Volume Enclosed by Example Subdivision Surfaces by Jan Hakenberg - May 5th, this document is available at vixra.org and hakenberg.de Abstract Simple meshes such as the cube, tetrahedron, and tripod frequently appear in the literature to illustrate the concept of subdivision. The formula for the volume enclosed by subdivision surfaces has only recently been identified. We specify simple meshes and state the volume enclosed by the corresponding limit surfaces. We consider the subdivision schemes Doo-Sabin, Midedge, Catmull-Clark, and Loop. Introduction The volume enclosed by the subdivision surface defined by a closed, orientable mesh M is the sum over all facets f vol S m f M = f œm t f i,j,k Ỳ i,j,k pxi py j pz k t f The coefficients Ỳ i,j,k consititute trilinear forms that are alternating. The coefficients are uniquely determined by the subdivision weights for topology type t f of a facet f. The points px i,py i,pz i for i =,,..., m f are the vertices of the mesh that determine the subdivision surface associated to facet f. For the derivation and proof of the formula, see [Hakenberg et al. ]. In this article, we state the volume enclosed by subdivision surfaces generated from various example meshes. Each of the four subdivision schemes is treated in a separate section Doo-Sabin Midedge Catmull-Clark Loop Each example has the following structure: We state the vertex coordinates and topology of the mesh. The specification is omitted if the mesh was already introduced previously. The subdivision iteration is illustrated up to a certain level. After one or two rounds of subdivision, the mesh topology admits the application of the volume formula. In that case, the facets are colored based on their contribution to the volume. We state the volume enclosed by the corresponding subdivision surface. In some cases, we are unable to obtain the volume in symbolic form, and revert to numeric precision. Then, the approximation of the volume by the piecewise linear surface from the first refinements are tabulated and plotted. Non-planar quads are triangulated in an arbitrary fashion. The results can help to validate implementations of the volume formula.
volume_enclosed_by_example_subdivision_surfaces_55.nb Doo-Sabin The Doo-Sabin subdivision scheme is published as [Doo/Sabin 978]. The algorithm applies to meshes with n-gons. One round of Doo-Sabin subdivision requires the contraction of each face in the mesh. The weights to contract an n-gon are w, w,..., w n-. Specifically, w = n n + 5, and w k = n p k + Cos for k =,,..., n -. n The weights are applied in a rotational fashion. For instance, a triangle is contracted using the mask,, centered at all vertices; 6 6 a quad is contractred using 9,,, ; 6 6 6 6 for a pentagon, the mask is, 5 + 5, 5-5, 5-5, 5 + 5 ; a hexagon is contracted using,,,,,. 6 6 After one round of subdivision, all newly introduced faces are quads. Cube Vertices Faces 6 5 8 6 5 6 8 7 7 8 7 5 Important: In order to color the facets based on their contribution to the global volume, we display the dual mesh. Required valences f Limit volume.696569586569 6 99.7867 7 5 9 7795 88 657 768.79.986.5566.6855....5.. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb Corner Mesh Vertices Faces 6 5 5 6 8 7 7 8 7 5 6 6 5 9 6 5 9 5 6 6 9 8 8 8 6 Required valences f,, 5 Limit volume.75687977975 7 9 98 6 55 566 756 5 857 9.67886.575 9 8 7 9 5 6 6 65 95 89 5 5 78 6.98.988....5..5..5..5. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb Tripod Vertices Faces 6 5 5 6 8 7 7 8 7 5 6 5 9 6 5 9 5 6 6 9 8 8 8 6 7 8 9 9 6 8 8 7 7 9 6 Required valences f,, 5, 6 Limit volume.878979769599668 57 799 98 6 55 566 756 5 8 9 76.7997 67.695 9 98 8 99 5 88 979 885 69 5 57 8.77.585....5..5..5..5. Level Tetrahedron Vertices Faces
volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f, Limit volume.6588859587565586 7 8.5667 Octahedron 6 7 6 5 5 6 69 97 99 66 8.687.88897.98....5...977 Level Vertices Faces 5 5 5 5 6 6 6 6 Required valences f, Limit volume.78965855988 9 7 88
6 volume_enclosed_by_example_subdivision_surfaces_55.nb 7 7 8 7 96 7 7 78 Regular Prism.59.8867.7.6757....5...6 Level Vertices Faces,,,, 5,, 5, 6,, 6, 5,,, 6 Required valences f, Limit volume.865765877989987787 8 9 6 976 7 8 9 5 7 7 8 88 68.855.75.57.7.656789....5.. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb 7 Torus Vertices.5.88678.655.658.655.658.5.88678.5.88678.655.658.655.658.5.88678.8665.8665.8665.8665...8665.8665.8665.8665 Faces 8 8 9 7 9 5 8 9 7 6 6 5 5 5 8 6 9 9 8 7 8 7 5 5 9 7 8 8 6 5 8 7 7 8 5 9 5 9 5 9 7 5 5 7 6 9 6 7 6 5 9 9 5 7 6 9 5 6 7 6 5 7 5 8 9 5 7 8
8 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 6, 7, 8 Limit volume.797.98.577.9.8.86.65.87.695....5.5..5..5. Level Print The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume. Required valences f, 5 Limit volume.995.65.687.5.567..7895..7...5..5..5. Level Bigguy The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume.
