Introduction to the mathematics of subdivision surfaces by Lars-Erik Andersson

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 Read more...

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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.

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