117. Subdivision surfaces2ReadingRecommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications, 1996, section 10.2.3Building complex modelsWe can extend the idea of subdivision from curves to surfaces…4Subdivision surfacesChaikin’s use of subdivision for curves inspired similar techniques for subdivision surfaces.Iteratively refine a control polyhedron (or control mesh) to produce the limit surfaceusing splitting and averaging steps.jjM∞→= limσ5Triangular subdivisionThere are a variety of ways to subdivide a poylgon mesh.A common choice for triangle meshes is 4:1 subdivision – each triangular face is split into four subfaces:6Loop averaging stepOnce again we can use masks for the averaging step:whereThese values, due to Charles Loop, are carefully chosen to ensure smoothness – namely, tangent plane or normal continuity.Note: tangent plane continuity is also known as G1continuity for surfaces.32))/2cos(23(45)()())(1()(2nnnnnnπβββα+−=−=1()()nnnnαα+++←+QQ QQL7Loop evaluation and tangent masksAs with subdivision curves, we can split and average a number of times and then push the points to their limit positions.whereHow do we compute the normal?ετπβcosi3n(n)= (n)= (2 i/n)(n)11112211 122()()() () ()() () ()nnnnnnnnnnn nnn nεεττ τττ τ∞∞∞−+++=+=+ ++=+++QQ QQTQQ QTQQ QLLL8Recipe for subdivision surfacesAs with subdivision curves, we can now describe a recipe for creating and rendering subdivision surfaces: Subdivide (split+average) the control polyhedron a few times. Use the averaging mask. Compute two tangent vectors using the tangent masks. Compute the normal from the tangent vectors. Push the resulting points to the limit positions. Use the evaluation mask. Render!9Adding creases without trim curvesIn some cases, we want a particular feature such as a crease to be preserved. With NURBS surfaces, this required the use of trim curves.For subdivision surfaces, we can just modify the subdivision mask:This gives rise to G0continuous surfaces (i.e., having positional but not tangent plane continuity)10Creases without trim curves, cont.Here’s an example using Catmull-Clark surfaces (based on subdividing quadrilateral meshes):11Face schemes4:1 subdivision of triangles is sometimes called a face scheme for subdivision, as each face begets more faces. An alternative face scheme starts with arbitrary polygon meshes and inserts vertices along edges and at face centroids:Catmull-Clark subdivision:Note: after the first subdivision, all polygons are quadilaterals in this scheme.12Subdivision can be equivalentto tensor-product patches!For a regular quadrilateral mesh, Catmull-Clark subdivision produces the same surface as tensor-product cubic B-splines!But – it handles irregular meshes as well.There are similar correspondences between other subdivision schemes and other tensor-product patch schemes.These correspondences can be proven (but we won’t do it…)13Vertex schemesIn a vertex scheme, each vertex begets more vertices. In particular, a vertex surrounded by n faces is split into n sub-vertices, one for each face: Doo-Sabin subdivision:The number edges (faces) incident to a vertex is called its valence. Edges with only once incident face are on the boundary. After splitting in this subdivision scheme, all non-boundary vertices are of valence 4. 14Interpolating subdivision surfacesInterpolating schemes are defined by splitting averaging only new verticesThe following averaging mask is used in butterfly subdivision: Setting t=0 gives the original polyhedron, and increasing small values of t makes the surface smoother, until t=1/8 when the surface is provably G1.There are several variants of Butterfly subdivision.15Next class: Projections & Z-BuffersTopics:- How do projections from 3D world to2D image plane work?- How does the Z-buffer visibility algorithm(used in today’s graphics hardware) work?Read:• Watt, Section 5.2.2 – 5.2.4, 6.3, 6.6 (esp. introand subsections 1, 4, and 8–10)Optional:• Foley, et al, Chapter 5.6 and Chapter 6• David F. Rogers and J. Alan Adams,Mathematical Elements for Computer Graphics, 2nd Ed., McGraw-Hill, New York, 1990,Chapter 2.• I. E. Sutherland, R. F. Sproull, and R. A.Schumacker, A characterization of ten hiddensurface algorithms, ACM Computing Surveys6(1): 1-55, March
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