36,1],affine_spac:0,height:[42,34,26,41],hyperbolicgeodesicpd:37,evaluated_on:27,permut:[0,35,26,42,6,15,4],unsimplifi:24,assum:[35,1,26,5,6,27,17,14,20,47],interior_facet:17,speak:[3,39],question:5,mathieu:26,degener:[35,5,26,46,29,10,34],union:[5,4,26,42,36,47],cube_du:26,short_nam:3,three:[0,1,26,5,6,42,36,23],been:[6,35,16],much:[0,35,4,26,42,15,36,23],interest:[],basic:[47,26],cell24:[0,15,16],wiliam:5,hyperplanearrang:[47,48,10,9],set_engin:[35,5,17],is_trivi:[5,17],poly_con:23,ani:[1,3,5,6,7,8,42,13,14,17,22,23,26,27,29,30,34,35,36,37,41,40,43,46],lift:[35,5,22,18],pezzo:1,is_polyhedron:26,z3squar:0,hex_point:34,ident:[1,37,26,5,19,4,45,34],latex_dual_nam:22,gnu:42,properti:[0,36,23,34,41],normal_form:42,standardalgorithm:14,euclidean:[],calcul:[35,3,17],pairwis:[6,36],hopkin:48,e_rai:[36,5],change_r:[47,10,32],anchor:46,pyramid:[35,26],positive_circuit:35,space_dimens:0,axes_width:34,matrix_spac:14,disjoint:42,kwarg:[44,3],cone:[],jesu:35,sever:[1,4,24,6,40,42,36,23,47],n_bounded_region:47,minkowski_differ:26,perform:[5,24,43,42,36,45,22,33,34],suggest:[36,5],make:[0,1,3,5,6,11,42,14,15,16,17,20,22,23,26,36,33,34,35,43,40,46,47],transpar:40,figsiz:6,drawback:24,roland:42,split:[1,34,40],n_ambient_vrepresent:33,lexicographic_triangul:[35,17],point_zord:40,complet:[0,1,37,3,42,40,43,36,22],is_empti:[30,26],lattice_euclidean_group_el:39,vobj:27,hand:[36,1,27],polyhedra_zz_ppl_with_categori:[30,8,41],rais:[0,1,3,5,10,36,42,15,17,18,19,20,23,26,29,30,35,45,37,39,41,47],portal:26,unimodular:41,refin:[0,1,23,36],tung:1,generator_thick:40,tune:[6,42],n_facet:[16,26],thu:[1,42,5],min:15,coord_index_of:34,inherit:[10,27],tho:11,thi:[0,1,3,4,5,6,7,9,10,11,42,14,15,16,17,19,20,22,23,24,26,27,29,30,32,33,34,35,36,37,39,40,41,43,44,45,46,47],everyth:[6,36,42,40],moreov:[22,42],maxima:42,is_toriclattic:22,newtonpolygon_el:44,identifi:[0,22,20,42],combinatorial_automorphism_group:26,just:[0,5,46,14,7,42,43,36,32,39,22,23],toral:36,newton:[],is_extrem:14,bounding_box:[0,34,26],yet:[36,1,9,40],previous:[6,35],shoud:10,hyperconifold:17,hal:1,easi:[6,36,42],hao:1,polyhedron_zz:[31,41],conjug:3,spread:[24,9],point_ord:35,els:[40,26],save:[36,1,23,26],explanatori:42,jsmath:45,opt:9,applic:[35,42,43,26,41],mayb:40,preserv:[0,35,3,26,42,19,10],c5_20:26,all_facet_equ:42,repelling_fixed_point:19,background:3,shadow:[35,26],poset:[5,26,43,36,33,47],has_user_basi:22,measur:[35,26],specif:[35,37,3,40,19,11],arbitrari:[],transvers:26,manual:[33,42,40,17,20],nfacet:42,cube_nonpolyhedr:1,internal_rai:23,"_linear_subspac":7,dutour:26,underli:[0,1,3,26,27,17,11,23],www:[35,5,23,41],right:[46,45,6,4,18,44,23,34],old:47,dilated_cell24:15,interv:42,maxim:[0,35,26,42,14,15,36,23],relative_orthogonal_quoti:5,flatten:[36,1,4],indirect:[5,17,9,22],double_description_inhomogen:[14,7],octaplex:4,ymin:40,ico:4,bottom:[36,5,26,43],fachbereich:16,subclass:[14,23],circ:[42,3],ichim:5,fannotisomorphicerror:[36,18],condit:[1,23,18,42,5],p2_123:36,plu:5,peel:15,fgp_modul:22,base_project:0,stefan:6,adjacency_graph:17,convex:[],vertices_list:26,hyperbolicmodelhm:[24,3],"super":24,chapter:5,alexand:42,finiteposet:[36,26,43],slightli:[0,34,26],coars:1,toriclattice_sublattice_with_basi:22,marshal:[35,46,23,43],soc:42,"float":[4,46,34,26],encod:[],bound:[0,46,37,3,26,42,15,36,40,11,47],ppl:[],down:[6,15,43,41],phi_:1,ussr:36,precomput:42,primitive_preimage_con:1,label_fonts:9,deal:[6,26],suffici:[24,3,6,15,7,36,18],orthogonal_sublattic:5,support:[35,46,13,26,1,27,17,30,36,23,5,47],is_directed_acycl:26,transform:[0,5,26,42,6,19,35,36],why:[36,46,47,35,17],avail:[35,4,44,26,42,40,7,9,36,45,22,23],label_zord:40,fraction:[5,20,32],acknowledg:42,overhead:0,stopiter:[20,26],delta_polar:42,head:40,freemodule_submodule_with_basis_pid:22,form:[0,1,42,4,26,5,15,17,9,36,18,40,23,47],offer:16,forc:[5,40,27],zz3:22,skeleton_point:[36,42],k_extend:36,"true":[0,1,2,3,4,5,6,9,10,11,42,14,15,16,17,18,20,22,23,26,27,29,30,32,33,34,35,36,37,41,40,19,44,45,46,47],is_strictly_convex:5,reset:40,moveto:34,simplex_union:26,maximum:[15,23,42],pseudolin:[],arccosh:[19,24,11,3,37],classif:[36,42,19],featur:[0,5,36,42],projectionfuncschlegel:34,ambient_vector_spac:[2,32],"abstract":[37,3,19,20,22,11],tournament:4,common_refin:36,bogart:23,is_lattice_polytop:[36,4,26,41],komei:[14,46,23],check:[1,3,5,6,11,42,14,15,17,18,22,23,26,45,4,32,33,36,41,40,19,44,47],pipe:42,linear_express:[0,10,32],minimized_gener:0,is_equival:[36,1,17,5],when:[35,46,4,1,41,39,42,36,5],refactor:[46,47],role:[1,42],test:[0,1,2,3,4,5,6,10,11,42,13,14,15,16,17,18,22,23,26,27,29,28,30,32,34,35,36,37,38,41,40,48,44,47],write_palp_matrix:42,intend:[37,39,40,36,22,23],benefici:36,subdivid:[36,1],base_projection_matrix:0,yperplan:[],intent:36,consid:[35,46,26,1,42,36,20,22,23,5],hirzebruch:5,uniformli:[3,26],subdivis:[36,1],lattice_automorphism_group:0,abscissa:44,p_rr:15,pseudo:43,"__iter__":20,restrict_to_regular_triangul:35,ignor:[42,8,26],time:[0,1,42,4,26,5,6,15,48,35,36,45,40,44,47,34],push:35,parameter_indic:23,chain:36,skip:[36,42,15,41,30,31],global:[40,23],affine_transform:42,invent:23,signific:[1,45],n_hrepresent:26,hyperboloid:[19,24,11,3,37],row:[5,26,45,14,7,42,47],backend_field:[46,13],decid:[36,22,26,40],middl:[35,1],depend:[0,5,42,26,46,40,39,35,36,22,23],system:[42,23,16,17],random_matrix:18,graph:[4,37,26,42,48,16,17,36,18,47],intermedi:[1,22],rightmost:36,flippabl:23,sourc:[4,43],string:[0,46,25,26,24,45,40,41,8,42,9,30,19,36,47,22,11,34,48,23],asymptot:37,triangul:[],domain_fan:1,word:[35,5,26,46],exact:[46,13,4,26,42,14,40,36,34],dim:[0,1,42,4,26,5,14,41,7,8,35,9,36,45,20,46,33,23],latticepolytopeclass:42,solver:26,did:36,dimenison:48,n_vrepresent:[8,26],n_vertic:[0,4,8,26],iter:[0,5,2,35,4,26,46,15,27,8,10,36,18,45,20,44,47,34,48,42,32],unsupport:[22,10],farthest_point:35,representaton:26,quick:[36,5],is_h:27,linear_part:[2,10],round:40,dict:40,nvertic:42,prevent:[1,22],slower:[42,13,7],empir:4,core:47,g_semiord:48,multiindic:36,sign:[5,26,27,41,10,22],cost:[0,43],run:[0,42,13,38,26,41,14,15,16,28,9,30,31,22],equivari:36,explictli:15,appear:[6,36,23,44],distance_enumer:47,toriclattice_ambi:22,uniform:[4,41],current:[5,37,4,26,24,40,19,42,30,45,36,46,44,11,34,23],maximilian:[42,41],gfan_polyhedral_con:23,get_background_graph:3,nsub:5,suppress:42,padic:44,deriv:[0,47,20,27],subspac:[],gener:[],toriclattice_quotient_el:22,coeffici:[35,5,42,26,46,6,15,41,7,10,36,14,32,22,47,27],label_i:40,satisfi:[35,1,42,26,27,15,17,23,34],slow:[46,15],normalize_rai:5,along:[36,5,34,29,42],wait:26,box:[0,15,47,26,40],rhombic_dodecahedron:4,wall_alpha:40,plotter:[],set_rai:40,behav:[36,1,47,45,5],extrem:[14,5,46],commonli:[],is_torsion_fre:22,mobiu:19,vector_space_morph:26,activ:[14,23],modul:[0,1,3,4,5,6,7,8,10,11,12,14,16,17,18,20,22,24,26,27,29,30,32,33,36,37,43,40,19,42,44,45,46,47],prefer:[45,40,23,43,42],parma:[],plot_point:40,leav:[36,4],p_wirefram:26,basis_matrix:2,toriclattice_sublattic:22,instal:[35,16,8],should:[0,1,2,5,6,7,9,10,11,42,14,17,20,22,23,26,27,36,33,35,45,41,40,19,43,47],classify_cone_2d:[36,5],memori:[22,45,42],univers:[30,5,7],perm:[6,18],perl:16,live:[36,1,34,32,5],is_centr:47,scale_color:23,ppid:34,h_7_3:4,reorder:15,permutahedron:[4,26],is_essenti:47,peopl:23,output_format:[5,45],polyhedron_rdf:[38,21],include_point:40,ambientvectorspac:10,support_contain:36,line3d:26,effort:3,ray_label:40,graphic:[35,5,37,3,26,40,29,48,17,9,10,36,47,11,34,23],prepar:42,constraint:[0,14,26],uniqu:[0,5,37,35,26,46,6,17,36,18,20,22],radon:47,facet_norm:[5,23,42],can:[0,1,3,4,5,6,7,8,9,10,11,42,15,17,19,20,22,23,24,26,27,29,30,32,33,34,35,36,37,39,40,41,43,45,46,47],points_pc:[0,42,26],purpos:[0,1,46,14,41,7,22],fulton:5,is_latticepolytop:42,a_row:14,tikzpictur:34,recycl:[30,26],cardin:[0,45,26],orgin:5,alwai:[0,5,42,35,26,46,14,40,27,10,44,45,22,47],differenti:11,multipl:[4,42,47,10],symmetry_gener:23,attract:[3,19],write:[5,25,26,42,16,23,34],small_rhombicuboctahedron:[4,34],pure:42,map:[0,1,3,26,39,15,35,4,24,5],product:[5,26,41,14,42,36,45,22,47],smith_form:41,reduc:[17,36,20,22,23,47],still:[36,5,23,24,22],max:15,is_equ:27,cours:[1,42,10,36,45,22,5],face_dimens:26,mai:[35,46,19,44,26,1,40,7,42,43,30,18,36,24,22,5],drastic:[36,5],data:[0,5,42,3,26,46,6,16,8,35,36,45,23,34],ray0:5,ray1:[5,40],ray2:[5,40],zero_set:14,explicit:[44,5,22],get_point:[19,24,11,3,37],produc:[40,23,17,42],quadrant:[1,26,5,15,40,34],inform:[],"switch":[35,5,45,26,42],coatom:[],combin:[36,15,47,26,42],subcollect:45,callabl:[36,40],blowup:1,talk:[46,47],superscript:[22,26],lsg:5,zmax:40,n_rho:5,poylhedra:8,random_geodes:[37,3],graduat:1,"_element_constructor_":10,"_vertex_pool":30,plot_wal:40,equip:3,revisit:14,conjunct:40,group:[],thank:43,apex:[34,17],geo:41,jim:4,toru:[36,1,42],sage_matrix_to_maxima:42,ssc:5,n_in_m:22,main:[0,1,23,16,22],is_smooth:[36,5],sign_vector:47,non:[0,1,42,35,4,26,46,15,43,29,36,40,47,5],recal:[36,35,17,42],generator_color:40,edge_thick:42,initi:[0,1,37,3,42,5,14,40,19,17,36,45,20,24,47,22,11,23,35],disabl:34,triangulations_list:35,half:[46,37,3,26,1,15,19,36,24,11,34,5],nov:47,now:[1,26,5,6,40,42,36,22,23],discuss:[42,34],nor:[30,5,36,26,22],introduct:[5,45],outweigh:24,term:[42,41,14,7,10,36,32,23,47],name:[1,42,3,24,40,48,8,10,30,36,32,22,47],perspect:34,revers:[44,18,47,20],revert:40,separ:[35,40,47,45,42],magazin:4,attributeerror:[36,5,40],subpolygons_of_polar_p1xp1:29,compil:[15,42],p_fine:35,domain:[0,1,42,26,39,8,17,30,18,36,32,5],hasse_diagram:[36,5,43],tikz3:34,tikz2:34,citat:26,replac:[36,5,41],individu:[47,16,10],continu:[5,16,18,15],restrict_to_connected_triangul:35,significantli:[36,5,42],individi:36,as_polynomial_map:1,operand:[22,10],happen:[1,22,5],shown:[42,9,45],polyhedron_rdf_cdd:[30,21],"3rd":42,space:[],toricplott:40,storag:27,perm_con:18,parallelogram:4,rays_list:26,correct:[36,5,45,42,22],sikir:26,tropical_intersect:23,fan2d:36,hull:[0,5,42,35,4,26,44,46,8,41,21,16,10,28,30,15,33,47,34,38,27],runtimeerror:41,envelop:26,simplex_to_int:20,zombi:35,base_region:47,get_isometri:[11,3,19],org:[5,34,26,42],care:[1,26,5,36,20,44],graphics3dgroup:[34,26],cuboctah:36,reisner:[36,23],toriclatticeel:22,arnaud:46,new_rai:36,pentaki:4,zzdb:8,turn:[4,5],person:23,place:[35,1,29,42],polyhedra_rdf_cdd:30,principl:[36,5],think:[36,5,26,22],first:[0,1,3,5,6,10,42,14,15,17,23,26,45,29,4,32,33,34,35,36,37,43,44,46,47],origin:[0,1,37,25,26,46,14,15,41,7,17,10,30,36,40,44,47,34,5,42,35],directli:[5,42,14,40,19,17,9,10,30,45,36,22,47],onc:[5,42,4,46,6,29,36],arrai:45,neighbors_in:5,yourself:[35,27],isometry_test:3,new_fan:36,tach_str:34,submit:36,bradshaw:42,toriclatticefactori:22,ring:[],custom:16,as_polyhedron:[46,33],lexicograph:[35,23],size:[42,2,6,40,10,9,4,14,37,11,47],given:[0,1,3,5,6,7,9,36,11,42,15,20,22,23,24,25,26,27,4,34,35,30,37,41,40,19,43,45,46,47,48],sheet:1,convent:[44,40],nef_x:42,whitney_data:47,p_regular:35,citi:47,quadrant_bl:1,necessarili:[35,1,37,26,5,6,15,29,41,10,17,36,20,42],f_1:19,f_2:19,reduced_affine_vector:20,proposit:[5,42],conveni:[46,37,24,19,36,45,11],defined_affin:35,hue:9,linearexpress:[10,32],reflection_involut:37,copi:44,braun:[0,5,46,40,35,36,20,47],specifi:[0,1,42,3,4,26,44,5,15,40,17,10,36,43,20,33,46,22,11,34,35],is_ident:19,echelonized_basis_matrix:42,newton_polygon:44,mostli:[40,43],domin:1,than:[0,5,13,3,26,46,7,10,42,