The Catmull–Clark

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

is a technique used in

3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for t ...

to create curved surfaces by using

subdivision surface In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface) is a curved surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface, ...

modeling A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

. It was devised by

Edwin Catmull Edwin Earl "Ed" Catmull (born March 31, 1945) is an American computer scientist who is the co-founder of Pixar and was the President of Walt Disney Animation Studios. He has been honored for his contributions to 3D computer graphics, including th ...

and

Jim Clark James Clark Jr. OBE (4 March 1936 – 7 April 1968) was a British Formula One racing driver from Scotland, who won two World Championships, in 1963 and 1965. A versatile driver, he competed in sports cars, touring cars and in the Indianapol ...

in 1978 as a generalization of

bi-cubic In mathematics, bicubic interpolation is an extension of cubic interpolation (not to be confused with cubic spline interpolation, a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular ...

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

B-spline In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be express ...

surfaces to arbitrary

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

. In 2005, Edwin Catmull, together with

Tony DeRose Tony may refer to: People and fictional characters * Tony (given name), including a list of people and fictional characters * Gregory Tony (born 1978), American law enforcement officer * Motu Tony (born 1981), New Zealand international rugby lea ...

and

Jos Stam Jos Stam (born 28 December 1965 in The Hague, Netherlands) is a researcher in the field of computer graphics, focusing on the simulation of natural physical phenomena for 3D-computer animation. He achieved technical breakthroughs with the simulati ...

, received an

Academy Award for Technical Achievement The Technical Achievement Award is one of three Scientific and Technical Awards given from time to time by the Academy of Motion Picture Arts and Sciences. (The other two awards are the Scientific and Engineering Award and the Academy Award of Mer ...

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.

Recursive evaluation

Catmull–Clark surfaces are defined

recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...

, using the following ''refinement scheme.'' Start with a

mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, e ...

of an arbitrary

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

. All the vertices in this mesh shall be called ''original points''. * For each face, add a ''face point'' ** Set each face point to be the

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...

of all ''original points'' for the respective face Catmull-Clark Recursive Step 1

* For each edge, add an ''edge point''. ** Set each edge point to be the average of the two neighbouring face points ''(AF)'' and the ''midpoint'' of the edge ''(ME)''

\frac

* For each ''original point'' (''P)'', take the average (''F)'' of all ''n'' (recently created) face points for faces touching ''P'', and take the average ''(R)'' of all ''n'' edge midpoints for original edges touching ''P'', where each edge midpoint is the average of its two endpoint vertices (not to be confused with new ''edge points'' above). (Note that from the perspective of a vertex ''P'', the number of edges neighboring ''P'' is also the number of adjacent faces, hence ''n'') **Move each ''original point'' to the new ''vertex point''

\frac

(This is the

barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important co ...

of ''P'', ''R'' and ''F'' with respective weights (''n'' − 3), 2 and 1) Catmull-Clark Recursive Step 3

*Form edges and faces in the new mesh **Connect each new ''face point'' to the new ''edge points'' of all original edges defining the original face Catmull-Clark Recursive Step 4

** Connect each new ''vertex point'' to the new ''edge points'' of all original edges incident on the original vertex Catmull-Clark Recursive Step 5

** Define new faces as enclosed by edges Catmull-Clark recursive step 6

Properties

The new mesh will consist only of

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

s, 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

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. It can be shown that the limit surface obtained by this refinement process is at least

\mathcal^1

at extraordinary vertices and

\mathcal^2

everywhere else (when ''n'' indicates how many derivatives are continuous, we speak of

\mathcal^n

continuity). After one iteration, the number of extraordinary points on the surface remains constant.

Exact evaluation

The limit surface of Catmull–Clark subdivision surfaces can also be evaluated directly, without any recursive refinement. This can be accomplished by means of the technique of

(1998). This method reformulates the recursive refinement process into a

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

problem, which can be solved directly by means of

matrix diagonalization In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...

Software using the algorithm

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* AC3D * Anim8or *

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* CATIA (Imagine and Shape) *

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Hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...

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* LightWave 3D, version 9 * Makehuman *

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* MODO *

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- Grasshopper 3D Plugin - Weaverbird Plugin *

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- Requires a Plugin. *

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* Strata 3D CX *

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* Zbrush

Recursive evaluation

Properties

Exact evaluation

Software using the algorithm

See also

References

Further reading