24 Temmuz 2007 Salı

Catmull-Clark Subdivision Surfaces (1978)

Catmull Clark Wikipedia

Symbolcraft Catmull-clark

The Catmull-Clark algorithm is used in subdivision surface modeling to create smooth surfaces. It was devised by Edwin Catmull (of Pixar) and Jim Clark, and won an Academy Award for Technical Achievement in 2006.

When a polyhedron is subdivided with the Catmull-Clark method, new vertices (called "face points") are placed at the center of each original face, new "edge points" are similarly placed at the center of each original edge, and then new edges are added to connect the new edge points to the new adjacent face points. The positions of the vertices are calculated as follows:

* The face points are positioned as the average of the positions of the face's original vertices;
* The edge point locations are calculated as the average of the center point of the original edge and the average of the locations of the two new adjacent face points;
* The old vertices are repositioned according to the equation:

Q 2R S(n-3)
--- + ---- + --------
n n n


o Q is the average of the new face points surrounding the old vertex,
o R is the average of the midpoints of the edges that share the old vertex,
o S is the old vertex point, and
o n is the number of edges that share the old vertex.

"Recursively generated B-spline surfaces on arbitrary topological surfaces" by E. Catmull and J. Clark (Computer-Aided Design 10(6):350-355, November 1978).

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