36,37,11,23,27,35],serv:[5,46],wide:16,render_0d:34,integralraycollect:[36,5],"912008cm":34,face_interior:35,archimedean:4,were:[1,5,40,36,22,23,47],posit:[35,1,42,26,46,6,40,17,9,10,4,14,45,33,22,23,5],seri:[44,47],pre:15,sai:[36,1,45],ehrhart:41,"_gfan_gen":23,polyhedra_qq_ppl:30,argument:[0,5,42,26,46,6,40,41,27,8,35,43,36,45,33,22],dash:37,notimplementederror:[42,37,26,11,34,23],bistellar:[35,26],n_sigma:5,gkz:[35,17],engin:[35,17,26],squar:[0,35,42,4,26,41,19,29,36,33,23,34],characteristic_polynomi:[47,48],chessboard:47,"_palp":42,note:[0,1,3,5,6,7,11,42,14,15,10,18,20,22,23,25,26,27,36,32,34,35,4,39,40,44,45,46,47],ideal:[0,5,37,3,26,42,8,17,30,18,36,32,11,23],denomin:10,harald:[42,41],take:[1,3,4,26,46,6,40,19,42,43,36,20,5],interior:[0,1,42,3,26,5,41,27,17,36,47,11,34,23,35],functori:22,wonder:36,noth:[5,3,26,42,14,22],begin:[45,34,25,26],sure:[6,5,11,36,42],trace:19,normal:[35,1,42,26,5,15,41,10,17,36,45,40,23,34,47],multipli:[22,10,32],is_minkowski_summand:26,sagemath:[5,34,42],compress:20,random_vector:5,sqrt5:[4,26],pair:[0,1,42,35,4,26,5,14,15,41,29,36,47],show_wal:40,latex:[45,22,23,34,40],latticepolytop:[0,5,36,26,42],synonym:[36,35,42],fan_morph:1,color_list:40,later:40,quantiti:19,dodecahedron:[4,33],poly_r:23,axi:[1,34,19,42],sigma:[36,1,5],poly_x:42,slope:44,projective_spac:42,hyperboloidgiven:11,show:[42,37,26,6,40,17,11,34],delta:42,line_opt:34,ambient_hrepresent:[33,26],fibrat:[0,1,41],concurr:42,threshold:26,corner:23,codomain:[36,1,26,18,39],ns2:22,rotat:[34,26],onli:[0,1,2,48,5,6,7,9,42,14,15,10,20,22,23,26,27,29,36,32,34,35,4,37,39,40,41,43,44,45,46,47],explicitli:[35,5,3,42,40,36,45,20,22],ratio:4,transact:47,integer_kernel:42,distance_affin:35,parametr:9,black:[40,34,9],analyz:[42,41],pointsets_mod_automorph:0,is_simplex:[0,26],polyhedronfac:[33,26],overwritten:37,variou:[5,23,34,26,46],get:[1,4,5,40,42,36,45,33,22],star_generator_indic:36,secondari:[35,17],cannot:[35,5,37,25,26,27,45,15,42,36,18,3,40,22],count_onli:15,gen:[5,4,32,22,23,47],requir:[35,1,26,42,6,43,17,10,36],gel:35,prime:[42,47,34,26],tikz:34,is_incid:27,yield:[0,35,41,6,18,23],email:42,twelv:4,xmax:40,test_p:26,where:[0,1,2,48,4,26,46,6,15,40,35,43,36,45,32,47,33,22,23,5,42],a_con:14,kernel:[1,42],integral_length:42,stanlei:[36,4,23,47],contan:36,spars:22,pointset:0,has_good_reduct:47,parallelotop:[4,5,15],concern:14,infinit:[1,37,3,26,46,44],timeit:[0,26],detect:1,polyhedra_qq_cdd_with_categori:30,review:[36,1,5],enumer:[35,5,26,46,6,15,41,27,17,14,20,23],label:[0,42,40,48,9,36],are_adjac:14,enough:[0,5,36,26,15],between:[],"import":[0,1,2,3,4,5,6,7,8,9,10,42,14,15,17,18,19,20,21,22,23,24,25,26,27,29,30,32,33,34,36,37,39,40,41,44,45,46,48],vlist:34,across:24,cdd_vrepresent:[25,26],parent:[],latt:41,screen:42,punto:[35,20],cycl:36,highlight:23,come:[36,10],tneg:35,simplex_vertic:26,random_isometri:3,polyhedron_cdd:21,domain_dim:39,nill:42,region:[40,23,47],contract:17,tutori:36,cyclegraph:48,improv:[36,5,35,46],improp:29,spane:5,among:[6,35,23,27],rescal:[46,10],minimi:5,add_hyperplan:47,set_palp_dimens:42,hyperbolicmodelkm:[24,3],pole:1,poi:11,hyperplan:[],v2h:7,poli:[0,35,42,15,23,34],hobj:27,lidia:41,burel:24,marc:26,stretch:26,invert:[19,3,29],shim:20,invers:[19,20,26],mark:[42,23],hilbert_seri:8,preimage_con:1,reflex:[],valueerror:[0,1,37,3,26,5,6,15,41,10,42,30,45,19,36,47,40,22,11,23,35],curvatur:3,suscept:46,normalfan:[36,1],andrei:[46,1,40,43,42,36,45,22,5],lectur:[14,26],those:[1,26,6,40,14,47],"case":[35,1,42,26,46,6,15,43,8,17,10,36,45,20,40,23,34,5,47],atom_to_coatom:43,sagemanifold:3,superflu:46,cast:26,invok:5,exist:[0,5,37,26,40,35,29,36],invoc:41,"0x7f26f23370c8":23,pertain:3,adjac:[5,26,27,14,17,36,34],advantag:[5,42],stdout:42,metric:[35,3],canon:[36,1,4,5],uncov:1,ambient:[0,5,2,4,46,7,8,10,42,14,17,18,20,21,22,23,26,27,29,30,32,33,34,35,36,48,45,47],mani:[36,24,23,4,42],base_rdf:[38,21],gfan:23,develop:16,d12:4,author:[],edwin:16,exchang:[35,26,41],same:[35,1,37,4,26,39,6,40,7,10,43,36,45,20,47,33,46,22,23,5,42],poincar:[11,19,37,47],binari:[6,26],is_isomorph:[35,5,4,29,36,18,47],document:[35,5,26,46,6,41,42,36,22,23,47],finish:43,closest:10,linear_equivalence_id:36,companion:47,capabl:23,"_pivot_inequ":7,trivial_con:[36,5],extern:[42,26],hyperbolicpoint:[11,3,37],defn:1,cone_of_fan:36,appropri:[5,3,42,40,19,36,45,11],choos:40,assosi:36,embed_in_reflexive_polytop:0,n_integral_point:0,subcomput:23,without:[35,5,3,26,46,14,40,42,36,45,22],model:[],dimension:[0,1,48,4,5,8,9,10,42,15,17,18,21,22,23,24,26,27,28,29,30,33,34,35,36,38,39,40,41,44,46,47],render_wirefram:26,polygen:44,manifold:3,execut:[42,41],fgp_element:22,dihedr:26,tip:35,resp:44,ray_thick:40,puntos2000:35,rest:[36,1,42],four_simplex:26,isometri:[],aspect:42,speed:[5,26,42,15,30,45,36],kirkman_icosahedron:4,blow:1,palpread:8,is_vertex:[46,27],miscellan:47,sheldon:42,except:[35,1,42,26,39,40,17,36,18,45,44,23,5],littl:[14,1,22,36,5],all_opt:40,pindex_color:42,render3d:23,hrepresent:[35,46,26,27,14,15,10,33],pitman:4,real:[46,3,14,19,10,4,20,11,34],psi:1,around:[0,9],read:[6,42,34,8],kirkman:4,all_nef_partit:42,penta:[34,26],point_configur:[35,17,20],is_separating_hyperplan:47,color:[42,37,3,26,40,9,36,44,11,34,23],mod:47,isomorphsm:29,legend_3d:9,required_atom:43,satisfied_as_equ:15,pseudolinearrang:6,generator_system:[0,15],integ:[0,1,2,48,4,5,6,7,8,10,42,14,15,16,17,18,20,21,22,23,26,27,30,32,33,34,35,36,41,40,43,44,45,46,47],linial:48,newtonpolygon:44,chamber:[1,40],either:[35,46,37,3,4,26,1,45,15,41,42,9,30,19,36,20,33,22,5],output:[0,1,2,3,4,5,8,9,10,42,14,15,17,18,20,22,23,26,27,29,30,32,33,34,35,36,37,41,40,19,43,44,45,46,47,48],ambient_vrepresent:[33,26],h_indic:33,affine_dimens:0,freemodule_submodule_pid:22,hly02:1,perm_rai:18,inequalitycollect:15,simplici:[35,5,26,41,17,36,20,23],octant:5,enumerate_simplic:17,hamptonio:23,nonzero:36,mysteri:26,definit:[0,1,42,26,5,14,16,35,36,6],iff:36,vdinv:15,four_rai:5,inject:1,base_r:[0,46,7,36,13,14,15,10,20,22,26,27,28,4,32,35,30,38,41,48,31,47],is_surject:1,complic:[5,26,42,36,46,47],permutationgroup:[0,35,26],refer:[],has_ip_properti:[0,41],boundary_point_in_model:3,gfan_polyhedral_fan:23,ration:[],schuermann:26,is_rai:27,great:4,n_equat:26,immut:[5,45,42,36,46,22],image1:34,"throw":1,lattice_filt:40,polar_p2_polytop:[0,29],face_vector:47,comparison:10,simplest:22,central:[35,48,17,26],simplicialcomplex:17,render_3d:34,is_face_of:5,splitlin:34,degre:[0,5,2,44,41,40,7,16,10,36,22,23,34],input_r:23,stand:[47,10,26],neighbor:[36,5,27,4,46],act:[0,35,3,26,18,22],"2x2":19,autp24:26,morphism:[],routin:[35,26],izvestiya:36,effici:[14,1,36,42],elementari:4,number_of_vari:23,orthogonal_project:10,terminolog:6,affine_lattice_polytop:0,fibration_gener:[0,41],your:[35,1,42,5,40,48,17,36,22],topcom:[35,5,41,17,26],hep:42,ymax:40,aren:19,hex:34,overwrit:11,ks98:42,interfac:[],low:[36,47],lot:[42,23],"242192cm":34,strictli:[0,1,26,27,40,43,42,36,5],hodg:[42,8],repr_obj:27,polytop:[],edge_length:4,tupl:[0,1,2,48,5,9,10,42,14,15,17,18,20,22,23,26,45,29,4,32,33,34,35,36,41,43,44,47],bundl:[36,1],regard:47,alain:14,"_abcol":[36,5,42],satur:[0,1,26,5,15,35,36,22],facefan:[36,1],restrict_to_fine_triangul:35,inject_vari:[23,47],faster:[35,46,4,43,42,36],notat:[5,42,7,10,32,33,23,47],tripl:[14,1,35,26],possibl:[0,1,42,4,26,46,6,15,43,29,36,45,20,34,5],"default":[0,1,48,5,6,8,9,10,11,42,15,17,18,20,22,23,24,26,45,4,33,34,35,36,41,40,43,44,46,47],minkowski:[46,26,42,41,30,27],adjacency_matrix:[34,26],index_set:40,hyperbolic_geodes:37,embed:[0,5,42,39,14,7,29,36],icositetrachoron:4,minkowski_sum:[42,26],expect:[36,1,40,5],creas:17,creat:[1,37,3,26,5,6,40,19,16,17,9,10,36,45,20,47,33,22,23,42],certain:[36,42,47],argre:36,braid:[47,48],osnabrueck:5,decreas:[36,5,26],file:[],render_wireframe_3d:34,encompass:26,prevarieti:23,fill:[36,23,34,26],incorrect:1,denot:[35,47],sublattic:[1,22,40,5],mathematik:[5,16],reduct:47,polyhedra_bas:30,field:[35,5,2,48,4,26,46,14,41,7,13,9,10,30,36,20,22,23,47,32],prism:26,wai:[0,1,42,4,26,46,14,40,35,36,6,45,22,47],geometr:[1,47,26],you:[0,1,2,3,5,7,8,9,10,42,13,14,15,16,17,20,22,23,25,26,27,29,36,32,33,35,4,41,40,45,46,47],christof:5,hilbert_coeffici:5,polar:[42,26,45,41,29,36],create_kei:22,sequenc:[6,36,42],symbol:[46,19,24],toriclattice_gener:22,vertex:[0,5,46,7,8,42,13,14,15,16,10,20,25,26,27,29,28,4,33,34,35,30,38,41,43,44,47],mrho:5,schlegel_project:[34,26],ordered_vertic:29,track:34,reduced_projective_vector_spac:20,geodes:[],render_2d:34,"_test_pickl":[30,15,31,41],spend:45,directori:[42,8],descript:[],linearly_independent_vertic:42,scroll:34,goe:6,mimic:6,derrick:24,reduced_groebner_bas:23,potenti:[1,29,26,15],cpr:[1,40],represent:[],all:[0,1,48,4,5,6,7,9,11,42,13,14,15,17,18,20,22,23,24,26,27,29,30,32,33,35,36,37,41,40,19,43,44,45,46,47],dist:[19,24,11,3,37],consider:[36,15,42],face_codimens:35,illustr:[5,3,42,40],alg:14,hyperbolic_spac:[19,24,11,3,37],katz:42,parallelotope_point:15,scalar:[42,47,10,32,26],deprecationwarn:[5,34,42],follow:[0,1,42,35,4,26,5,6,15,41,7,29,9,43,36,14,40,22,34],disk:[19,24,11,3,37],extra:[46,26,1,42,36,5],poly_test:26,vertex_label:0,dot_product:26,rewrot:23,hyperbolic_point:11,white:40,facet_adjacency_matrix:26,articl:[4,26,41],program:[42,23,26,41],plot_gener:40,read_palp_matrix:42,numberfield:[46,13],subcon:5,norm:[4,37,10],vrepresentation_spac:[30,26],perpendicular_bisector:37,vert:[34,26],line2d:34,generating_con:[36,1,5],fals:[0,1,2,3,5,9,10,11,42,14,15,17,18,20,21,22,23,26,27,36,34,35,4,37,41,40,19,44,45,46,47,48],faq:46,normaliz:5,latte_int:[4,41],projet:34,fan:[],affinesubspac:2,fall:22,veri:[1,4,5,14,15,8,42,36],ticket:[36,1,4,26,5],orthonorm:4,cartesian_product:[36,5,45],keep_symmetr:42,laun:[19,24,11,3,37],positive_coord:46,four_cub:[4,26],list:[0,5,2,48,4,46,6,7,8,9,10,42,13,14,15,16,17,18,20,21,22,23,26,27,29,30,32,34,35,36,37,41,40,19,43,44,31,47],hyp:47,arithmeticerror:[22,42],adjust:[1,40],small:[35,5,4],cddr:26,codimens:[0,1,42,26,5,35,36,22],dimens:[0,1,2,3,4,5,6,7,8,9,10,11,42,14,15,16,17,19,20,21,22,23,26,45,29,30,32,33,34,35,36,37,39,40,41,46,47,48],read_all_polytop:42,number_of_column:42,automorphism_group:[36,26],tex:34,zero:[5,42,4,26,46,14,40,7,10,9,30,36,32,22,23,47,27],compute_vertex_con:41,design:[36,5,23,45,22],pass:[35,42,2,26,27,40,41,8,17,43,36,18,45,20,47,34,32],further:[36,5,42,26,46],what:[35,42,37,36,20,33,22],primitive_collect:36,richard:[4,47],section:[0,42,17,43],crush:42,delet:[30,47,26],abbrevi:[24,11,19,37],version:[0,1,37,3,4,42,5,15,19,47,17,36,45,20,24,40,46,22,11,23,35],mytriangul:17,intersect:[5,2,26,37,42,6,7,10,36,33,22,23,47],consecut:[44,42],deepli:26,method:[],expon:[23,16],hyperbolicspac:24,full:[0,5,42,4,26,41,14,15,10,36,22,23,34],convexrationalpolyhedralcon:[36,5,43],"_polynomial_system":23,midpoint:37,behaviour:[6,1,36],ambient_spac:[0,30,47,10,26],toric_varieti:[1,42,41,40,17,36,18,5],modifi:[40,42],legend:9,valu:[0,5,44,26,42,6,15,43,8,17,9,36,45,20,40,22,23,47],hass:[],search:[12,45],another_n:22,divisor:[36,1,42],bremner:26,indetermin:47,cross_polytop:[1,4,26,27,41,42,36,33,34],amount:[4,9],doctest:[35,5,26,42,15,17,9,45,34],pick:[6,46,36,35,14],action:19,quotient:[0,5,22],via:[35,1,3,26,42,6,10,20,34],shorthand:[33,26],hrep:[13,38,26,41,15,28,21,31],render_vertices_3d:34,primit:[0,1,37,35,26,5,15,29,17,9,10,36,47,40,11,34,42],is_pointcollect