By Lars-Erik Andersson
This can be an creation to the mathematical idea which underlies subdivision surfaces, because it is utilized in special effects and animation. Subdivision surfaces permit a dressmaker to specify the approximate kind of a floor that defines an item after which to refine it to get a extra helpful or beautiful model. a large amount of mathematical idea is required to appreciate the features of the resulting surfaces, and this booklet explains the cloth rigorously and conscientiously. The textual content is extremely available, establishing subdivision equipment in a distinct and unambiguous hierarchy which builds perception and realizing. the cloth isn't constrained to questions on the topic of regularity of subdivision surfaces at so-called impressive issues, yet supplies a wide dialogue of many of the equipment. it truly is accordingly a superb training for extra complex texts that delve extra deeply into exact questions of regularity. Read more...
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Extra info for Introduction to the mathematics of subdivision surfaces
28/29 (left). 8)/29 is replaced by n−1 V = αn∗ V + βn∗ n−1 EjL + γn∗ j=0 FjL . 9) j=0 The Repeated Averaging method corresponds to using a smoothing stencil with αn∗ replaced by 1/4, βn∗ replaced by 1/(2n), and γn∗ replaced by 1/(4n) (again, see Exercise 7/48 ). ” Catmull–Clark variant (in-place formulation) The method as described above involves Jacobi-manner smoothings, including one which is redundant, but an in-place version of the method can also be given. 10) where S = V, R 1 n Q = = 1 n n−1 j=0 1 (V + Ej ), 2 n−1 Fj .
Then, a single (d − 1 = 1) averaging substep is executed, producing a new array of points. 20/23 (right): the new control point for each black square is calculated as the average of the four neighbouring points computed in the ﬁrst substep. Thus, for example, the control point for the lower left black square is easily veriﬁed to be 9 3 3 1 pk,l + pk+1,l + pk,l+1 + pk+1,l+1 . 21/23 . The single step of the biquadratic Lane–Riesenfeld algorithm LR(2 × 2), just described, is repeated as often as desired, producing a ﬁner and ﬁner mesh at ✐ ✐ ✐ ✐ ✐ ✐ ✐ 24 book 2010/3/3 page 24 ✐ Chapter 1.
The deﬁnition of this second computation is referred to as the speciﬁcation of the smoothing rule for the method. Normally there is a fairly small number of cases for the local vertex-edgeface topology in the reﬁned mesh, and to describe a step in the subdivision method it is suﬃcient to deﬁne how to compute the new control points for each such case. This can be done by specifying a stencil for each case. Stencils are primarily used to deﬁne or illustrate a subdivision rule. 19/21 . 2 below. 14 This means that the original unmodiﬁed data are is somehow kept available, perhaps by making a copy of the data, or by reading the input data, transforming it, and writing the output without overwriting any of the input data.