Subdivision Surface

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Subdivision Surface

• Jun Murakawa
• 830 Advanced Topic presentation
• May 12, 2005

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Outline

• Motivation
• What is subdivision surface?
• Subdivision schemes
• Loop scheme
• Implementation
• Demo
• Whats next?
• References

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Motivation (1)

• Flat polygons are efficient to render through
  pipeline.
• Phong shading gives smooth shaded colors on the
  surface.

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Motivation (2)
However, polygon edges always exist and are
especially visible at silhouettes. We want
perfectly smooth surface!
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Motivation (3)

• Polynomial curve/surface representation
• Bézier
• B-Spline
• NURBS

Perfectly smooth surface patches, but have severe
topological restrictions. Need smooth surfaces
for arbitrary topology meshes.
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What is subdivision surface?

• Initial control mesh
• Successive refinement by subdivision
• Smooth limit surface

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Advantages of subdivision surface

• Arbitrary topology
• Scalability
• Uniform representation
• Numerical stability
• Code simplicity

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How to subdivide?
Triangle
Quadrilateral
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Subdivision schemes
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Loop scheme

• Splitting Add edge points to all edges
• Averaging Each vertex is moved with a weighted
  average or neighbor vertices
• Reconnect edges

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1. Edge point
A new edge point
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2. Averaging
For a vertex v which has n neighboring vertices
New vertex position
Where,

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3. Reconnect edges
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Implementation

• Top level mesh
• Requirements
• No more than two polygons share an edge
• All polygons sharing a vertex form an open or
  closed neighborhood of the vertex
• Efficient adjacency query
• Representation
• Winged edge, Quad edge, Half edge
• Polygon based

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Half Edge data structure
It can answer following adjacency questions (and
more) for O(1). Which vertices are adjacent to
this vertex? Which faces use this vertex? Which
faces border this edge?
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Adjacency query example
Which vertices are adjacent to vertex 0?
vertex0 -gt edge1 edge1 -gt vertex1 edge1 -gt edge2
(pair) edge2 -gt vetex2 edge2 -gt edge3
(next) edge3 -gt vertex3
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Hierarchical meshes

• Represent all levels of meshes generated by
  subdivision
• Quad tree
• Array
• Enable multi-resolution modeling

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Quad tree
Level 0 initial mesh
Level 1 subdivided
Level 2 subdivided
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Demo (1)
Level 0
Level 1
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Demo (2)
Level 2
Level 1
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Whats next?

• Multi-resolution modeling
• Operations such as cut and paste
• Hardware implementation
• Directly evaluate a limit surface from the top
  level mesh
• Triangle/Quad mixed mesh

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References (1)

• Subdivision Surface
• Zorin, D. Shroder, P. Subdivision for Modeling
  and Animation. SIGGRAPH 2000 Course Notes
• http//www.mrl.nyu.edu/publications/subdiv-course2
  000/
• Farin, G. Curves and surfaces for CAGD. Fifth
  edition. 1999.
• Loop, C. Smooth Subdivision Surfaces Based on
  Triangles. Masters thesis, University of Utah,
  Department of Mathematics, 1987.

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References (2)

• Lib3ds 3ds file loader
• http//lib3ds.sourceforge.net/
• Half edge data structure
• http//www.flipcode.com/articles/article_halfedge.
  shtml