:45,deprec:[4,5,34,42],vlabel:42,coercion:[10,26],select:[35,42,26,14,17,36,34],wirefram:[34,26],v_indic:33,distinct:[0,5,26,41,36,34],is_integr:27,hilbert_basi:5,two:[1,3,46,6,36,42,14,18,19,22,26,45,29,4,34,5,35,30,37,39,41,43,47],coverag:35,poly_2d:0,almost:4,toric_plott:[36,5,40,22],formul:1,taken:[35,46,4],vrep2hrep:7,more:[35,1,42,3,4,26,5,6,40,41,17,29,30,14,36,47,46,22,11,43,23],rhombicuboctahedron:4,diamond:[36,1,42],desir:[4,5,40,39],birat:1,henc:[6,36,4,26,44],an_el:[30,35,47,32],moduleel:32,show_vertic:42,stick:6,pfetsch:26,particular:[35,5,42,39,6,40,41,10,17,36,18,33,43],known:[36,4,26,40],hyperplanearrangements_with_categori:10,nomal:5,cach:[5,42,14,15,10,36,45],cprfanotoricvariety_field:40,none:[0,1,2,3,4,5,8,42,13,15,16,17,20,21,22,23,25,26,27,30,31,33,34,35,36,37,41,40,43,44,45,46,48],endpoint:[37,3,26],hour:36,h11:8,plot_lattic:40,outlin:34,det:[36,5,47,18],remain:[1,37,4,26,42,14,36],del:[44,1],learn:10,incidence_matrix:26,def:[26,41],dual:[5,42,4,26,41,14,17,36,45,22],elementary_divisor:[5,18],icosahedron:[4,34,26],adjust_opt:40,challeng:5,share:[5,16,45],gawrilow:16,sphere:[40,11,34,26],minimum:44,lattice_dim:[36,5],set_legend_opt:9,is_linear:47,add_inequ:14,huge:4,projectiv:35,newton_polytop:44,divid:1,rather:[46,11,3,26,37],anoth:[0,5,37,19,26,41,6,7,10,36,22,23,42],direct_sum:22,is_lin:27,latticepolygon_ppl_class:[8,29],zero_matrix:1,acta:5,simpl:[5,26,46,16,17,36],gale_transform:[36,35,23,26],sandwich:47,point_color:[22,42,40],algebra:[14,1,4,42,46],variant:5,reflect:19,plane:[0,1,37,3,4,26,5,6,41,42,9,36,14,19,24,46,11,34],associ:[1,26,5,42,36,22,23],"short":[42,3,9],vector_integer_dens:[46,27],sch_proj:26,ideal_endpoint:37,caus:[36,42],zerodivisionerror:26,hyperbolicmodeluhp:[24,3],hodge_numb:42,spheric:40,matrixspac:14,dual_lattic:[5,42],multivari:[36,23],is_fan:36,gkz_phi:17,help:42,hyperplane_legend:9,soon:16,paper:[6,40,43,42],through:[1,2,26,6,17,9,4,37,47,34],always_use_fil:42,all_:42,reducedgroebnerbasi:23,free_modul:22,itp:42,paramet:[1,4,26,5,40,42,9,36,22,23],edge_color:[42,34],style:34,boum:41,is_eulerian:26,inward:[5,23,17],indegre:4,subpolygons_of_polar_p2:29,relev:40,html:23,kleindisk:24,stephen:17,might:[46,26,42],finer:47,unique_represent:[35,24,3,30,32,44,47],good:[36,5,23,35,47],"return":[0,1,2,3,4,5,6,8,9,10,11,42,14,15,16,17,18,19,20,22,23,24,25,26,27,29,30,32,33,34,35,36,37,39,40,41,43,44,45,46,47,48],normal_form_pc:42,show3d:42,z_2:5,unneccessarili:35,framework:[1,45,16,36,22,5],sublattice_compl:5,maxdet:41,uhp:[19,24,11,3,37],cone_contain:36,fcylind:34,equilater:[4,46],bounded:26,bigger:[36,5,26,40],super_categori:24,latticepolytope_ppl_class:[0,29],level:[5,43,40],robbin:4,hyperbolicgeodesicuhp:37,easili:[36,5,23,42,22],achiev:5,radian:37,minmial:5,varchenko_matrix:47,category_realization_of_par:24,found:[6,15,39],truncat:[4,40,26],valuat:44,darmstadt:16,bistellar_flip:35,weight:23,monoton:6,idea:[36,45,26],realli:[26,6,36,22,23,47],connect:[35,5,3,26,17,20,47],hyperplanearrangementel:47,beyond:40,orient:[37,3,26,19,36,47],admiss:35,compute_vertic:42,robert:42,polyhedron_qq_cdd:[30,21],convex_hul:[35,42,16,26],print:[0,1,5,6,7,8,42,15,22,23,25,26,45,4,33,34,35,36,41,40,43,46],involutori:11,occurr:[6,45,41,27],unori:46,file_nam:42,benjamin:42,fiber:[0,1,41],advanc:[36,1,35],gf2:23,minkowsi:46,translation_length:19,polyhedra:[],guess:[6,5,36],fan_isomorphic_necessary_condit:18,reason:[36,22,45,10,26],base:[],ask:[36,5,37,35],basi:[0,5,2,4,42,14,7,10,36,45,22,23],launch:26,square_from_vertic:46,omit:[36,1,45,42,15],print_numb:42,american:[1,47,42],phi_b:1,circuit:[35,20],ehrhart_polynomi:[4,41],phi_i:1,pivotedinequ:7,base_extend:[30,22,26],distance_f:35,g_shi:48,major:47,obviou:36,upper:[1,37,3,24,19,11,34],feel:36,famou:[],schenck:1,misc:[34,3],number:[0,1,3,5,6,8,9,42,14,15,17,20,23,26,27,29,4,32,33,34,36,41,40,19,44,45,46,47],cfm:26,done:[36,5,23,45,42],hyperplanearrangementlibrari:48,nend:25,miss:[36,5,16],model_name_nam:3,matric:[37,14,19,4,18,47],is_bound:[0,3],differ:[0,1,42,35,4,26,5,6,15,27,10,9,29,36,18,47,40,22,11,23],hyperbolic_isometri:[37,19],points1:42,points2:42,unrestrict:0,ideal_to_gfan_format:23,"097442cm":34,expand:[40,16],accept:[36,35],reflexivepolytop:42,scheme:[1,40],store:[6,42],assign:4,is_zero:5,relationship:17,behind:45,groebner:[],eqn:[46,13,25,26,27,15,41,21,31],park:[4,47],vertex_gener:[46,26,27,4,33,34],part:[35,1,2,42,5,40,7,10,36,45,22,27],uniquerepresent:[35,24,3,30,32,44,47],sub_reflexive_polygon:29,three_d:23,consult:41,sold:4,gmail:23,produit:44,kind:[35,47,26,39],contrari:44,doubli:[47,19],cyclic:[36,18,34,4,29],gotten:0,remov:[0,46,26,42,35,36,20,47],greg:[19,24,11,3,37],keep_project:42,arrang:[],polyhedron_bas:[35,5,13,38,26,41,15,28,21,31],grei:40,randomli:42,"4ti2":41,point_test:3,gale:[36,35,26],math_mod:40,affine_hul:26,"null":26,option:[0,1,5,8,9,11,42,15,16,10,22,23,26,27,4,33,34,35,36,37,41,40,44,45,46],lie:[35,15,11,19,4],built:[4,42,8],equival:[0,1,26,46,6,42,36,47,5],randint:6,self:[0,1,2,3,5,10,36,11,42,14,17,20,22,23,24,26,27,29,4,34,35,30,37,41,40,19,44,45,47],npart:42,click:34,also:[0,1,3,5,6,11,42,15,17,20,22,23,26,4,32,33,35,36,37,40,19,47],build:[6,36],filled_poli:34,is_groebner_basi:23,analogu:4,point_in_model:3,distribut:[4,42,3,41],contained_simplex:35,goldman:24,previou:[6,1,40,36,5],reach:[4,42],chart:1,font_siz:40,most:[0,1,3,5,6,11,42,14,15,10,18,20,22,23,26,45,29,4,34,35,36,37,41,40,19,47],plai:[1,42],rho:5,alpha:40,projection_func_ident:34,rhombic:4,level_set:[36,5,26,43],charg:26,tpo:35,pointconfiguration_bas:[35,20],clear:[22,10,45],cover:[1,19,5],pushing_triangul:35,roughli:35,ext:[],exp:[37,19],render_1d:34,n_line:26,cls11:1,r_col:14,relative_star_gener:1,cdd:[],render_outline_2d:34,carefulli:5,vaden:24,nlinear:25,polyhedronrepresent:27,miller:47,font:[40,9],sigma1:5,find:[],sigma2:5,penalti:[45,43],cell:[4,16],cone3d:5,coerc:[47,26],indexerror:42,p_newstar:35,pretti:42,buckybal:[4,34,26],ieq:[0,46,13,25,26,27,15,41,10,21,31,33,34],solut:46,polyhedron_qq_ppl:[30,31],factor:[36,1,47,44,10],iec:15,to_linear_program:26,"3_21_polytop":4,hit:[36,5],unus:14,express:[],poit:42,ambient_dim:[35,46,4,26,27,41,42,30,20,21,33,23],nativ:35,longest:15,finit:[],cgi:41,barthelemi:46,ucdavi:[35,41],render_fill_2d:34,arxiv:[1,26,42,41,48,4],dimension_of_homogeneity_spac:23,common:[35,46,37,26,6,16,17,10,36,23],polar_p1xp1_polytop:[0,29],wrote:23,coarsest:1,set:[0,5,3,46,6,9,11,42,14,15,17,18,24,26,45,4,34,35,36,43,40,19,44,47,48],dump:[22,23,42],printout:23,decompos:41,calabi:1,see:[0,1,3,4,5,6,7,9,42,14,15,17,18,20,22,23,26,29,30,33,34,35,36,41,40,43,46,47,48],sec:4,pfi0:23,arg:[0,5,4,44,45,6,36,22,23,34],graphics_opt:[11,3,37],outward:[36,15,23,17],analog:35,cube1:34,cube4:34,fand:35,someth:[5,40,10,36,47,34],fano:[1,40],deloera:35,smallest:[35,1,26,42,36,46],gk13:42,subscript:40,experi:24,altern:[36,5,47,45,40],ngenerating_con:[36,1],triangulation_render_2d:17,positive_xi:5,mixedintegerlinearprogram:26,numer:[46,5,23,42,12],complement:[5,2,26,47],six_hundred_cel:4,frozenset:[0,5,45,15,17,36,18],distinguish:[35,46,26,1,40,43,17,36,33,5],vectorspac:2,install_packag:[35,8],both:[35,46,37,26,1,15,43,42,36,45,22,5],toward:15,last:[0,1,3,5,6,11,42,14,15,10,18,20,22,23,26,45,29,4,34,35,36,37,41,40,19,44,47],opac:[34,26,9],groton:24,facet_const:42,alon:26,coordinate_r:22,tuwien:42,freemodule_ambient_pid:22,fan1:[36,18],fan0:18,fan3:36,fan2:[36,18],fan4:36,unwind:15,pdf:6,whole:[35,1,26,27,41,42,36,33,34,5],polyhedra_qq_cdd:30,load:[22,23,42],regular_polygon:[4,34,26,41],simpli:[5,3],point:[],instanti:[14,11,19,20,27],schlegel_edge_indic:26,mobius_transform:[37,19],linearexpressionmodul:[10,32],generator_zord:40,suppli:[23,16],averag:26,mistak:[36,42,45],zeta:1,completegraph:48,mathbb:[22,42],winfri:5,initial_point:[35,26],becom:[1,45],first_vertex:27,empti:[0,5,2,25,4,26,27,15,29,7,17,10,30,18,36,47,37,41,43,42],p600:4,numerical_approx:[4,37,3,26],isometry_in_model:3,whitnei:[35,47],imag:[1,37,3,5,19,42,36,44,11,34],convert:[5,37,3,4,26,42,7,10,36,19,47,22,11,34,23],append:42,gap:[5,26],coordin:[0,1,3,5,6,7,9,10,11,12,14,15,17,20,22,23,24,26,27,36,32,34,35,4,37,40,48,42,44,45,46,47],understand:[1,5],func:36,demand:14,thath:35,repetit:[36,42,44],poincare_polynomi:47,gmp:26,look:[6,1,36,42,5],packag:[35,5,26,41,16,42,23,47],solid:[4,5,34,26],straight:6,bill:24,histor:26,"while":[35,1,13,3,26,5,6,40,7,42,43,36,45,24,46,22,23],unifi:45,n_region:[47,48],abov:[0,1,37,26,46,6,40,41,42,36,45,22,5],shift:[35,23,17,26],loos:34,"0010082v2":1,pack:[36,5],codomain_fan:1,hilb:5,common_perpendicular:37,vol:47,c_polyhedron:[0,5,15,39],placing_triangul:35,echelon:[0,18,7,22],is_asymptotically_parallel:37,itself:[5,37,3,26,42,6,36,33,22,23,34],virtu:14,point_label:0,vanish:[36,17],fixed_geodes:19,hyperbolicisometrypd:19,open:[5,3,42,15,17,34],minim:[35,1,3,26,46,15,7,23,5],cdd_hrepresent:[25,26],inequality_int:15,quadrant_con:1,belong:[4,37,42,19,36,20,11],hyperplane_arrang:[2,26,48,10,9,47],ldot:4,hrepresentation_spac:[30,26],n_simplex:4,continued_fraction_list:5,higher:[35,4,26,42,6,7,29,36,34],transitivetourna:4,optim:[36,5,22,42,15],vrep:[13,38,26,41,28,21,31],consumpt:45,moment:42,user:[22,2,3,40,24],koch:5,chang:[5,26,42,40,16,10,36,45,32,22,47],focal:[34,26],fanmorph:[1,18],lower:[0,1,26,5,40,42,36],symmetry_involut:11,lib:[0,15],lines_list:26,entri:[0,5,42,3,26,46,15,17,36,20,40,35,33,47,32],u_2:40,u_1:40,u_0:40,expens:5,numarr:6,bigraphicalarrang:48,latticepolytopeerror:39,double_descript:[14,7],explan:[40,47],construct:[0,1,3,5,7,10,36,11,42,17,22,27,4,32,35,30,37,43,40,19,44,45,46,47,48],h21:8,set_immut:[0,15,26,22],shape:[40,23,39],combinat:36,inequality_gener:[46,27,26,15],cut:[46,26,40],liu:1,pts1:[0,26],polyhedron_qq:[31,28,21],pts2:[0,26],theoret:[26,39],rgb:40,nef_partit:42,input:[0,1,2,3,4,5,6,7,8,9,10,11,42,13,14,15,16,17,18,20,21,22,23,26,27,29,30,31,32,33,34,35,36,37,41,40,19,43,44,45,46,47,48],subsequ:[40,32],part_of_point:42,bim:41,ns_sat:22,stein:[42,23,16],format:[],big:[36,42,43],subfield:46,simplex_point:15,inequ:[0,5,13,35,26,46,14,15,41,7,10,33,23,42,27],catalan:[47,48],"658184cm":34,bounded_region:47,bit:[5,6,15,10,36,14,20,40,22,47],characterist:[23,10,47],latticepolytope_ppl:[0,29,39],is_bundl:1,formal:36,truncated_cuboctahedron:4,another_simplex:26,skark:[42,41],rgbcolor:[23,26],four_cross:4,collect:[],princip:[0,5,26,42,8,17,30,18,32],is_inject:1,sub_polytop:29,tropicalprevarieti:23,n_rai:26,often:46,palp:[],creation:[5,26,42,14,30,36,22],some:[46,37,4,26,44,1,15,42,30,36,32,24,40,22,11,5,23],back:[42,26,6,40,7,17,22,34],zz1:23,is_inequ:27,ncol:[14,9],mirror:[36,42],polyhedron_ppl:31,surpris:4,scale:[22,34,16,17,40],irrational_prim:41,restricted_automorphism_group:[0,35,26],plot_ray_label:40,per:[45,26],minkowski_decomposit:41,mathemat:[1,4,42,16,36,47,5],larg:[35,15,8,26,42],proj:[0,34,26],leftmost:36,prod:[1,15],sublattice_quoti:5,machin:15,cone2:[36,5],cone3:[36,5],cone1:[36,5],perpendicular:[42,23,37],last_slop:44,surject:1,step:5,weil:42,example2:23,delta_i:42,automorph:[0,35,26,42],bdry_opt:3,subtract:26,hilbert:5,shing:1,max_abs_coordin:15,transpos:[42,47],prove:41,identity_matrix:[1,3,5,15,19,42,18,22,11],plot_:40,inhomogen:41,block:42,compactli:36,emphasi:[36,5],digraph:[4,26],within:[16,26],integral:41,npoint:42,sub_polytope_gener:0,ensur:[35,5,