volume_enclosed_by_example_subdivision_surfaces_55.nb 9 Required valences f,, 5, 6 Limit volume.798...65.8565.58.7....6.8. Level Midedge The Midedge subdivision scheme is specified by [Peters/Reif 997]. The algorithm applies to meshes with n-gons. One round of Midedge subdivision requires the contraction of each face in the mesh. In the simplest-pure configuration, the weights to contract an n-gon are w =, w =, w =,..., w n- =, w n- =. The weights are applied in a rotational fashion. For instance, a triangle is contracted using the mask,, centered at all vertices; a quad is contractred using,,,. After one round of subdivision, all newly introduced faces are quads. Cube The vertex coordinates and topology of the mesh are specified in a section above. Important: In order to color the facets based on their contribution to the global volume, we display the dual mesh.
volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f Limit volume.6965566598876795659 99 665.88 7 98 56 7 9 5 96 5 6 9 56 Corner Mesh.8868.7.58.578....5.. Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5 Limit volume.697999765765 8 576 59 9 575.6889 9.67558 8 8 88 88 68 9 6.88955.96756....5..5..5..5. Level Tripod The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5, 6 Limit volume.87558698987695685
volume_enclosed_by_example_subdivision_surfaces_55.nb 7 6 95 57 8 7 756 6.85 67.89 9 9 79 88 55 57 786.8.8....5..5..5..5. Level Tetrahedron The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume.785699589685 99 5 8.966 Octahedron 6 5 96 7 7 869 98 89 77 58 9.96.8758.96...5..5.9 Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume.98755969887578
volume_enclosed_by_example_subdivision_surfaces_55.nb 99 7 8 98 7 7 98 96 5 58 9 Regular Prism.5.55.7879.768...5..5.5 Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume.666685567968 8 Torus 9 8 55 89 65 5 88 87 55.965.79.96988.7.596...5..5 Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 6, 7, 8 Limit volume
volume_enclosed_by_example_subdivision_surfaces_55.nb.57.98.665.566.8856.55.6758.5.678....5...5..5..5. Level Print The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume. Required valences f, 5 Limit volume.987.65.69776. 69.5.8..7...5..5..5. Level Bigguy The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume.
volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 5, 6 Limit volume.797...75.855.576977.7....6.8. Level Catmull-Clark The Catmull-Clark subdivision scheme is published as [Catmull/Clark 978]. The algorithm applies to meshes with quads. Weights are specified for the insertion of a face midpoint, and edge midpoint, as well as the repositioning of a vertex. The weights are subject to normalization so that their sum adds up to. Cube The vertex coordinates and topology of the mesh are specified in a section above.
volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f Limit volume.755798778788876 559786585858957758596885788567597656657787875856 9695/8855878956695565755697557865859595759 889979869668998.678 5.89 Corner 8 6 555 887 87 78 58 9 7 8 7 69 88 8 6 76 8 69 5 69 7 5 758 58.568.5557.79...5..5 Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5 Limit volume.787659787699656 885688857958777666965575978989798578795658857995879975 9969878699665769569756699988996978587666966678567557659 658875599858995785899686757776675885757768 85655787787769977587879977996897968799987/ 77858965665969795695685959797668585859766677968 58688995596596787859576556987857778988669 966766589988986597796678995885867658696758886678659 99855586588667885655659656868998685798659977768.78 587.795 Tripod 798 65 7 66 6 6 5 56 97 8 5 896 5 8 9 855 599 6 99 56 9 7 57 585 8.5769.6.5569...5..5 Level The vertex coordinates and topology of the mesh are specified in a section above.
