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Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Subdivision overview
CS 4621 Lecture 1
1 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Subdivision rules for curves
• New vertex positions are linear combinations of old
positions
ODD EVEN
Subdivision curves
[Schröder & Zorin SIGGRAPH 2000 course 23]
Subdivision surfaces
[Schröder & Zorin SIGGRAPH 2000 course 23]
Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Generalizing from curves to surfaces
• Two parts to subdivision process
• Subdividing the mesh (computing new topology)
– For curves: replace every segment with two segments
– For surfaces: replace every face with some new faces
• Positioning the vertices (computing new geometry)
– For curves: two rules (one for odd vertices, one for even )
• New vertex’s position is a weighted average of positions
of old vertices that are nearby along the sequence
– For surfaces: two kinds of rules (still called odd and even)
• New vertex’s position is a weighted average of positions
of old vertices that are nearby in the mesh
5 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Subdivision of meshes
• Quadrilaterals
– Catmull-Clark 1978
• Triangles
– Loop 1987
[Schröder & Zorin SIGGRAPH 2000 course 23] 6
Loop regular rules
[Schröder & Zorin SIGGRAPH 2000 course 23]
Catmull-Clark regular rules
Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Full Catmull-Clark rules (quad mesh)
[Schröder & Zorin SIGGRAPH 2000 course 23] 13 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop Subdivision Example
control polyhedron
Loop Subdivision Example
refined
control polyhedron
Loop Subdivision Example
odd
subdivision mask
Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop Subdivision Example
subdivision level 1
17 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop Subdivision Example
even
subdivision mask
(ordinary vertex)
Loop Subdivision Example
subdivision level 1
Loop Subdivision Example
even
subdivision mask
(extraordinary vertex)
Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop Subdivision Example
subdivision level 4
25 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop Subdivision Example
limit surface
Relationship to splines
• In regular regions, behavior is identical
• At extraordinary vertices, achieve C^1
– near extraordinary, different from splines
• Linear everywhere
– mapping from parameter space to 3D is a linear combination
of the control points
– “emergent” basis functions per control point
• match the splines in regular regions
• “custom” basis functions around extraordinary vertices
Loop vs. Catmull-Clark
[Schröder & Zorin SIGGRAPH 2000 course 23]
Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop vs. Catmull-Clark
[Schröder & Zorin SIGGRAPH 2000 course 23] 29 Cornell CS4621 Fall 2008 •!Lecture 1 © 2008 Steve Marschner •
Loop vs. Catmull-Clark
Loop
(after splitting faces)
Catmull-Clark [Schröder & Zorin SIGGRAPH 2000 course 23]
Loop with creases
[Hugues Hoppe]
Catmull-Clark with creases
[DeRose et al.
SIGGRAPH 1998]