26,43,40,36],pentagon:4,trunc_quadr:46,inclus:[0,1,47,36,5],institut:16,riegler:42,element_class:[8,10,41],combinator:[14,47,26],fast:[],vertices_satur:0,brun:5,cube_deform:36,hyperbolicisometryuhp:19,arithmet:[46,26,22],includ:[1,42,3,12,40,41,27,29,9,36,45,44,23,5,47],image_con:1,forward:[6,41],is_parallel:37,repeatedli:15,edge_label:36,sqrt3:[46,13],sqrt2:[14,4],simplic:[35,26,17,20],vertex_facet_pairing_matrix:42,decomposit:[35,42,20],interiror:3,link:45,translat:[2,26,41,6,19,34],return_satur:15,atom:[],linestyl:37,ns1:22,line:[0,1,5,6,7,9,10,42,13,15,16,17,21,23,25,26,27,30,31,33,34,36,37,40,44,45,46,47],info:23,consist:[1,5,9,10,36,11,42,14,17,20,23,26,29,4,34,35,30,37,41,40,48,47],dual_modul:45,p_star:35,hrep_gener:26,similar:[1,26,46],curv:[6,1,23],example_id:23,constant:[46,42,3,27,14,7,10,17,32,47],armstrong:48,polyhedralcon:23,chan:4,dualiti:[42,26],repres:[0,5,2,3,47,37,46,6,19,10,36,33,11,34,23],polyhedra_zz_ppl:30,sort_slop:44,semiord:[47,48,9],find_transl:41,fullplan:5,rainbow:[40,42],parabol:19,polygon:[],nedergaard:23,titl:43,invalid:[36,40],positive_point:22,cyclic_sort_vertices_2d:34,cyclic_polytop:[4,26],nice:[40,42],assertionerror:14,draw:[42,26,40,11,34,23],as_sum_of_permut:4,create_object:22,polynomialr:[36,23,16],william:[42,23,16],meaning:27,eval:[15,26,27],lent:24,elementmethod:24,exclude_sandwich:47,rgbtupl:42,star_cent:35,algorithm:[],reiner:47,visual:[47,3,16],notion:[46,40,22],discrimin:35,"_cone":23,translated_polyhedron:41,far:[14,42,34,3,26],hyperbolicisometri:19,hello:[26,9],oop:36,z1z1:23,sector:40,hampton:[35,46,23,43],code:[0,5,26,24,40,43,42,36,22,23,47],partial:[35,26,43],edg:[46,4,26,42,15,48,17,36,40,23,34],scratch:36,zmin:40,preserves_orient:[11,3,19],restrict_to_star_triangul:35,ellipt:19,p_list:23,edu:[35,41],is_compact:[47,26],upperhalfplan:24,toric:[],compact:[0,46,26,41],cython:[],groebnerfan:23,iii:47,send:1,birkhoff_polytop:4,p_all:35,base_rai:0,sens:[46,1,14,40,43,42,5],subfan:1,datafil:16,equations_list:26,show_vindic:42,uniquefactori:22,fan_2d_echelon_form:18,hasse_diagram_from_incid:[46,26,43],volum:[35,1,26,42,14,17,4,23],implicitli:25,kitti:26,ppl_lattice_polytop:[0,29,39],orthant:[5,23],span:[0,5,42,4,26,27,15,41,35,36,45,20,22,47],is_univers:[30,26],michael:16,pair_class:14,"try":[5,26,42,6,40,43,36,23],interval_in_qq2:36,traverse_boundari:42,amantzaf:41,pleas:[5,42,6,36,22,23],impli:[1,41],smaller:[0,5,26,27,6,40,42],kapranov:35,one_fac:5,natur:[40,45],stereograph:[34,26],tikz_pic:34,n_ambient_hrepresent:33,blanklin:[18,34],polyhedron_par:27,equation_gener:[26,27],setofalllatticepolytopesclass:42,make_par:48,victor:42,compat:[],index:[35,1,26,12,6,15,41,27,42,36,45,20,33,47,34,5],compar:[7,18,3,44],polyfil:23,"_compute_fac":42,han:4,affin:[],access:[],hyperbolicisometrykm:19,vrep_gener:26,perimet:29,bergeron:46,hyperbolicmodelpd:[24,3],first_coordinate_plan:14,len:[0,1,42,35,4,26,46,15,41,8,29,30,18,36,20,40,23,5,47],closur:[5,17],let:[0,1,42,26,5,6,40,41,35,43,36,22,47],sink:[4,26],lex:23,pointcollect:[5,45,8,42,36,18],vertic:[0,5,48,4,46,6,7,8,10,42,13,15,16,17,20,21,23,25,26,27,29,30,33,34,35,36,39,40,41,44,45,31,47],sinc:[0,1,3,5,45,27,16,42,36,19,22],show_lattic:40,housegraph:48,survei:42,convers:[0,5,26,27,6,7,22],birkhoff:[4,16],dilat:[36,26,41],larger:[30,23,26],reus:[30,26],repel:[3,19],skeleton_show:42,implement:[46,3,7,10,36,11,42,13,14,15,17,20,23,24,26,29,4,32,35,30,37,43,40,19,44,47],hexagon:4,configur:[],show_facet:42,face_lattic:[5,43,26,33],appli:[42,9],approxim:[36,4,40],yuen:4,submodul:[36,5,22],nabla_i:42,"boolean":[0,1,3,46,9,42,14,15,10,18,20,26,27,29,4,34,5,35,36,41,40,44,47],image2:34,piecewis:[35,17,26],project_point:4,from:[0,1,2,3,4,5,6,7,8,9,10,42,14,15,17,18,19,20,21,22,23,24,25,26,27,29,30,32,33,34,35,36,37,39,40,41,44,45,46,47,48],zip:6,commun:41,syntact:[47,32],toriclattic:[1,42,15,36,22,5],doubl:[],flat:35,next:[0,46,42,44,26,27,15,8,35,4,18,20,33],connectedtriangulationsiter:20,few:[35,42],reduce_dimens:42,preserve_orient:3,ineq:27,commut:4,sort:[1,3,43,6,36,44,47],batyrev:[36,42],rdf:[],pfi:23,isometry_group:3,is_toriclatticequoti:22,about:[],skip_palp_matrix:42,trail:22,ray_basi:36,iif:6,bisector:37,inequality_collect:15,quadrangl:26,is_complet:[36,1,37],sage_trac:5,trivialfan3:1,trivialfan2:1,account:36,retriev:42,alia:[35,24,3,44,26,27,14,10,30,32,22,47],xfig:23,testsuit:[13,38,26,41,15,28,30,31,22],erwin:42,thin:[0,34],meet:[6,15,43],control:23,list_of_poli:23,image3:34,felsner_matrix:6,process:[36,42,25],"329367cm":34,base_qq:[31,28,21],high:[36,5],xmin:40,acm:26,technisch:16,projection_point:34,np1:44,swap_ineq_to_front:15,dualiz:41,surfac:1,sit:[36,1,47,5],ambient_modul:[22,32],line1:27,plot_rai:40,instead:[1,4,26,46,6,40,41,42,43,30,36,22,34,5],np2:44,sucess:35,cartier:42,is_domin:1,frac:5,overridden:40,singular:36,perm4:4,notebook:23,redund:[46,41],to_model:[11,19,37],debruijn:26,birkhaus:35,essenti:[36,35,23,47,9],halfspac:[5,27],correspond:[0,1,37,3,4,26,44,5,6,27,17,43,36,20,47,46,22,23,34,42,35],element:[],issu:[23,26],region2:47,region1:47,cut_frac:26,allow:[1,13,3,26,46,45,15,7,8,10,29,44,19,24,22,34],render_points_2d:34,vertex_color:[42,34],fastest:42,least:[1,42,43,26,5],discard_fac:36,top:[5,26,43,40,36,44,23],equinumer:4,prefix:23,c5_20_fl:26,chien:1,hyperplane_label:9,chosen:[35,5,26,46,6,40,36,22,34],plot_hyperplan:9,reduced_project:20,zaslavski:47,therefor:[46,1,40,36,5],random_inequ:14,p4mirror:15,bim_bam_boum:41,circuits_support:35,greater:[36,42,44],"__getitem__":20,nonneg:[35,5,4,26,22],python:[],auto:[6,35,26],overal:[35,10],chaincomplex:36,devel:23,mention:[44,26,43],p12:26,autp:26,front:[15,34],hyperplane_opac:9,another_nam:22,trac:[0,1,4,26,5,42,36,34],anyth:[35,5,42,26,27,40,10,36,45,22,47],beginnig:15,plot_label:40,mode:[40,23,45],aspect_ratio:[35,17,9],bubblesortgraph:4,subset:[0,1,4,26,43,6,35,36,45,47],vector_rational_dens:20,subseq:42,hly:1,domain_con:1,int_to_simplex:20,strip:46,line_set:5,is_bir:1,"static":45,zero_map:39,fine:[35,26,17,20],our:[1,3,42,43,36,46,22],special:[1,13,26,7,29,36],out:[36,1,42,35,46],variabl:[0,26,15,48,10,36,32,23,47],h_4_2:4,matrix:[0,1,2,3,5,6,7,10,42,14,15,17,18,20,22,26,27,4,35,36,37,39,19,45,47,48],quadrilater:4,vrepresent:[35,46,26,27,15,8,10,30,33,34],rep:26,secondary_polytop:35,categori:[24,3,19],polytopes_db_4d:8,suitabl:[0,5,42,26,46,40,17,30,20],rel:[1,26,5,42,36,33,23,47],lattic:[],coordinate_vector:22,ref:8,affine_subspac:2,red:[37,3,26,9,22,11],statist:42,span_of_basi:22,random_el:[3,6,36,18,32,11],volker:[0,5,26,46,40,35,36,20,47],insid:[36,5,23,42,15],polynomial_system:23,katsura:23,index_shift:42,hyperbolicplan:[19,24,11,3,37],distance_between_region:47,dictionari:[0,42,26,40,48,23,34],preimage_fan:1,achil:26,dualli:7,afterward:6,keep_product:42,bam:41,transposit:[6,35,4],expected_charpoli:47,could:42,qqbar:4,put:40,edge_trunc:26,david:[1,47,26,42],length:[5,37,26,27,6,40,19,17,9,10,4,44,42],outsid:[4,34,3,26],petersengraph:47,south:1,softwar:[23,16],isometry_from_fixed_point:[3,19],finest:1,rand01:16,already_echelon:22,felser:6,m_perm:47,c5_10_fl:26,initialform:23,refus:25,p2c:36,"long":[0,5,42,26,24,15,48,35,9,4,47,34],rectangular_box_point:15,seq:6,strict:[5,26],finitefield:47,licens:42,ewgenij:16,show_hyperboloid:37,messag:[15,42],are_satisfi:15,attach:44,is_affin:20,"final":[0,5,26,46,6,42,36],zelevinski:35,error:[46,37,19,1,15,39,42,4,11,23],hard:[1,23,42],from_data:16,positive_dual_point:22,"0x7f8fc5367aa0":[],fuzzi:26,kasprzyk:42,poly_egg:26,vertex_adjacency_matrix:[34,26],random_point:3,exactli:[1,5,6,42,36,45,22,34],is_isometry_group_project:3,update_graph:[11,37],structur:[1,2,3,5,7,8,10,11,42,16,17,20,22,23,24,26,27,30,32,33,34,35,36,37,39,40,44,45,47],polyhedron_field:[30,13],rania:24,"function":[],minimal_total_degree_of_a_groebner_basi:23,all_cached_data:42,viewer:[34,26],ultraparallel:37,k_normal:36,clearli:[4,40,26],have:[1,46,6,9,10,42,14,15,16,17,20,22,24,26,45,4,5,36,37,41,40,43,44,47],close:[1,42,5],need:[0,5,42,26,46,6,15,40,16,35,36,45,33,22,23,47],n_inequ:[4,46,26],gosset_3_21:4,cddlib:[46,25,41],render_4d:34,hyperbolic_model:[3,19],facet_opac:42,mix:[5,23,26,22],render_points_1d:34,discret:[44,5,16],which:[1,3,4,5,6,9,10,11,42,15,16,17,20,22,23,24,25,26,29,30,32,34,35,36,37,43,40,19,46,47,48],prepare_next_to_inner_loop:15,polyhedron_zz_ppl:[30,31],singl:[0,1,42,35,26,46,40,17,9,36,32,22,47,5],hyperbolicpointuhp:[11,3],textur:34,unless:[5,4,27,40,42,36,46,33],c2_z2:5,who:23,ppl_lattice_polygon:[8,29],hyperbolicgeodesichm:37,eight:26,amongst:27,segment:[42,37],"class":[],homogen:[35,14,7,36,20,23],dens:[14,1],rationalpolyhedralfan:[36,23,17,43],gather:[6,4],request:[36,1,40,42,5],face:[],inde:[36,42],univari:41,determin:[0,5,37,3,4,26,41,40,19,35,9,36,45,33,22,23,42],farlei:17,ppoint:34,irrat:[26,41],fact:[5,3,26,39,15,36,22],relative_quoti:5,platon:4,inessenti:[47,9],text:45,verbos:[35,1,46,41,36,21,23],spanning_point:15,is_boundari:[11,3],longer:[0,5,36,26],trivial:[1,40,26,5],anywai:42,homomorph:22,dp7:18,dp6:[36,1,40],outdegre:4,locat:8,nois:6,groebner_con:23,recent_work:35,fan_isomorphism_gener:18,anticip:40,dp8:[36,1,18,5],initial_pair:14,suppos:45,bindable_class:3,flexibl:16,cube:[4,42,26,41,15,16,17,10,36,47,34],latticepolytopesnotisomorphicerror:[29,39],subpolygons_of_polar_p2_112:29,disagre:36,joint:[23,43],theori:[42,26],weight_vector:23,p1xp1:[36,1,29],increas:[36,5,42,26,40],zero_sum_project:4,triangl:[35,46,26,27,42,4,33,23,47],dimnsion:0,enabl:[15,3,42],organ:42,typeset:22,bisect:37,fetter2012:4,label_offset:9,dual_nam:22,arrangeh:47,trunc_cub:26,integr:[],partit:[35,42,47],contain:[0,1,3,4,5,7,8,10,11,42,14,15,17,18,19,20,22,23,25,26,27,29,30,33,34,35,36,37,39,40,41,43,44,45,46,47],shi:[47,48],positive_integer_rel:42,coatom_to_atom:43,latticepolytopenoembeddingerror:[0,39],view:[1,34,42],end1:37,conform:3,end2:37,modulo:[0,22,23,47],farthest:35,charpoli:47,frame:[42,9],accord:[6,34,27,17,26],flip:[35,26],p24:[4,26],multipart:26,n_hyperplan:47,hyperbolicgeodes:37,outpu:45,rambau:35,appreci:23,rpf:23,impos:35,wire:[6,34],correctli:[1,34,42,40,5],pattern:6,boundari:[0,1,37,3,42,46,27,17,36,19,11],tend:35,state:[30,40,34,26,42],quickest:42,number_of_row:42,ray_gener:[46,33,26,27],progress:[42,23],simple_polytop:26,induc:[36,1,35,26,42],neither:[30,5,26,22],p2a:36,kei:[0,42,3,43,6,4,22,23],p2b:36,itertool:41,moebiu:47,line_gener:[33,26,27],entir:[30,5,19,26],zonotop:47,subgroup:[0,5,35,26],exclam:42,amer:[24,42],ker:42,addit:[1,26,5,41,8,42,43,30,45,36,23,47],"6sec":4,show_gener:40,freemodule_generic_pid:22,nabla:42,equal:[0,1,37,35,4,26,44,46,15,27,16,17,9,36,22,23,5,47],starshap:35,etc:[35,42,44,45],instanc:[0,35,26,6,48,10,14,21,47],prodon:14,equat:[0,46,42,35,26,27,6,15,7,17,9,10,33,47],cartesianproduct:47,freeli:42,rohombicuboctahedron:4,comment:[42,23],bn08:42,wall:[40,23],pentakis_dodecahedron:4,complementari:[5,46],indici:42,walk:23,free_module_morph:1,universitat:16,respect:[0,1,42,35,26,5,10,9,20,22,23,47],linear_transform:26,torsion:[36,5,22],quit:[36,40,26],lattice_polytop:[36,1,42,26,5],yau:1,tropical_basi:23,relative_interior_point:23,compon:[35,5,47,42,24],make_simplici:36,besid:9,treat:[42,7],immedi:[6,46,26],all_fac:42,barycent:26,doubledescriptionpair:14,coincid:[22,41],nonregular:35,poincaredisk:24,assert:[5,4,14,36,18,47],togeth:[36,5,40,10,26