Subdivision%20Surfaces

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Subdivision Surfaces

• Jeff Ballard
• Nick Rasmussen

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Overview

• Motivation for Subdivision Surfaces
• Subdivision Schemes
• Interesting Properties
• Discussion of Papers

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Traditional Modeling Techniques

• Polygons
• Splines
• Implicit Surfaces

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Polygons - Pro

• Fast
• Arbitrary Topology
• Easy Stitching

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Polygons - Con

• C0 Continuity
• High Counts for Complex Surfaces
• Edits Hard to do Globally

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B-Splines - Pro

• Mathematically Elegant
• Well Understood
• C2 Everywhere

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B-Splines - Con

• Needs Socking to Prevent Cracks
• Sharp/Curved Edges Hard
• High CV Count Creatures

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NURBS - Pro

• CV Weights gt Sharp Edges, Finer Control
• Trim Curves
• Widespread Support (sort-of )

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NURBS - Con

• Trim Curves Susceptible to Numerical Error
• Very Hard to Sock with Trim Curves
• Difficult to Implement
• Greater Sophistication Needed for Modeling Tools

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Implicit Surfaces - Pro

• Blobby Objects
• Mathematically Trivial

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Implicit Surfaces - Con

• Poor Control
• Hard to Model Details
• Achieving Correct Polyginization

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Why Subdivision Surfaces?

• Single Surface for All Modeling Operations
• Multi-Resolution Edits
• Extensions to Support Sharp Edges
• Arbitrary Topology
• No Cracks

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Subdivision Terminology

• Vertices
• Regular/Extraordinary
• Odd/Even
• Face/Edge
• Edges
• Boundaries and Creases
• Control Mesh

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Overview Of Subdivision Schemes

• Corner Cutting
• Vertex Insertion
• Interpolating
• Approximating

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Corner Cutting

• Old Vertices are Discarded
• Common Schemes
• Doo-Sabin
• Mid-Edge

2 steps
4 steps
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Interpolating

• Even Vertices Remain Stationary
• Inflates Out to Limit Surface (bad)
• Common Schemes
• Modified Butterfly (triangle based)
• Kabbelt (quad based)

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Approximating

• Even Vertices are Moved
• Converges in to Limit Surface
• Convergence Faster than Interpolating Schemes
• Better Mathematical Properties
• Common Schemes
• Loop (triangle based)
• Catmull-Clark (quad based)

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Subdivision ExampleCatmull-Clark

• Quad-Based
• Approximating Subdivision Surface
• Isomorphic to B-Spline Patch Around Regular
  Vertices

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Catmull-Clark Demonstration
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Interesting Properties

• Explicit Evaluation
• Subdivide Scalar Parameters
• Snap to Limit Surface

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Big Wins

• No Cracks

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Big Wins (cont.)

• View-Dependant Mesh Refinement
• Operate on only one Geometry
• Polygon Reduction on Control Mesh
• Pawn Geometry

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Papers

• Interactive Multiresolution Mesh Editing
• Zorin et. al.
• Non-Uniform Recursive Subdivision Surfaces
• Sederberg et. al.
• Exact Evaluation of Loop Subdivision Surfaces
• Jos Stam

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Papers (cont.)

• Exact Evaluation of Catmull-Clark Subdivision
  Surfaces
• Jos Stam
• Subdivision Surfaces in Character Animation
• DeRose et. al.

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Interactive Multiresolution Mesh Editing

• Uniform Editing On All Detail Levels
• Edit At One Resolution and Propagate Changes

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Interactive Multiresolution Mesh Editing (cont.)

• Subdivide Surfaces Only Where it Matters

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Non-Uniform Recursive Subdivision Surfaces

• Interesting, But Missed the Ball
• NURSSesNURBS as Catmull-ClarkB-Splines
• We Dont Want/Need NURBS

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Exact Evaluation of Catmull-Clark Subdivision
Surfaces

• Subdivision Surfaces Without Subdividing
• Precompute Eigenbasis Functions
• Produce the Limit Surface Directly

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Subdivision Surfaces in Character Animation

• Wonderful Overview of Subdivision Surfaces
• The Right Way to do Edge Weights
• Fits Well With RenderMan

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Subdivision in Action
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A Bugs Life
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Geris Game
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Conclusions