The CatmullâClark algorithm is a technique used in 3D computer graphics to create curved surfaces by using subdivision surface modeling. It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology. [1]
In 2005, Edwin Catmull, together with Tony DeRose and Jos Stam, received an Academy Award for Technical Achievement for their invention and application of subdivision surfaces. DeRose wrote about "efficient, fair interpolation" and character animation. Stam described a technique for a direct evaluation of the limit surface without recursion.
CatmullâClark surfaces are defined recursively, using the following refinement scheme. [1]
Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points.
The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look "smoother" (i.e. less "jagged" or "pointy") than the old mesh. Repeated subdivision results in meshes that are more and more rounded.
The arbitrary-looking barycenter formula was chosen by Catmull and Clark based on the aesthetic appearance of the resulting surfaces rather than on a mathematical derivation, although they do go to great lengths to rigorously show that the method converges to bicubic B-spline surfaces. [1]
It can be shown that the limit surface obtained by this refinement process is at least at extraordinary vertices and everywhere else (when n indicates how many derivatives are continuous, we speak of continuity). After one iteration, the number of extraordinary points on the surface remains constant.
The limit surface of CatmullâClark subdivision surfaces can also be evaluated directly, without any recursive refinement. This can be accomplished by means of the technique of Jos Stam (1998). [3] This method reformulates the recursive refinement process into a matrix exponential problem, which can be solved directly by means of matrix diagonalization.
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The CatmullâClark algorithm is a technique used in 3D computer graphics to create curved surfaces by using subdivision surface modeling. It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology. [1]
In 2005, Edwin Catmull, together with Tony DeRose and Jos Stam, received an Academy Award for Technical Achievement for their invention and application of subdivision surfaces. DeRose wrote about "efficient, fair interpolation" and character animation. Stam described a technique for a direct evaluation of the limit surface without recursion.
CatmullâClark surfaces are defined recursively, using the following refinement scheme. [1]
Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points.
The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look "smoother" (i.e. less "jagged" or "pointy") than the old mesh. Repeated subdivision results in meshes that are more and more rounded.
The arbitrary-looking barycenter formula was chosen by Catmull and Clark based on the aesthetic appearance of the resulting surfaces rather than on a mathematical derivation, although they do go to great lengths to rigorously show that the method converges to bicubic B-spline surfaces. [1]
It can be shown that the limit surface obtained by this refinement process is at least at extraordinary vertices and everywhere else (when n indicates how many derivatives are continuous, we speak of continuity). After one iteration, the number of extraordinary points on the surface remains constant.
The limit surface of CatmullâClark subdivision surfaces can also be evaluated directly, without any recursive refinement. This can be accomplished by means of the technique of Jos Stam (1998). [3] This method reformulates the recursive refinement process into a matrix exponential problem, which can be solved directly by means of matrix diagonalization.
This section needs additional citations for
verification. (April 2013) |
{{
cite web}}
: CS1 maint: archived copy as title (
link)