],x_0:15,x_1:15,hyperplane_color:9,present:[46,26,24,6,15,43,42,40,33],relative_interior_contain:[5,26],multi:4,irregular:[35,17],a_real:24,align:45,rectangular:[0,15,26,40],unsuit:40,defin:[0,1,4,5,6,7,8,42,13,14,15,10,18,20,21,23,24,26,27,28,30,33,34,35,36,38,41,40,44,46,47],mixed_volum:23,sebastien:46,ultra:37,inner_product_matrix:14,bucki:4,helper:[],srang:[34,26],irreduc:5,motiv:47,avi:26,substanti:[36,5],scienc:14,uniti:41,proj_matrix:0,welcom:5,cross:[6,4,26,42],sqrt:[5,13,3,4,26,46,14,15,37,36,24,42],member:42,handl:[35,46,1,22,34,5],toricvarieti:[36,8],largest:[36,1,23,35,5],is_prop:5,start:[35,5,37,3,26,42,6,15,19,10,43,45,22,23,47],loera:35,phi:[1,17],ball:[4,42],http:[35,5,26,41,6,42,23,34],again:[40,26,41],expans:5,is_simplici:[36,5,23,26],v_list:34,upon:46,effect:26,octagon:4,phy:[1,41],decoupl:24,cyclicpermutationgroup:18,one_con:5,off:[35,15,26,40],center:[35,40,34,26,9],reduced_affine_vector_spac:20,nevertheless:42,well:[1,42,46,6,15,29,45,24,40,22,5],thought:[6,26],octahedron:[36,33,4,26,42],exampl:[],command:[35,5,26,41,16,8,42,9,36,40,23],integral_point:[0,26,41,15,8,4],undefin:19,usual:[0,5,42,44,26,46,40,8,35,9,36,22],distanc:[35,24,37,3,26,42,47],less:[0,46,4,42,40,29,35,10,36],obtain:[35,4,42,26,27,6,40,17,36,22,47,34],catgori:35,standarddoubledescriptionpair:14,rekha:23,adv:1,novoseltsev:[46,1,40,43,42,36,45,22,5],rapid:24,initial_form_system:23,box_min:[0,15,26],latex_nam:22,halfplan:[5,15],smith:[15,26],add:[5,42,26,46,14,40,41,10,36,32,22,23],valid:[5,3,46,14,36,22,11],cube_proj:34,result_nam:42,c12:5,semigroup:[30,5],print_cach:15,adj:4,match:[10,26],reflexive4dhodg:8,perkinson:[47,48],wall_color:40,face_constructor:43,max_degre:23,ridden:27,realiz:[24,42],v_gen:26,know:[36,1,33,42,5],facial:42,stanley_reisner_id:[36,23],press:[5,23,26],pitsta:4,desc:[42,16],n_cube:4,insert:[1,45,15],like:[35,5,3,42,40,19,36,45,22,47,34],all_polar:42,a_simplex:26,fubini:35,point_siz:[40,9,42],show_rai:40,necessari:[35,46,26,1,15,42,36,18,40,47,34,5],page:[35,12,42,6,17,14],cone_lattic:36,therein:42,extra_opt:40,exceed:15,new_stat:42,poly_4d:0,shea:16,"482887cm":34,interact:[35,24,2,23],smoothli:36,proper:[1,42,5],guarante:[36,5],is_full_dimension:[0,26],drop:[42,26],matrix2:4,echelonized_basi:22,librari:[],great_rhombicuboctahedron:4,hypersurfac:1,is_solid:5,intersection_poset:47,lead:[36,1,42,24,22],gfan_gen:23,avoid:[6,5,42,14],freemodulemorph:1,overlap:40,jeremi:24,summand:[36,44,41,26,27],a_matrix:14,nonempti:[5,47],duplic:36,is_reflex:[42,41],codomain_dim:39,to_rationalpolyhedralfan:23,reduced_affin:20,prefix_check:23,triangulation_render_3d:17,usag:40,nathann:6,stellar:36,virtual_rai:36,simplex:[0,35,26,42,15,17,4,20,23,34,47],although:[36,46,3,43],offset:9,reset_opt:40,multidimension:35,diagonal_matrix:[37,3,19],simpler:22,ascend:15,actual:[35,5,42,15,40,36,18,45,20,33,22,47],echelon_form:18,column:[35,5,26,42,6,15,7,17,14,45,20,47],mapsto:44,ander:23,constructor:[35,1,19,4,26,46,40,7,17,30,18,36,20,33],discard:[36,5,44],tristram:23,projectionfuncstereograph:34,polar_p2_112_polytop:[0,29],own:[0,5,26,40,48,35,36,22],societi:[1,47,42],primal:[5,41],automat:[46,3,44,1,40,42,36,20,22,5],"_object":[34,26],latexexpr:34,rectifi:1,n_point:20,orkisdodecahedron:4,nabla_polar:42,tikz1:34,pictur:[42,34],assumpt:[36,1],homolog:[36,17],representative_point:26,trigger:14,inner:[14,15,42],"var":[19,26],r_by_sign:14,overflowerror:15,pointconfigur:[35,5,26,17,20],felsner:6,north:1,ray_color:40,unexpect:22,triangular:26,defining_id:8,neutral:30,klyachko:36,overflow:15,highest:23,bug:46,pproj:34,count:[15,41,34,20,27],bdry:36,made:[6,5,36,4],josh:35,tab:[4,34],displac:26,whether:[0,1,3,5,9,11,42,14,15,10,20,22,23,26,27,36,33,35,41,40,44,46,47],wish:40,smooth:[36,5,15],displai:[6,26,9],troubl:16,point_collect:[45,8],record:3,below:[0,1,26,5,6,40,9,36,34],limit:[15,42],semigroup_gener:[5,15],otherwis:[35,1,37,3,4,26,44,5,45,40,41,42,9,36,19,20,22,47,34],problem:[6,1,36,24,14],backend_ppl:[31,46,8,41],c5_10:26,evalu:[15,26,32,27],"int":[15,20],dure:[1,42,40,36,23,5],orbit:[0,1,36,42,5],filenam:[16,23,8,26,42],indirectli:40,show_edg:42,novemb:26,myn:36,permutohedron:4,pir:26,piv:7,joswig:16,oplu:22,mutual:22,set_gener:42,wall_zord:40,maximal_total_degree_of_a_groebner_basi:23,detail:[0,1,42,35,4,26,46,6,40,48,41,17,43,36,47,34,5],virtual:36,other:[1,2,5,6,10,11,42,17,22,23,24,26,36,32,34,35,45,37,41,40,19,43,46,47],bool:[19,37,11,3,23],futur:[36,42],rememb:[6,36,16,42],varieti:[0,1,5,40,16,42,36,45,22,23],find_isomorph:[18,29],theor:1,repeat:[44,40,41],star:[35,1,26,17,20,22],integral_points_not_interior_to_facet:0,jung:5,irrational_all_prim:41,semir:26,test_rend:23,triangulation_max_height:41,cdd_file_format:25,cuboctahedron:[36,4,27,17,26],involut:37,palp_databas:8,data_basenam:8,philipp:24,toriclattice_quoti:22,eof:42,nrow:14,dihedralgroup:35,reliabl:42,soeger:5,probabl:[5,26,43,14,45,22,23],preimag:1,auxiliari:33,perhap:[5,42],half_plan:[5,26,46],invari:[0,1,47,42,22],interior_point:26},objtypes:{"0":"py:module","1":"py:method","2":"py:function","3":"py:class","4":"py:attribute","5":"py:exception","6":"py:staticmethod","7":"py:classmethod"},objnames:{"0":["py","module","Python module"],"1":["py","method","Python method"],"2":["py","function","Python function"],"3":["py","class","Python class"],"4":["py","attribute","Python attribute"],"5":["py","exception","Python exception"],"6":["py","staticmethod","Python static method"],"7":["py","classmethod","Python class method"]},filenames:["sage/geometry/polyhedron/ppl_lattice_polytope","sage/geometry/fan_morphism","sage/geometry/hyperplane_arrangement/affine_subspace","sage/geometry/hyperbolic_space/hyperbolic_model","sage/geometry/polyhedron/library","sage/geometry/cone","sage/geometry/pseudolines","sage/geometry/polyhedron/double_description_inhomogeneous","sage/geometry/polyhedron/palp_database","sage/geometry/hyperplane_arrangement/plot","sage/geometry/hyperplane_arrangement/hyperplane","sage/geometry/hyperbolic_space/hyperbolic_point","index","sage/geometry/polyhedron/backend_field","sage/geometry/polyhedron/double_description","sage/geometry/integral_points","sage/geometry/polytope","sage/geometry/triangulation/element","sage/geometry/fan_isomorphism","sage/geometry/hyperbolic_space/hyperbolic_isometry","sage/geometry/triangulation/base","sage/geometry/polyhedron/backend_cdd","sage/geometry/toric_lattice","sage/rings/polynomial/groebner_fan","sage/geometry/hyperbolic_space/hyperbolic_interface","sage/geometry/polyhedron/cdd_file_format","sage/geometry/polyhedron/base","sage/geometry/polyhedron/representation","sage/geometry/polyhedron/base_QQ","sage/geometry/polyhedron/ppl_lattice_polygon","sage/geometry/polyhedron/parent","sage/geometry/polyhedron/backend_ppl","sage/geometry/linear_expression","sage/geometry/polyhedron/face","sage/geometry/polyhedron/plot","sage/geometry/triangulation/point_configuration","sage/geometry/fan","sage/geometry/hyperbolic_space/hyperbolic_geodesic","sage/geometry/polyhedron/base_RDF","sage/geometry/polyhedron/lattice_euclidean_group_element","sage/geometry/toric_plotter","sage/geometry/polyhedron/base_ZZ","sage/geometry/lattice_polytope","sage/geometry/hasse_diagram","sage/geometry/newton_polygon","sage/geometry/point_collection","sage/geometry/polyhedron/constructor","sage/geometry/hyperplane_arrangement/arrangement","sage/geometry/hyperplane_arrangement/library"],titles:["Fast Lattice Polytopes using PPL.","Morphisms between toric lattices compatible with fans","Affine Subspaces of a Vector Space","Hyperbolic Models","Library of commonly used, famous, or interesting polytopes","Convex rational polyhedral cones","Pseudolines","Double Description for Arbitrary Polyhedra","Access the PALP database(s) of reflexive lattice polytopes","Plotting of Hyperplane Arrangements","Hyperplanes","Hyperbolic Points","Geometry","The Python backend","Double Description Algorithm for Cones","Cython helper methods to compute integral points in polyhedra.","Polytopes","A triangulation","Find isomorphisms between fans.","Hyperbolic Isometries","Base classes for triangulations","The cdd backend for polyhedral computations","Toric lattices","Groebner Fans","Interface to Hyperbolic Models","Generate cdd <tt class=\"docutils literal\"><span class=\"pre\">.ext</span></tt> / <tt class=\"docutils literal\"><span class=\"pre\">.ine</span></tt> file format","Base class for polyhedra","H(yperplane) and V(ertex) representation objects for polyhedra","Base class for polyhedra over <span class=\"math\">\\(\\QQ\\)</span>","Fast Lattice Polygons using PPL.","Parents for Polyhedra","The PPL (Parma Polyhedra Library) backend for polyhedral computations","Linear Expressions","A class to keep information about faces of a polyhedron","Functions for plotting polyhedra","Triangulations of a point configuration","Rational polyhedral fans","Hyperbolic Geodesics","Base class for polyhedra over <tt class=\"docutils literal\"><span class=\"pre\">RDF</span></tt>.","Lattice Euclidean Group Elements","Toric plotter","Base class for polyhedra over <span class=\"math\">\\(\\ZZ\\)</span>","Lattice and reflexive polytopes","Hasse diagrams of finite atomic and coatomic lattices.","Newton Polygons","Point collections","Polyhedra","Hyperplane Arrangements","Library of Hyperplane Arrangements"],objects:{"sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelKM":{point_in_model:[3,1,1,""],isometry_in_model:[3,1,1,""],get_background_graphic:[3,1,1,""],boundary_point_in_model:[3,1,1,""]},"sage.geometry":{linear_expression:[32,0,0,"-"],hasse_diagram:[43,0,0,"-"],polytope:[16,0,0,"-"],fan_isomorphism:[18,0,0,"-"],toric_lattice:[22,0,0,"-"],integral_points:[15,0,0,"-"],pseudolines:[6,0,0,"-"],fan_morphism:[1,0,0,"-"],point_collection:[45,0,0,"-"],fan:[36,0,0,"-"],cone:[5,0,0,"-"],newton_polygon:[44,0,0,"-"],lattice_polytope:[42,0,0,"-"],toric_plotter:[40,0,0,"-"]},"sage.geometry.toric_lattice.ToricLattice_generic":{saturation:[22,1,1,""],span:[22,1,1,""],direct_sum:[22,1,1,""],Element:[22,4,1,""],construction:[22,1,1,""],span_of_basis:[22,1,1,""],intersection:[22,1,1,""],quotient:[22,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelPD":{point_in_model:[3,1,1,""],get_background_graphic:[3,1,1,""],boundary_point_in_model:[3,1,1,""],isometry_in_model:[3,1,1,""]},"sage.geometry.fan.Cone_of_fan":{star_generator_indices:[36,1,1,""],star_generators:[36,1,1,""]},"sage.geometry.polyhedron.double_description.StandardDoubleDescriptionPair":{add_inequality:[14,1,1,""]},"sage.geometry.polyhedron.backend_cdd":{Polyhedron_RDF_cdd:[21,3,1,""],Polyhedron_QQ_cdd:[21,3,1,""],Polyhedron_cdd:[21,3,1,""]},"sage.geometry.triangulation.point_configuration":{PointConfiguration:[35,3,1,""]},"sage.geometry.point_collection.PointCollection":{index:[45,1,1,""],set:[45,1,1,""],matrix:[45,1,1,""],dual_module:[45,1,1,""],basis:[45,1,1,""],output_format:[45,6,1,""],column_matrix:[45,1,1,""],module:[45,1,1,""],dim:[45,1,1,""],cardinality:[45,1,1,""],dimension:[45,1,1,""],cartesian_product:[45,1,1,""]},"sage.geometry.triangulation.base.PointConfiguration_base":{dim:[20,1,1,""],simplex_to_int:[20,1,1,""],base_ring:[20,1,1,""],point:[20,1,1,""],reduced_projective_vector_space:[20,1,1,""],int_to_simplex:[20,1,1,""],reduced_affine_vector_space:[20,1,1,""],n_points:[20,1,1,""],ambient