Subdivision Surfaces and Curves: Chaikin's Algorithm and NURBS, Slides of Calculus

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Subdivision Surfaces

Subdivision Curves

Idea: Repeatedly refine control polygon: P 1 → P 2 → P 3 Curve will be limit of infinite process

Averaging Mask

Rather than average with nearest neighbor, apply weighted averaging mask during averaging step: r = (…, r-1, r 0 , r 1 , …) Chaikin’s algorithm: r = (1/2, 1/2)

Averaging Example

A B C A B C Split → Average → a b A B C a b c

Extending to Surfaces

Subdivision curves extend to surfaces Used in all major 3D modeling programs Preserves lower polymeshs while allowing for high-quality models

NURBS

• Non-uniform rational basis splines
• Patches generated from curves
• Model curves and surfaces
• Intuitive control points
• Efficient evaluation https://www.youtube.com/watch?v=m9U_XmnHQMU

Subdivisions for Modeling

https://www.youtube.com/watch?v=cUcif7nH4FM

Approximating Schemes

Limit surfaces approximate initial meshes Generated control points not on surface Examples:

• Doo-Sabin
• Catmull-Clark
• Loop

Blue vertices and yellow edges show topological relationship to subdivision

Vertex Schemes

Vertices create more vertices: A vertex surrounded by n faces is split into n sub-vertices (one per face) Note: valence is number of edges incident to a vertex extraordinary vertices do not have standard valence of topology (generally unavoidable)

Catmull-Clark Scheme

For each face, create face point averaging original vertices For each edge, create edge point averaging original end points and neighboring face points For each face point, connect the face point to each edge point of the face

Catmull-Clark Scheme

Move the original vertex (O) based on the valence (n) based on faces and edges Average of all created face points: F Average of all edge midpoints: E newPosition = O ( n − 3 ) + F + 2 E n Weight mask based on valence:

Finding the Limit

Possible to evaluate limit of Catmull-Clark surfaces without explicit subdivision

• Patches have same limit surface regardless of valence after subdivision
• Can be evaluated analytically as an eigenbasis http://www.dgp.toronto.edu/people/stam/ reality/Research/pdf/sig98.pdf

Loop Scheme

Subdivides triangles into smaller triangles (4:1 subdivision) Each face is split into four subfaces based on weight mask