By Lars-Erik Andersson

This is often an advent to the mathematical thought which underlies subdivision surfaces, because it is utilized in special effects and animation. Subdivision surfaces allow a fashion designer to specify the approximate kind of a floor that defines an item after which to refine it to get a extra invaluable or beautiful model. a large amount of mathematical thought is required to appreciate the features of the resultingRead more...

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**Sample text**

This paper can be viewed as the origin of two distinct and major ﬁelds involving the use of splines: geometrical modelling [30, 51, 127] and statistical data smoothing [165]. Subdivision algorithms for curves and surfaces in Bernstein form (B´ezier curves and surfaces) were developed starting around 1960 by de Casteljau and B´ezier. The recurrence relations for B-splines were discovered by de Boor, Cox, and Mansﬁeld in 1972, and one of the most important algorithms for B-splines, from our point of view, was published by Lane and Riesenfeld in 1980 [81].

Let M be a locally planar mesh without boundary. 2/10 ), and the number of edges incident at a vertex (the valence) by n. Then, a regular triangular mesh is deﬁned as one for which all faces have e = 3 and all interior vertices have valence n = 6. Similarly, a regular quadrilateral mesh is deﬁned9 as one for which all faces have e = 4 and all interior vertices have valence n = 4. 2. If a mesh is considered to be triangular, but it is not a regular triangular mesh, then any vertex with n = 6 is called an extraordinary vertex , and any face with e = 3 is called an extraordinary face.

Two examples of logical meshes. (3, 2, 1, 0). 7/11 (right). The faces of the two meshes are not indicated in the ﬁgure, but these faces (and their edge sets, which implicitly specify the way in which the faces are linked to the vertices and edges) must be speciﬁed. 7/11 (right) the direction of the through hole is ambiguous: it could join the sides as shown in the ﬁgure, or it could join the opposite pair of sides; alternatively, the hole could go from top to bottom, or even be absent. For subdivision, we are interested in logical meshes that satisfy the further condition of local planarity [124], [176, Sec.