_dim:[20,1,1,""],points:[20,1,1,""],is_affine:[20,1,1,""]},"sage.geometry.hyperplane_arrangement.plot":{plot:[9,2,1,""],plot_hyperplane:[9,2,1,""],legend_3d:[9,2,1,""]},"sage.rings.polynomial.groebner_fan.InitialForm":{rays:[23,1,1,""],cone:[23,1,1,""],internal_ray:[23,1,1,""],initial_forms:[23,1,1,""]},"sage.geometry.polyhedron.plot":{render_2d:[34,2,1,""],Projection:[34,3,1,""],ProjectionFuncSchlegel:[34,3,1,""],render_3d:[34,2,1,""],ProjectionFuncStereographic:[34,3,1,""],projection_func_identity:[34,2,1,""],render_4d:[34,2,1,""],cyclic_sort_vertices_2d:[34,2,1,""]},"sage.geometry.triangulation.element.Triangulation":{plot:[17,1,1,""],point_configuration:[17,1,1,""],gkz_phi:[17,1,1,""],adjacency_graph:[17,1,1,""],simplicial_complex:[17,1,1,""],enumerate_simplices:[17,1,1,""],interior_facets:[17,1,1,""],fan:[17,1,1,""],boundary:[17,1,1,""],normal_cone:[17,1,1,""]},"sage.geometry.hyperplane_arrangement.arrangement.HyperplaneArrangementElement":{restriction:[47,1,1,""],add_hyperplane:[47,1,1,""],unbounded_regions:[47,1,1,""],rank:[47,1,1,""],n_hyperplanes:[47,1,1,""],cone:[47,1,1,""],is_linear:[47,1,1,""],plot:[47,1,1,""],is_central:[47,1,1,""],intersection_poset:[47,1,1,""],union:[47,1,1,""],region_containing_point:[47,1,1,""],hyperplanes:[47,1,1,""],whitney_data:[47,1,1,""],regions:[47,1,1,""],change_ring:[47,1,1,""],deletion:[47,1,1,""],n_bounded_regions:[47,1,1,""],whitney_number:[47,1,1,""],characteristic_polynomial:[47,1,1,""],poincare_polynomial:[47,1,1,""],n_regions:[47,1,1,""],is_separating_hyperplane:[47,1,1,""],is_essential:[47,1,1,""],sign_vector:[47,1,1,""],doubly_indexed_whitney_number:[47,1,1,""],essentialization:[47,1,1,""],face_vector:[47,1,1,""],distance_between_regions:[47,1,1,""],has_good_reduction:[47,1,1,""],vertices:[47,1,1,""],bounded_regions:[47,1,1,""],varchenko_matrix:[47,1,1,""],distance_enumerator:[47,1,1,""],dimension:[47,1,1,""]},"sage.geometry.lattice_polytope":{ReflexivePolytope:[42,2,1,""],all_facet_equations:[42,2,1,""],integral_length:[42,2,1,""],all_cached_data:[42,2,1,""],is_LatticePolytope:[42,2,1,""],is_NefPartition:[42,2,1,""],projective_space:[42,2,1,""],convex_hull:[42,2,1,""],all_faces:[42,2,1,""],NefPartition:[42,3,1,""],all_nef_partitions:[42,2,1,""],LatticePolytope:[42,2,1,""],all_polars:[42,2,1,""],sage_matrix_to_maxima:[42,2,1,""],read_palp_matrix:[42,2,1,""],write_palp_matrix:[42,2,1,""],always_use_files:[42,2,1,""],filter_polytopes:[42,2,1,""],LatticePolytopeClass:[42,3,1,""],cross_polytope:[42,2,1,""],ReflexivePolytopes:[42,2,1,""],read_all_polytopes:[42,2,1,""],set_palp_dimension:[42,2,1,""],all_points:[42,2,1,""],SetOfAllLatticePolytopesClass:[42,3,1,""],positive_integer_relations:[42,2,1,""],skip_palp_matrix:[42,2,1,""],minkowski_sum:[42,2,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic.HyperbolicGeodesic":{intersection:[37,1,1,""],dist:[37,1,1,""],complete:[37,1,1,""],angle:[37,1,1,""],length:[37,1,1,""],update_graphics:[37,1,1,""],end:[37,1,1,""],midpoint:[37,1,1,""],is_ultra_parallel:[37,1,1,""],is_complete:[37,1,1,""],start:[37,1,1,""],is_asymptotically_parallel:[37,1,1,""],common_perpendicula:[37,1,1,""],is_parallel:[37,1,1,""],graphics_options:[37,1,1,""],perpendicular_bisector:[37,1,1,""],model:[37,1,1,""],endpoints:[37,1,1,""],reflection_involution:[37,1,1,""],to_model:[37,1,1,""],ideal_endpoints:[37,1,1,""]},"sage.geometry.cone":{ConvexRationalPolyhedralCone:[5,3,1,""],normalize_rays:[5,2,1,""],is_Cone:[5,2,1,""],Cone:[5,2,1,""],IntegralRayCollection:[5,3,1,""],classify_cone_2d:[5,2,1,""]},"sage.geometry.polyhedron.plot.Projection":{render_2d:[34,1,1,""],render_3d:[34,1,1,""],tikz:[34,1,1,""],render_solid_3d:[34,1,1,""],schlegel:[34,1,1,""],render_fill_2d:[34,1,1,""],render_vertices_3d:[34,1,1,""],render_1d:[34,1,1,""],render_0d:[34,1,1,""],coord_indices_of:[34,1,1,""],show:[34,1,1,""],render_points_1d:[34,1,1,""],render_points_2d:[34,1,1,""],coordinates_of:[34,1,1,""],render_line_1d:[34,1,1,""],render_wireframe_3d:[34,1,1,""],render_outline_2d:[34,1,1,""],coord_index_of:[34,1,1,""],identity:[34,1,1,""],stereographic:[34,1,1,""]},"sage.geometry.triangulation":{point_configuration:[35,0,0,"-"],base:[20,0,0,"-"],element:[17,0,0,"-"]},"sage.geometry.integral_points":{loop_over_parallelotope_points:[15,2,1,""],InequalityCollection:[15,3,1,""],Inequality_int:[15,3,1,""],parallelotope_points:[15,2,1,""],rectangular_box_points:[15,2,1,""],Inequality_generic:[15,3,1,""],ray_matrix_normal_form:[15,2,1,""],print_cache:[15,2,1,""],simplex_points:[15,2,1,""]},"sage.geometry.polyhedron.representation.PolyhedronRepresentation":{count:[27,1,1,""],index:[27,1,1,""],vector:[27,1,1,""],polyhedron:[27,1,1,""]},"sage.geometry.fan_morphism":{FanMorphism:[1,3,1,""]},"sage.geometry.polyhedron.representation.Line":{evaluated_on:[27,1,1,""],is_line:[27,1,1,""],type:[27,1,1,""]},"sage.geometry.toric_lattice.ToricLattice_quotient":{dimension:[22,1,1,""],is_torsion_free:[22,1,1,""],rank:[22,1,1,""],gens:[22,1,1,""],dual:[22,1,1,""],coordinate_vector:[22,1,1,""],Element:[22,4,1,""],base_extend:[22,1,1,""]},"sage.geometry.hyperplane_arrangement.hyperplane":{AmbientVectorSpace:[10,3,1,""],Hyperplane:[10,3,1,""]},"sage.geometry.polyhedron.ppl_lattice_polytope":{LatticePolytope_PPL_class:[0,3,1,""],LatticePolytope_PPL:[0,2,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_point.HyperbolicPointUHP":{symmetry_involution:[11,1,1,""],show:[11,1,1,""]},"sage.geometry.polyhedron.face.PolyhedronFace":{polyhedron:[33,1,1,""],dim:[33,1,1,""],n_ambient_Vrepresentation:[33,1,1,""],rays:[33,1,1,""],ambient_Hrepresentation:[33,1,1,""],vertex_generator:[33,1,1,""],lines:[33,1,1,""],vertices:[33,1,1,""],ambient_dim:[33,1,1,""],ambient_Vrepresentation:[33,1,1,""],n_ambient_Hrepresentation:[33,1,1,""],line_generator:[33,1,1,""],as_polyhedron:[33,1,1,""],ray_generator:[33,1,1,""]},"sage.geometry.polyhedron.representation.Inequality":{interior_contains:[27,1,1,""],contains:[27,1,1,""],type:[27,1,1,""],is_inequality:[27,1,1,""]},"sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ":{polar:[41,1,1,""],fibration_generator:[41,1,1,""],is_lattice_polytope:[41,1,1,""],ehrhart_polynomial:[41,1,1,""],has_IP_property:[41,1,1,""],is_reflexive:[41,1,1,""],Minkowski_decompositions:[41,1,1,""],find_translation:[41,1,1,""]},"sage.geometry.polyhedron.library":{project_points:[4,2,1,""],Polytopes:[4,3,1,""],zero_sum_projection:[4,2,1,""]},"sage.geometry.toric_plotter.ToricPlotter":{include_points:[40,1,1,""],plot_generators:[40,1,1,""],plot_lattice:[40,1,1,""],set_rays:[40,1,1,""],plot_walls:[40,1,1,""],plot_points:[40,1,1,""],adjust_options:[40,1,1,""],plot_ray_labels:[40,1,1,""],plot_labels:[40,1,1,""],plot_rays:[40,1,1,""]},"sage.geometry.polyhedron":{backend_field:[13,0,0,"-"],base_ZZ:[41,0,0,"-"],parent:[30,0,0,"-"],double_description_inhomogeneous:[7,0,0,"-"],double_description:[14,0,0,"-"],base_QQ:[28,0,0,"-"],backend_cdd:[21,0,0,"-"],library:[4,0,0,"-"],ppl_lattice_polygon:[29,0,0,"-"],lattice_euclidean_group_element:[39,0,0,"-"],plot:[34,0,0,"-"],base:[26,0,0,"-"],cdd_file_format:[25,0,0,"-"],ppl_lattice_polytope:[0,0,0,"-"],constructor:[46,0,0,"-"],representation:[27,0,0,"-"],palp_database:[8,0,0,"-"],face:[33,0,0,"-"],backend_ppl:[31,0,0,"-"],base_RDF:[38,0,0,"-"]},"sage.geometry.polyhedron.double_description":{DoubleDescriptionPair:[14,3,1,""],StandardDoubleDescriptionPair:[14,3,1,""],Problem:[14,3,1,""],random_inequalities:[14,2,1,""],StandardAlgorithm:[14,3,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_point":{HyperbolicPointUHP:[11,3,1,""],HyperbolicPoint:[11,3,1,""]},"sage.geometry.polyhedron.constructor":{Polyhedron:[46,2,1,""]},"sage.geometry.hyperplane_arrangement":{plot:[9,0,0,"-"],affine_subspace:[2,0,0,"-"],library:[48,0,0,"-"],hyperplane:[10,0,0,"-"],arrangement:[47,0,0,"-"]},"sage.geometry.polyhedron.parent.Polyhedra_field":{Element:[30,4,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic":{HyperbolicGeodesicKM:[37,3,1,""],HyperbolicGeodesicHM:[37,3,1,""],HyperbolicGeodesic:[37,3,1,""],HyperbolicGeodesicPD:[37,3,1,""],HyperbolicGeodesicUHP:[37,3,1,""]},"sage.geometry.hyperplane_arrangement.library.HyperplaneArrangementLibrary":{bigraphical:[48,1,1,""],graphical:[48,1,1,""],G_Shi:[48,1,1,""],G_semiorder:[48,1,1,""],linial:[48,1,1,""],Catalan:[48,1,1,""],semiorder:[48,1,1,""],coordinate:[48,1,1,""],Shi:[48,1,1,""],braid:[48,1,1,""],Ish:[48,1,1,""]},"sage.geometry.polyhedron.parent.Polyhedra_RDF_cdd":{Element:[30,4,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_interface.HyperbolicPlane":{UpperHalfPlane:[24,4,1,""],UHP:[24,4,1,""],KleinDisk:[24,4,1,""],KM:[24,4,1,""],PoincareDisk:[24,4,1,""],a_realization:[24,1,1,""],PD:[24,4,1,""],HM:[24,4,1,""],Hyperboloid:[24,4,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_isometry.HyperbolicIsometry":{classification:[19,1,1,""],inverse:[19,1,1,""],matrix:[19,1,1,""],fixed_geodesic:[19,1,1,""],fixed_point_set:[19,1,1,""],translation_length:[19,1,1,""],repelling_fixed_point:[19,1,1,""],preserves_orientation:[19,1,1,""],is_identity:[19,1,1,""],model:[19,1,1,""],attracting_fixed_point:[19,1,1,""],to_model:[19,1,1,""],axis:[19,1,1,""]},"sage.geometry.polyhedron.library.Polytopes":{regular_polygon:[4,1,1,""],icosahedron:[4,1,1,""],pentakis_dodecahedron:[4,1,1,""],flow_polytope:[4,1,1,""],cyclic_polytope:[4,1,1,""],permutahedron:[4,1,1,""],hypercube:[4,1,1,""],buckyball:[4,1,1,""],Birkhoff_polytope:[4,1,1,""],small_rhombicuboctahedron:[4,1,1,""],n_simplex:[4,1,1,""],Kirkman_icosahedron:[4,1,1,""],cube:[4,1,1,""],cross_polytope:[4,1,1,""],hypersimplex:[4,1,1,""],simplex:[4,1,1,""],cuboctahedron:[4,1,1,""],great_rhombicuboctahedron:[4,1,1,""],n_cube:[4,1,1,""],rhombic_dodecahedron:[4,1,1,""],parallelotope:[4,1,1,""],dodecahedron:[4,1,1,""],six_hundred_cell:[4,1,1,""],Gosset_3_21:[4,1,1,""],twenty_four_cell:[4,1,1,""]},"sage.geometry.hyperbolic_space":{hyperbolic_point:[11,0,0,"-"],hyperbolic_isometry:[19,0,0,"-"],hyperbolic_interface:[24,0,0,"-"],hyperbolic_geodesic:[37,0,0,"-"],hyperbolic_model:[3,0,0,"-"]},"sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd":{Element:[30,4,1,""]},"sage.geometry.pseudolines.PseudolineArrangement":{transpositions:[6,1,1,""],permutations:[6,1,1,""],show:[6,1,1,""],felsner_matrix:[6,1,1,""]},"sage.geometry.triangulation.base.Point":{reduced_projective_vector:[20,1,1,""],point_configuration:[20,1,1,""],reduced_affine_vector:[20,1,1,""],reduced_affine:[20,1,1,""],index:[20,1,1,""],affine:[20,1,1,""],reduced_projective:[20,1,1,""],projective:[20,1,1,""]},"sage.geometry.polyhedron.representation.Equation":{interior_contains:[27,1,1,""],is_equation:[27,1,1,""],contains:[27,1,1,""],type:[27,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_interface.HyperbolicModels":{super_categories:[24,1,1,""],ElementMethods:[24,3,1,""],ParentMethods:[24,3,1,""]},"sage.geometry.hyperplane_arrangement.hyperplane.Hyperplane":{polyhedron:[10,1,1,""],plot:[10,1,1,""],orthogonal_projection:[10,1,1,""],point:[10,1,1,""],normal:[10,1,1,""],primitive:[10,1,1,""],linear_part_projection:[10,1,1,""],linear_part:[10,1,1,""],intersection:[10,1,1,""],dimension:[10,1,1,""]},"sage.geometry.polyhedron.base_ZZ":{Polyhedron_ZZ:[41,3,1,""]},"sage.geometry.toric_lattice":{ToricLattice_sublattice_with_basis:[22,3,1,""],ToricLattice_quotient_element:[22,3,1,""],is_ToricLattice:[22,