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Subdivision Surfaces
Last Time?
- Curves & Surfaces
- Continuity Definitions
- Interpolation vs. Approximation Splines
- Cubic Bezier & BSpline
Today
- Spline Surfaces / Patches
- Tensor Product
- Bilinear Patches
- Bezier Patches
- Trimming Curves
- Subdivision Surface “Zoo”
- Seams In Subdivision
- Misc. Mesh/Surface Vocabulary
- “Piecewise Smooth Surface Reconstruction”
Tensor Product
- Of two vectors:
- Similarly, we can define a surface as the tensor product of two curves....
Farin, Curves and Surfaces for Computer Aided Geometric Design
Bilinear Patch Bilinear Patch
- Smooth version of quadrilateral with non-planar vertices... - But will this help us model smooth surfaces? - Do we have control of the derivative at the edges?
Today
- Spline Surfaces / Patches
- Tensor Product
- Bilinear Patches
- Bezier Patches
- Trimming Curves
- Subdivision Surface “Zoo”
- Seams In Subdivision
- Misc. Mesh/Surface Vocabulary
- “Piecewise Smooth Surface Reconstruction”
Bicubic Bezier Patch
Editing Bicubic Bezier Patches
Curve Basis Functions
Surface Basis Functions
Bicubic Bezier Patch Tessellation
- Given 16 control points and a tessellation resolution, we can create a triangle mesh
resolution: 5x5 vertices
resolution: 11x11 vertices
resolution: 41x41 vertices
Modeling with Bicubic Bezier Patches
- Original Teapot specified with Bezier Patches
- But it’s not "watertight": it has intersecting surfaces at spout & handle, no bottom, a hole at the spout tip, a gap between lid & base
Trimming Curves for Patches
Shirley, Fundamentals of Computer Graphics
Loop Subdivision
Shirley, Fundamentals of Computer Graphics
Loop Subdivision
SIGGRAPH 2000 course notes Subdivision for Modeling and Animation (page 70)
Catmull Clark Subdivision
"Subdivision Surfaces in Character Animation", DeRose, Kass & Truong, SIGGRAPH 1998
Questions?
Justin Legakis
Today
- Spline Surfaces / Patches
- Subdivision Surface “Zoo”
- Seams In Subdivision
- Misc. Mesh/Surface Vocabulary
- “Piecewise Smooth Surface Reconstruction”
Seams don’t Subdivide as Expected
Misc. Mesh/Surface Vocabulary
- Genus: The maximum number of disjoint simple closed curves which can be cut from an orientable surface of genus g without disconnecting it is g. - Homeomorphic/Topological equivalence : a continuous stretching and bending of the object into a new shape
Misc. Mesh/Surface Vocabulary
http://en.wikipedia.org/wiki/Image:Mug_and_Torus_morph.gif
- Dihedral Angle :
- the angle between the planes of two triangular faces
- “looking down the edge” between two faces, the angle between the faces.
Misc. Mesh/Surface Vocabulary Today
- Spline Surfaces / Patches
- Subdivision Surface “Zoo”
- Seams In Subdivision
- Misc. Mesh/Surface Vocabulary
- “Piecewise Smooth Surface Reconstruction”
Reading for Today
- Hoppe et al., “Piecewise Smooth Surface Reconstruction” SIGGRAPH 1994
Piecewise Smooth Surface Reconstruction
- From input: scanned mesh points
- Estimate topological type (genus)
- Mesh optimization (a.k.a. simplification)
- Smooth surface optimization
