Volume Enclosed by Example Subdivision Surfaces

Transcription

1 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.

2 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,,, ; for a pentagon, the mask is, 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 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 Level

3 volume_enclosed_by_example_subdivision_surfaces_55.nb Corner Mesh Vertices Faces Required valences f,, 5 Limit volume Level

4 volume_enclosed_by_example_subdivision_surfaces_55.nb Tripod Vertices Faces Required valences f,, 5, 6 Limit volume Level Tetrahedron Vertices Faces

5 volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f, Limit volume Octahedron Level Vertices Faces Required valences f, Limit volume

6 6 volume_enclosed_by_example_subdivision_surfaces_55.nb Regular Prism Level Vertices Faces,,,, 5,, 5, 6,, 6, 5,,, 6 Required valences f, Limit volume Level

7 volume_enclosed_by_example_subdivision_surfaces_55.nb 7 Torus Vertices Faces

8 8 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 6, 7, 8 Limit volume 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 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.

9 volume_enclosed_by_example_subdivision_surfaces_55.nb 9 Required valences f,, 5, 6 Limit volume 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.

10 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f Limit volume Corner Mesh Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5 Limit volume Level Tripod The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5, 6 Limit volume

11 volume_enclosed_by_example_subdivision_surfaces_55.nb Level Tetrahedron The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume Octahedron Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume

12 volume_enclosed_by_example_subdivision_surfaces_55.nb Regular Prism Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, Limit volume Torus Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 6, 7, 8 Limit volume

13 volume_enclosed_by_example_subdivision_surfaces_55.nb 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 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.

14 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 5, 6 Limit volume 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.

15 volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f Limit volume / Corner Level The vertex coordinates and topology of the mesh are specified in a section above. Required valences f,, 5 Limit volume / Tripod Level The vertex coordinates and topology of the mesh are specified in a section above.

16 6 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f,, 5, 6 Limit volume / Tetrahedron 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

17 volume_enclosed_by_example_subdivision_surfaces_55.nb 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 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

18 8 volume_enclosed_by_example_subdivision_surfaces_55.nb 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 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.

19 volume_enclosed_by_example_subdivision_surfaces_55.nb 9 Required valences f, 5 Limit volume 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 Level

20 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 Cos p n. Tetrahedron The vertex coordinates and topology of the mesh are specified in a section above. Required valences f, 6 Limit volume Octahedron Level The vertex coordinates and topology of the mesh are specified in a section above.

21 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f, 6 Limit volume Regular Prism 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

22 volume_enclosed_by_example_subdivision_surfaces_55.nb Twisted Prism Level Vertices Faces Required valences f,, 5, 6 Limit volume Level

23 volume_enclosed_by_example_subdivision_surfaces_55.nb Twisted Cube Vertices Faces Required valences f, 5, 6 Limit volume 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.

24 volume_enclosed_by_example_subdivision_surfaces_55.nb Required valences f, 6 Limit volume Twisted Corner Level Vertices Faces

25 volume_enclosed_by_example_subdivision_surfaces_55.nb 5 Required valences f, 5, 6, 7, 8 Limit volume Level The volumes are stated only in numeric precision. For instance at subdivision level in symbolic form, the volume is Sin p p + 88 Sin 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 Level

26 6 volume_enclosed_by_example_subdivision_surfaces_55.nb Regular Torus Vertices Faces Required valences f 6 Limit volume Level

27 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 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 Level

28 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