2,1,""],ToricLatticeFactory:[22,3,1,""],ToricLattice_quotient:[22,3,1,""],ToricLattice_sublattice:[22,3,1,""],is_ToricLatticeQuotient:[22,2,1,""],ToricLattice_generic:[22,3,1,""],ToricLattice_ambient:[22,3,1,""]},"sage.geometry.polyhedron.double_description_inhomogeneous":{Vrep2Hrep:[7,3,1,""],PivotedInequalities:[7,3,1,""],Hrep2Vrep:[7,3,1,""]},"sage.geometry.polyhedron.representation.Ray":{evaluated_on:[27,1,1,""],is_ray:[27,1,1,""],type:[27,1,1,""]},"sage.geometry.linear_expression.LinearExpressionModule":{ambient_vector_space:[32,1,1,""],gens:[32,1,1,""],random_element:[32,1,1,""],Element:[32,4,1,""],change_ring:[32,1,1,""],ngens:[32,1,1,""],ambient_module:[32,1,1,""],gen:[32,1,1,""]},"sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl":{Element:[30,4,1,""]},"sage.geometry.polyhedron.double_description.Problem":{A:[14,1,1,""],dim:[14,1,1,""],initial_pair:[14,1,1,""],pair_class:[14,4,1,""],A_matrix:[14,1,1,""],base_ring:[14,1,1,""]},"sage.geometry.polyhedron.ppl_lattice_polytope.LatticePolytope_PPL_class":{pointsets_mod_automorphism:[0,1,1,""],bounding_box:[0,1,1,""],n_integral_points:[0,1,1,""],embed_in_reflexive_polytope:[0,1,1,""],fibration_generator:[0,1,1,""],base_rays:[0,1,1,""],contains:[0,1,1,""],integral_points_not_interior_to_facets:[0,1,1,""],restricted_automorphism_group:[0,1,1,""],contains_origin:[0,1,1,""],base_projection:[0,1,1,""],is_simplex:[0,1,1,""],sub_polytope_generator:[0,1,1,""],integral_points:[0,1,1,""],affine_space:[0,1,1,""],vertices_saturating:[0,1,1,""],base_projection_matrix:[0,1,1,""],has_IP_property:[0,1,1,""],n_vertices:[0,1,1,""],lattice_automorphism_group:[0,1,1,""],vertices:[0,1,1,""],ambient_space:[0,1,1,""],is_bounded:[0,1,1,""],is_full_dimensional:[0,1,1,""],affine_lattice_polytope:[0,1,1,""]},"sage.geometry.polyhedron.double_description.DoubleDescriptionPair":{are_adjacent:[14,1,1,""],first_coordinate_plane:[14,1,1,""],zero_set:[14,1,1,""],verify:[14,1,1,""],is_extremal:[14,1,1,""],cone:[14,1,1,""],dual:[14,1,1,""],matrix_space:[14,1,1,""],R_by_sign:[14,1,1,""],inner_product_matrix:[14,1,1,""]},"sage.rings.polynomial.groebner_fan":{ideal_to_gfan_format:[23,2,1,""],prefix_check:[23,2,1,""],GroebnerFan:[23,3,1,""],TropicalPrevariety:[23,3,1,""],max_degree:[23,2,1,""],ReducedGroebnerBasis:[23,3,1,""],ring_to_gfan_format:[23,2,1,""],verts_for_normal:[23,2,1,""],PolyhedralFan:[23,3,1,""],PolyhedralCone:[23,3,1,""],InitialForm:[23,3,1,""]},"sage.geometry.polyhedron.base_RDF":{Polyhedron_RDF:[38,3,1,""]},"sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl":{Element:[30,4,1,""]},"sage.geometry.cone.IntegralRayCollection":{plot:[5,1,1,""],rays:[5,1,1,""],dim:[5,1,1,""],nrays:[5,1,1,""],lattice:[5,1,1,""],dual_lattice:[5,1,1,""],lattice_dim:[5,1,1,""],ray:[5,1,1,""],cartesian_product:[5,1,1,""]},"sage.geometry.polyhedron.parent.Polyhedra_base":{Hrepresentation_space:[30,1,1,""],universe:[30,1,1,""],an_element:[30,1,1,""],ambient_space:[30,1,1,""],Vrepresentation_space:[30,1,1,""],ambient_dim:[30,1,1,""],zero:[30,1,1,""],recycle:[30,1,1,""],some_elements:[30,1,1,""],base_extend:[30,1,1,""],empty:[30,1,1,""]},"sage.geometry.polyhedron.parent":{Polyhedra_field:[30,3,1,""],Polyhedra_QQ_ppl:[30,3,1,""],Polyhedra:[30,2,1,""],Polyhedra_ZZ_ppl:[30,3,1,""],Polyhedra_QQ_cdd:[30,3,1,""],Polyhedra_RDF_cdd:[30,3,1,""],Polyhedra_base:[30,3,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic.HyperbolicGeodesicKM":{show:[37,1,1,""]},"sage.geometry.polyhedron.cdd_file_format":{cdd_Vrepresentation:[25,2,1,""],cdd_Hrepresentation:[25,2,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic.HyperbolicGeodesicUHP":{angle:[37,1,1,""],common_perpendicular:[37,1,1,""],show:[37,1,1,""],midpoint:[37,1,1,""],perpendicular_bisector:[37,1,1,""],intersection:[37,1,1,""],reflection_involution:[37,1,1,""],ideal_endpoints:[37,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelUHP":{point_in_model:[3,1,1,""],isometry_in_model:[3,1,1,""],boundary_point_in_model:[3,1,1,""],Element:[3,4,1,""],random_point:[3,1,1,""],isometry_from_fixed_points:[3,1,1,""],random_isometry:[3,1,1,""],get_background_graphic:[3,1,1,""]},"sage.geometry.polyhedron.base_QQ":{Polyhedron_QQ:[28,3,1,""]},"sage.rings.polynomial":{groebner_fan:[23,0,0,"-"]},"sage.geometry.hyperbolic_space.hyperbolic_interface.HyperbolicModels.ElementMethods":{dist:[24,1,1,""]},"sage.geometry.fan":{is_Fan:[36,2,1,""],Fan2d:[36,2,1,""],RationalPolyhedralFan:[36,3,1,""],NormalFan:[36,2,1,""],Fan:[36,2,1,""],FaceFan:[36,2,1,""],discard_faces:[36,2,1,""],Cone_of_fan:[36,3,1,""]},"sage.geometry.polyhedron.representation.Vrepresentation":{neighbors:[27,1,1,""],is_ray:[27,1,1,""],is_vertex:[27,1,1,""],adjacent:[27,1,1,""],is_V:[27,1,1,""],incident:[27,1,1,""],is_incident:[27,1,1,""],is_line:[27,1,1,""]},"sage.geometry.triangulation.base.ConnectedTriangulationsIterator":{next:[20,1,1,""]},"sage.geometry.polyhedron.representation.Hrepresentation":{neighbors:[27,1,1,""],A:[27,1,1,""],b:[27,1,1,""],adjacent:[27,1,1,""],is_incident:[27,1,1,""],incident:[27,1,1,""],eval:[27,1,1,""],is_H:[27,1,1,""],is_equation:[27,1,1,""],is_inequality:[27,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_model":{HyperbolicModelHM:[3,3,1,""],HyperbolicModelPD:[3,3,1,""],HyperbolicModelUHP:[3,3,1,""],HyperbolicModel:[3,3,1,""],HyperbolicModelKM:[3,3,1,""]},"sage.geometry.cone.ConvexRationalPolyhedralCone":{polyhedron:[5,1,1,""],relative_orthogonal_quotient:[5,1,1,""],contains:[5,1,1,""],strict_quotient:[5,1,1,""],interior_contains:[5,1,1,""],line_set:[5,1,1,""],dual:[5,1,1,""],ambient:[5,1,1,""],face_lattice:[5,1,1,""],plot:[5,1,1,""],facet_of:[5,1,1,""],is_solid:[5,1,1,""],orthogonal_sublattice:[5,1,1,""],cartesian_product:[5,1,1,""],Hilbert_coefficients:[5,1,1,""],Hilbert_basis:[5,1,1,""],is_isomorphic:[5,1,1,""],sublattice:[5,1,1,""],adjacent:[5,1,1,""],semigroup_generators:[5,1,1,""],relative_interior_contains:[5,1,1,""],is_proper:[5,1,1,""],facet_normals:[5,1,1,""],is_trivial:[5,1,1,""],relative_quotient:[5,1,1,""],is_simplicial:[5,1,1,""],is_equivalent:[5,1,1,""],intersection:[5,1,1,""],linear_subspace:[5,1,1,""],sublattice_complement:[5,1,1,""],is_strictly_convex:[5,1,1,""],ambient_ray_indices:[5,1,1,""],is_smooth:[5,1,1,""],facets:[5,1,1,""],lines:[5,1,1,""],is_face_of:[5,1,1,""],lattice_polytope:[5,1,1,""],faces:[5,1,1,""],embed:[5,1,1,""],sublattice_quotient:[5,1,1,""]},"sage.geometry.point_collection":{PointCollection:[45,3,1,""],is_PointCollection:[45,2,1,""]},"sage.geometry.polyhedron.base.Polyhedron_base":{n_Hrepresentation:[26,1,1,""],combinatorial_automorphism_group:[26,1,1,""],show:[26,1,1,""],is_lattice_polytope:[26,1,1,""],bounding_box:[26,1,1,""],vertex_digraph:[26,1,1,""],is_compact:[26,1,1,""],dilation:[26,1,1,""],hyperplane_arrangement:[26,1,1,""],vertex_graph:[26,1,1,""],vertex_generator:[26,1,1,""],graph:[26,1,1,""],facet_adjacency_matrix:[26,1,1,""],is_simplex:[26,1,1,""],to_linear_program:[26,1,1,""],equations_list:[26,1,1,""],is_simplicial:[26,1,1,""],gale_transform:[26,1,1,""],ambient_dim:[26,1,1,""],intersection:[26,1,1,""],n_facets:[26,1,1,""],dim:[26,1,1,""],triangulate:[26,1,1,""],Hrepresentation:[26,1,1,""],ambient_space:[26,1,1,""],n_rays:[26,1,1,""],Vrepresentation:[26,1,1,""],radius:[26,1,1,""],inequalities:[26,1,1,""],Hrep_generator:[26,1,1,""],equation_generator:[26,1,1,""],rays_list:[26,1,1,""],is_universe:[26,1,1,""],inequalities_list:[26,1,1,""],inequality_generator:[26,1,1,""],integral_points:[26,1,1,""],bipyramid:[26,1,1,""],vertices_list:[26,1,1,""],equations:[26,1,1,""],n_equations:[26,1,1,""],f_vector:[26,1,1,""],incidence_matrix:[26,1,1,""],Vrepresentation_space:[26,1,1,""],cdd_Vrepresentation:[26,1,1,""],is_full_dimensional:[26,1,1,""],faces:[26,1,1,""],edge_truncation:[26,1,1,""],base_extend:[26,1,1,""],projection:[26,1,1,""],Vrep_generator:[26,1,1,""],render_solid:[26,1,1,""],is_simple:[26,1,1,""],Minkowski_sum:[26,1,1,""],affine_hull:[26,1,1,""],plot:[26,1,1,""],Hrepresentation_space:[26,1,1,""],contains:[26,1,1,""],representative_point:[26,1,1,""],relative_interior_contains:[26,1,1,""],ray_generator:[26,1,1,""],field:[26,1,1,""],n_inequalities:[26,1,1,""],volume:[26,1,1,""],cdd_Hrepresentation:[26,1,1,""],vertex_adjacency_matrix:[26,1,1,""],translation:[26,1,1,""],n_vertices:[26,1,1,""],is_Minkowski_summand:[26,1,1,""],adjacency_matrix:[26,1,1,""],n_Vrepresentation:[26,1,1,""],"delete":[26,1,1,""],pyramid:[26,1,1,""],base_ring:[26,1,1,""],interior_contains:[26,1,1,""],convex_hull:[26,1,1,""],face_lattice:[26,1,1,""],rays:[26,1,1,""],lattice_polytope:[26,1,1,""],restricted_automorphism_group:[26,1,1,""],prism:[26,1,1,""],vertices_matrix:[26,1,1,""],polar:[26,1,1,""],product:[26,1,1,""],bounded_edges:[26,1,1,""],n_lines:[26,1,1,""],radius_square:[26,1,1,""],is_empty:[26,1,1,""],render_wireframe:[26,1,1,""],schlegel_projection:[26,1,1,""],center:[26,1,1,""],lines_list:[26,1,1,""],Minkowski_difference:[26,1,1,""],lines:[26,1,1,""],vertices:[26,1,1,""],line_generator:[26,1,1,""],dimension:[26,1,1,""]},"sage.geometry.polyhedron.lattice_euclidean_group_element":{LatticePolytopesNotIsomorphicError:[39,5,1,""],LatticePolytopeNoEmbeddingError:[39,5,1,""],LatticePolytopeError:[39,5,1,""],LatticeEuclideanGroupElement:[39,3,1,""]},"sage.rings.polynomial.groebner_fan.PolyhedralCone":{ambient_dim:[23,1,1,""],dim:[23,1,1,""],facets:[23,1,1,""],lineality_dim:[23,1,1,""],relative_interior_point:[23,1,1,""]},"sage.geometry.polyhedron.base":{is_Polyhedron:[26,2,1,""],Polyhedron_base:[26,3,1,""]},"sage.geometry.lattice_polytope.NefPartition":{Delta_polar:[42,1,1,""],hodge_numbers:[42,1,1,""],part_of:[42,1,1,""],Deltas:[42,1,1,""],nabla_polar:[42,1,1,""],nabla:[42,1,1,""],parts:[42,1,1,""],nparts:[42,1,1,""],part:[42,1,1,""],dual:[42,1,1,""],Delta:[42,1,1,""],part_of_point:[42,1,1,""],nablas:[42,1,1,""]},"sage.geometry.polyhedron.representation.Vertex":{is_vertex:[27,1,1,""],is_integral:[27,1,1,""],evaluated_on:[27,1,1,""],type:[27,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModel":{point_in_model:[3,1,1,""],isometry_in_model:[3,1,1,""],boundary_point_in_model:[3,1,1,""],name:[3,1,1,""],short_name:[3,1,1,""],is_isometry_group_projective:[3,1,1,""],get_geodesic:[3,1,1,""],get_isometry:[3,1,1,""],random_isometry:[3,1,1,""],get_point:[3,1,1,""],Element:[3,4,1,""],is_conformal:[3,1,1,""],random_geodesic:[3,1,1,""],is_bounded:[3,1,1,""],random_element:[3,1,1,""],random_point:[3,1,1,""],isometry_from_fixed_points:[3,1,1,""],dist:[3,1,1,""],bdry_point_test:[3,1,1,""],point_test:[3,1,1,""],isometry_test:[3,1,1,""]},"sage.rings.polynomial.groebner_fan.GroebnerFan":{mixed_volume:[23,1,1,""],number_of_variables:[23,1,1,""],render:[23,1,1,""],weight_vectors:[23,1,1,""],characteristic:[23,1,1,""],reduced_groebner_bases:[23,1,1,""],minimal_total_degree_of_a_groebner_basis:[23,1,1,""],buchberger:[23,1,1,""],gfan:[23,1,1,""],tropical_intersection:[23,1,1,""],tropical_basis:[23,1,1,""],ideal:[23,1,1,""],homogeneity_space:[23,1,1,""],render3d:[23,1,1,""],maximal_total_degree_of_a_groebner_basis:[23,1,1,""],polyhedralfan:[23,1,1,""],ring:[23,1,1,""],dimension_of_homogeneity_space:[23,1,1,""],number_of_reduced_groebner_bases:[23,1,1,""],interactive:[23,1,1,""]},"sage.geometry.polyhedron.double_description.StandardAlgorithm":{pair_class:[14,4,1,""],run:[14,1,1,""]},"sage.geometry.hyperplane_arrangement.affine