6 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 5, 6 Limit volume.55765957679988 8867559977956589658978656966978997856695955789 576956656899675555555656668756967788986995569887677 67676795569676995599965785997858869889975778956867956 766876885778987669566896786955858858556666889555569 567675695759768797687/667899975979598877 57677587998685859785697765868979589997888778668588 88895677885665988855678559799759989675679888876787755 888575668957689665795875987956885697857689788 77557756557788559556967775596887776565887679.9599 66.68 Tetrahedron 6 8 777 57 6 756 77 8 958 65 9 9 9 76 5 86 6 58 7 97 7 97 5 988 5 55 7 56.6687.6555.69...5..5 Level The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, the faces are quadrangulated as illustrated. Required valences f, Limit volume.7559699978755 9 66 6 5 57 9 8 57 58 7 6 8 5 9 89 86 5 85 7 789 95 68 68-956 6 8 98 9 59 8 57 6 857 9 5 68 5 6 7 65 687 557 6 97 87 8 797 8 87 88 5 785 86 8 58 65 6 67 8-6 88 9 8 9 958 9 9 86 8 8
volume_enclosed_by_example_subdivision_surfaces_55.nb 7 6 57 97 66 99 59 8 87 68 8 5 77 67 69 6 99 96.55.8669.7.5869....5..5..5..5. Level Octahedron The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, the faces are quadrangulated as illustrated. Required valences f, Limit volume.66966896999 9 9 58 7 957 5 9 5 57 7 97 556 555 89 676 85 87 9 9 7 85 958 668 55 8 68 8 75 8 88 5 6 98 96 556 9 677 768 88 5 85 7 68 5 67 7 66 96 86 667 7 69 7 7 87 88.7..598.79....5..5..5..5. Level Regular Prism The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, the faces are quadrangulated as illustrated. Required valences f, Limit volume.587898797877559
8 volume_enclosed_by_example_subdivision_surfaces_55.nb 8 59 7 97 7 5 86 887 95 76 597 67 96 989 78 96 69 89 9 659 8 87 666 858-5 785 8 5 5 5 8 88 7 9 55 69 8 76 89 76 8 7 9 9 58 5 996 78 8 8 6 78 87 587 8-69 99 59 5 6 86 68 5 8 65 5 5 7 78 8 67 757 6 878 656 78 7 777 968 967 69 597 696.76.756.6679.589....5..5..5..5. Level Torus The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, the faces are quadrangulated as illustrated. Required valences f,, 6, 7, 8 Limit volume.967.98.68.9897.585.9.7.9978.67....5.5..5..5. Level Print The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume.
volume_enclosed_by_example_subdivision_surfaces_55.nb 9 Required valences f, 5 Limit volume.59.65.6986.77.67.577.6766.69.596....5..5..5..5. Level Bigguy The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume. Required valences f,, 5, 6 Limit volume.576..588.6855.967.6.777..7...5..5..5. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb Loop The Loop subdivision scheme is published as [Loop 987]. The algorithm applies to meshes with triangles. The weights for the insertion of an edge midpoint, as well as the repositioning of a vertex that already existed in the input mesh are where b= n 5 8-8 + Cos p n. Tetrahedron The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, 6 Limit volume.556955958876889686 9 9 5 855 6 865 75 5 89.99 Octahedron 6 5 8 57 65 56 5 8 85 6 68 79 77 98 5 88 8.655.9868.7...5..5..57575 Level The vertex coordinates and topology of the mesh are specified in a section above.
volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f, 6 Limit volume.789695759785668 969 77 77 78 8 9 65 5 99 8 97 6 8 989 6 7 7 8 7 8 56 Regular Prism 8 7 67 8 86 7 757 897 5 99 5 67 776 75 96 955 8 98 59 8 98.6976.88.9859.876...5..5.595 Level The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, each quad is split into triangles as illustrated. Required valences f, 5, 6 Limit volume.85689778898 9 78 8 8 66 7 99 58 55 86 7 7 599 6 585 9 85 8 577 5 58 9-7 577 95 5 6 85 6 6 7 96 6 87 86 7 556 85 8 9 986 7 95 6 689 98 6 9 8 79 69 88 568 57 6 9 55-55 595 8 9 8 78 5 8 9 8 7 577 5 7 65 7 87 5 6 875 5 96 8 77 6 789 6 88 9 976 98 695 75 89 96-9 9 87 98 7 7 867 9 66 75 6 88
volume_enclosed_by_example_subdivision_surfaces_55.nb Twisted Prism 899 75 5 65 56 6 9 87 967 56 5 56 87 9 7 88 578 7 7 8 98 7 58 5 7 59 86 6.98.86.78....5..5878.5..5..5. Level Vertices Faces 5 5 6 5 6 6 6 6 Required valences f,, 5, 6 Limit volume.697986769 68 76 7 955 5 5 7 6 8 97 58 68 8 68 7 988 9 6 7-6 89 57 6 9 88 6 9 7 9 9 97 8 668 77 7 68 99 6 8 768 59 799 85 6 65 57 5 7 69 6 9 98 977 5 898-5 9 976 6 5 58 5 8 89 59 569 5 56 65 7 69 75 5 6 688 58 978 5 996 95 78 68 59 6 7 8 58 687 5 9 89 79 95 7-8 89 86 7 65 96 568 6 765 69 59 5 56 85 76.88898 6 5 5 576 59 68 57 5 5 88 9 888 9 779 5 87 96 985 5 6 5 99.97.677.57969....5...5..5..5. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb Twisted Cube Vertices Faces 6 6 5 8 6 6 5 6 8 5 8 7 7 8 7 7 7 5 Required valences f, 5, 6 Limit volume.7967995759555 5 86 66 68 8 78 6 7 5 8 8 77 8 876 767 589 5 95 6 8 89 75 8-96 79 987 86 79 69 669 656 99 9 78 78 65 5 98 7 77 886 95 9 6 7 5 85 597 8 6 67 85 5 6 5 975 679 6 98 79 66 7-76 575 88 788 88 56 5 56 9 75 5 98 7 75 5 58 87 5 86 86 797 85 8 775 78 5 75 8 77 9 5 7 87 66 6-8 95 9 655 85 55 59 965 68 8.6976 5 9 9 5 88 9 8 7 96 7 5 65 8 7 76 78 6 7 889 5 6 597 69 766 656.6.95.55987....5..5..5..5. Level Regular Cube The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, each quad is split into triangles as illustrated.
volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f, 6 Limit volume.675779655577796 57 56 5 968 759 56.7 87.8 5 9 7 8 69 55 68 5 56 86 75 96 87 7 877 96 9 Twisted Corner.957.8576.68....5.. Level Vertices Faces 6 6 5 5 6 8 5 8 7 7 8 7 7 7 5 6 6 5 6 5 9 6 5 5 9 5 6 9 6 6 9 6 8 8 8 6 8 6
volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f, 5, 6, 7, 8 Limit volume.85..879..7886.8696.58.8856.7.885.57...5..5 Level The volumes are stated only in numeric precision. For instance at subdivision level in symbolic form, the volume is 77 5 + 67 + 6 5 + 9 6 Sin p p + 88 Sin 76 56 Regular Corner The vertex coordinates and topology of the mesh are specified in a section above. For subdivision, each quad is split into triangles as illustrated. Required valences f, 6, 8, Limit volume.96.5877 9 96 75 5 9 5 8 9 9 85 56 756 6 5 956 6 6 96.56.7556..7...5..5..5. Level
6 volume_enclosed_by_example_subdivision_surfaces_55.nb Regular Torus Vertices Faces 5 5 6 6 6 7 7 7 8 8 5 8 9 5 5 9 6 9 6 6 7 7 7 8 8 8 9 9 9 5 5 5 6 6 6 5 5 5 6 6 6 5 Required valences f 6 Limit volume 8.559955867958599559 5 86 87 6 7 6 7.88 9.575 85 96 87 5 68.5.8688....5..5..5..5. Level
volume_enclosed_by_example_subdivision_surfaces_55.nb 7 Torus The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, 6, 7, 8 Limit volume.66.98.96866.7699.7.6.586.779.77....5..5..5..5. Level Hip_s The specification of the mesh is omitted. The example is included for the purpose of illustration, and to study the approximation of the volume. Required valences f,, 5, 6, 7, 8, 9, Limit volume.6997.77..7755.879.7.56..7...5..5..5. Level
8 volume_enclosed_by_example_subdivision_surfaces_55.nb References [Catmull/Clark 978] Catmull E., Clark J.: Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-Aided Design 6(6), 978 [Doo/Sabin 978] Doo. D., Sabin M.: Behaviour of recursive division surfaces near extraordinary points, Computer- Aided Design (6), 978 [Hakenberg et al. ] Hakenberg J., Reif U., Schaefer S., Warren J.: Volume Enclosed by Subdivision Surfaces, vixra:5., vixra.org, [Loop 987] Loop C.: Smooth subdivision surfaces based on triangles, Master s thesis, University of Utah, 987 [Peters/Reif 997] Peters J., Reif U.: The simplest subdivision scheme for smoothing polyhedra, 997