_subspace":{AffineSubspace:[2,3,1,""]},"sage.geometry.polytope.Polymake":{cube:[16,1,1,""],from_data:[16,1,1,""],rand01:[16,1,1,""],birkhoff:[16,1,1,""],associahedron:[16,1,1,""],reconfigure:[16,1,1,""],cell24:[16,1,1,""],convex_hull:[16,1,1,""]},"sage.geometry.polyhedron.double_description_inhomogeneous.Vrep2Hrep":{verify:[7,1,1,""]},"sage.geometry.hasse_diagram":{Hasse_diagram_from_incidences:[43,2,1,""]},"sage.geometry.polytope":{Polytope:[16,3,1,""],Polymake:[16,3,1,""]},"sage.geometry.fan_morphism.FanMorphism":{codomain_fan:[1,1,1,""],domain_fan:[1,1,1,""],index:[1,1,1,""],preimage_cones:[1,1,1,""],is_injective:[1,1,1,""],factor:[1,1,1,""],relative_star_generators:[1,1,1,""],is_bundle:[1,1,1,""],is_dominant:[1,1,1,""],primitive_preimage_cones:[1,1,1,""],is_surjective:[1,1,1,""],image_cone:[1,1,1,""],is_birational:[1,1,1,""],preimage_fan:[1,1,1,""],is_fibration:[1,1,1,""],kernel_fan:[1,1,1,""]},"sage.geometry.hyperplane_arrangement.affine_subspace.AffineSubspace":{linear_part:[2,1,1,""],intersection:[2,1,1,""],dimension:[2,1,1,""],point:[2,1,1,""]},"sage.geometry.hyperplane_arrangement.hyperplane.AmbientVectorSpace":{change_ring:[10,1,1,""],dimension:[10,1,1,""],Element:[10,4,1,""]},"sage.geometry.fan_isomorphism":{fan_isomorphism_generator:[18,2,1,""],fan_isomorphic_necessary_conditions:[18,2,1,""],fan_2d_cyclically_ordered_rays:[18,2,1,""],fan_2d_echelon_form:[18,2,1,""],find_isomorphism:[18,2,1,""],FanNotIsomorphicError:[18,5,1,""],fan_2d_echelon_forms:[18,2,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_isometry.HyperbolicIsometryPD":{preserves_orientation:[19,1,1,""]},"sage.geometry.polyhedron.palp_database":{PALPreader:[8,3,1,""],Reflexive4dHodge:[8,3,1,""]},"sage.geometry.toric_plotter":{sector:[40,2,1,""],label_list:[40,2,1,""],reset_options:[40,2,1,""],color_list:[40,2,1,""],ToricPlotter:[40,3,1,""],options:[40,2,1,""]},"sage.geometry.triangulation.point_configuration.PointConfiguration":{restrict_to_fine_triangulations:[35,1,1,""],set_engine:[35,7,1,""],convex_hull:[35,1,1,""],positive_circuits:[35,1,1,""],face_codimension:[35,1,1,""],contained_simplex:[35,1,1,""],distance_FS:[35,1,1,""],restrict_to_star_triangulations:[35,1,1,""],triangulations:[35,1,1,""],restricted_automorphism_group:[35,1,1,""],restrict_to_connected_triangulations:[35,1,1,""],exclude_points:[35,1,1,""],restrict_to_regular_triangulations:[35,1,1,""],an_element:[35,1,1,""],Element:[35,4,1,""],volume:[35,1,1,""],pushing_triangulation:[35,1,1,""],secondary_polytope:[35,1,1,""],farthest_point:[35,1,1,""],lexicographic_triangulation:[35,1,1,""],bistellar_flips:[35,1,1,""],distance_affine:[35,1,1,""],star_center:[35,1,1,""],circuits_support:[35,1,1,""],distance:[35,1,1,""],circuits:[35,1,1,""],Gale_transform:[35,1,1,""],triangulate:[35,1,1,""],placing_triangulation:[35,1,1,""],face_interior:[35,1,1,""],triangulations_list:[35,1,1,""]},"sage.geometry.polyhedron.lattice_euclidean_group_element.LatticeEuclideanGroupElement":{domain_dim:[39,1,1,""],codomain_dim:[39,1,1,""]},"sage.geometry.polyhedron.ppl_lattice_polygon":{sub_reflexive_polygons:[29,2,1,""],polar_P1xP1_polytope:[29,2,1,""],subpolygons_of_polar_P2_112:[29,2,1,""],polar_P2_112_polytope:[29,2,1,""],LatticePolygon_PPL_class:[29,3,1,""],subpolygons_of_polar_P2:[29,2,1,""],polar_P2_polytope:[29,2,1,""],subpolygons_of_polar_P1xP1:[29,2,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_interface.HyperbolicModels.ParentMethods":{dist:[24,1,1,""]},"sage.geometry.hyperplane_arrangement.arrangement.HyperplaneArrangements":{base_ring:[47,1,1,""],gens:[47,1,1,""],ngens:[47,1,1,""],Element:[47,4,1,""],change_ring:[47,1,1,""],ambient_space:[47,1,1,""],gen:[47,1,1,""]},"sage.geometry.polyhedron.backend_ppl":{Polyhedron_ppl:[31,3,1,""],Polyhedron_QQ_ppl:[31,3,1,""],Polyhedron_ZZ_ppl:[31,3,1,""]},"sage.geometry.newton_polygon":{NewtonPolygon_element:[44,3,1,""],ParentNewtonPolygon:[44,3,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_point.HyperbolicPoint":{is_boundary:[11,1,1,""],show:[11,1,1,""],update_graphics:[11,1,1,""],symmetry_involution:[11,1,1,""],coordinates:[11,1,1,""],graphics_options:[11,1,1,""],model:[11,1,1,""],to_model:[11,1,1,""]},"sage.geometry.hyperplane_arrangement.arrangement":{HyperplaneArrangementElement:[47,3,1,""],HyperplaneArrangements:[47,3,1,""]},"sage.rings.polynomial.groebner_fan.TropicalPrevariety":{initial_form_systems:[23,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_isometry":{HyperbolicIsometryKM:[19,3,1,""],HyperbolicIsometry:[19,3,1,""],mobius_transform:[19,2,1,""],HyperbolicIsometryUHP:[19,3,1,""],HyperbolicIsometryPD:[19,3,1,""]},"sage.geometry.newton_polygon.ParentNewtonPolygon":{Element:[44,4,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic.HyperbolicGeodesicPD":{show:[37,1,1,""]},"sage.geometry.pseudolines":{PseudolineArrangement:[6,3,1,""]},"sage.geometry.polyhedron.representation":{Equation:[27,3,1,""],Vertex:[27,3,1,""],Hrepresentation:[27,3,1,""],Vrepresentation:[27,3,1,""],Inequality:[27,3,1,""],PolyhedronRepresentation:[27,3,1,""],Line:[27,3,1,""],Ray:[27,3,1,""]},"sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis":{plot:[22,1,1,""],dual:[22,1,1,""]},"sage.geometry.fan.RationalPolyhedralFan":{support_contains:[36,1,1,""],cones:[36,1,1,""],generating_cones:[36,1,1,""],cone_containing:[36,1,1,""],plot:[36,1,1,""],vertex_graph:[36,1,1,""],contains:[36,1,1,""],subdivide:[36,1,1,""],is_isomorphic:[36,1,1,""],ngenerating_cones:[36,1,1,""],complex:[36,1,1,""],make_simplicial:[36,1,1,""],is_smooth:[36,1,1,""],is_simplicial:[36,1,1,""],oriented_boundary:[36,1,1,""],common_refinement:[36,1,1,""],is_equivalent:[36,1,1,""],primitive_collections:[36,1,1,""],Stanley_Reisner_ideal:[36,1,1,""],cartesian_product:[36,1,1,""],linear_equivalence_ideal:[36,1,1,""],isomorphism:[36,1,1,""],Gale_transform:[36,1,1,""],virtual_rays:[36,1,1,""],generating_cone:[36,1,1,""],cone_lattice:[36,1,1,""],embed:[36,1,1,""],is_complete:[36,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_geodesic.HyperbolicGeodesicHM":{show:[37,1,1,""]},"sage.geometry.triangulation.element":{Triangulation:[17,3,1,""],triangulation_render_3d:[17,2,1,""],triangulation_render_2d:[17,2,1,""]},"sage.rings.polynomial.groebner_fan.PolyhedralFan":{lineality_dim:[23,1,1,""],rays:[23,1,1,""],f_vector:[23,1,1,""],cones:[23,1,1,""],is_simplicial:[23,1,1,""],ambient_dim:[23,1,1,""],to_RationalPolyhedralFan:[23,1,1,""],dim:[23,1,1,""],maximal_cones:[23,1,1,""]},"sage.geometry.linear_expression":{LinearExpression:[32,3,1,""],LinearExpressionModule:[32,3,1,""]},"sage.geometry.toric_lattice.ToricLatticeFactory":{create_key:[22,1,1,""],create_object:[22,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_interface":{HyperbolicPlane:[24,3,1,""],HyperbolicSpace:[24,2,1,""],HyperbolicModels:[24,3,1,""]},"sage.geometry.triangulation.base":{PointConfiguration_base:[20,3,1,""],ConnectedTriangulationsIterator:[20,3,1,""],Point:[20,3,1,""]},"sage.geometry.polyhedron.ppl_lattice_polygon.LatticePolygon_PPL_class":{plot:[29,1,1,""],ordered_vertices:[29,1,1,""],find_isomorphism:[29,1,1,""],is_isomorphic:[29,1,1,""],sub_polytopes:[29,1,1,""]},"sage.geometry.lattice_polytope.LatticePolytopeClass":{origin:[42,1,1,""],distances:[42,1,1,""],skeleton:[42,1,1,""],point:[42,1,1,""],normal_form_pc:[42,1,1,""],show3d:[42,1,1,""],facet_normal:[42,1,1,""],normal_form:[42,1,1,""],is_reflexive:[42,1,1,""],facet_constants:[42,1,1,""],polyhedron:[42,1,1,""],index:[42,1,1,""],nef_x:[42,1,1,""],lattice:[42,1,1,""],dual_lattice:[42,1,1,""],plot3d:[42,1,1,""],facet_normals:[42,1,1,""],nef_partitions:[42,1,1,""],polar:[42,1,1,""],facet_constant:[42,1,1,""],parent:[42,1,1,""],vertices_pc:[42,1,1,""],affine_transform:[42,1,1,""],linearly_independent_vertices:[42,1,1,""],vertex:[42,1,1,""],vertex_facet_pairing_matrix:[42,1,1,""],ambient_dim:[42,1,1,""],nfacets:[42,1,1,""],edges:[42,1,1,""],dim:[42,1,1,""],traverse_boundary:[42,1,1,""],nvertices:[42,1,1,""],vertices:[42,1,1,""],poly_x:[42,1,1,""],skeleton_points:[42,1,1,""],points:[42,1,1,""],skeleton_show:[42,1,1,""],faces:[42,1,1,""],npoints:[42,1,1,""],points_pc:[42,1,1,""],facets:[42,1,1,""]},"sage.geometry.linear_expression.LinearExpression":{A:[32,1,1,""],b:[32,1,1,""],constant_term:[32,1,1,""],evaluate:[32,1,1,""],change_ring:[32,1,1,""],coefficients:[32,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelHM":{point_in_model:[3,1,1,""],boundary_point_in_model:[3,1,1,""],get_background_graphic:[3,1,1,""],isometry_in_model:[3,1,1,""]},"sage.rings.polynomial.groebner_fan.ReducedGroebnerBasis":{groebner_cone:[23,1,1,""],ideal:[23,1,1,""],interactive:[23,1,1,""]},"sage.geometry.polyhedron.face":{PolyhedronFace:[33,3,1,""]},"sage.geometry.polyhedron.backend_field":{Polyhedron_field:[13,3,1,""]},"sage.geometry.toric_lattice.ToricLattice_ambient":{ambient_module:[22,1,1,""],plot:[22,1,1,""],dual:[22,1,1,""],Element:[22,4,1,""]},"sage.geometry.polytope.Polytope":{graph:[16,1,1,""],facets:[16,1,1,""],cmd:[16,1,1,""],is_simple:[16,1,1,""],vertices:[16,1,1,""],write:[16,1,1,""],visual:[16,1,1,""],data:[16,1,1,""],n_facets:[16,1,1,""]},"sage.geometry.newton_polygon.NewtonPolygon_element":{plot:[44,1,1,""],last_slope:[44,1,1,""],slopes:[44,1,1,""],vertices:[44,1,1,""],reverse:[44,1,1,""]},"sage.geometry.hyperbolic_space.hyperbolic_isometry.HyperbolicIsometryUHP":{classification:[19,1,1,""],fixed_point_set:[19,1,1,""],translation_length:[19,1,1,""],repelling_fixed_point:[19,1,1,""],preserves_orientation:[19,1,1,""],attracting_fixed_point:[19,1,1,""]},"sage.geometry.toric_lattice.ToricLattice_quotient_element":{set_immutable:[22,1,1,""]},"sage.geometry.hyperplane_arrangement.library":{HyperplaneArrangementLibrary:[48,3,1,""],make_parent:[48,2,1,""]},"sage.geometry.integral_points.InequalityCollection":{are_satisfied:[15,1,1,""],prepare_next_to_inner_loop:[15,1,1,""],satisfied_as_equalities:[15,1,1,""],prepare_inner_loop:[15,1,1,""],swap_ineq_to_front:[15,1,1,""]},"sage.geometry.polyhedron.double_description_inhomogeneous.Hrep2Vrep":{verify:[7,1,1,""]}},titleterms:{cdd:[25,21],represent:27,over:[38,28,41],newton:44,isometri:19,find:18,toric:[1,40,22],cython:15,group:39,polyhedr:[36,5,21,31,12],configur:35,format:25,python:13,triangul:[35,17,20],lattic:[0,1,42,39,43,8,29,22],coatom:43,fan:[36,1,23,18],arbitrari:7,collect:45,vector:2,integr:15,palp:8,combinatori:12,intern:12,convex:5,hyperbol:[12,37,3,24,19,11],todo:[34,3,37],ring:46,librari:[4,31,48],compat:1,unbound:46,space:2,databas:8,affin:2,access:8,between:[1,18],method:[6,15],refer:6,polyhedra:[46,38,26,27,15,41,7,28,30,31,34],linear:32,parent:30,gener:25,ration:[36,5],subspac:2,base:[46,38,26,41,28,20],ertex:27,hass:43,about:33,plotter:40,keep:33,rdf:38,geometri:12,commonli:4,isomorph:18,polyhedron:33,point:[35,15,11,45],famou:4,encod:6,parma:31,ppl:[0,31,29],hyperplan:[47,48,10,9],backend:[31,12,13,21],plot:[34,9],doubl:[14,7],fast:[0,29],interfac:24,interest:4,reflex:[42,8],"function":34,polytop:[0,16,8,4,42],groebner:23,finit:43,appendix:46,atom:43,euclidean:39,arrang:[47,48,9],pseudolin:6,comput:[31,12,21,15],helper:15,indic:12,cone:[14,5],file:25,tabl:12,element:39,polygon:[44,29],author:6,express:32,object:27,diagram:43,yperplan:27,"class":[38,26,41,28,20,33],geodes:37,algorithm:14,descript:[14,7],face:33,inform:33,ext:25,exampl:6,model:[24,3],morphism:1}})