Download PDF by Jörg Peters: Subdivision surfaces

By Jörg Peters

ISBN-10: 3540764054

ISBN-13: 9783540764052

Since their first visual appeal in 1974, subdivision algorithms for producing surfaces of arbitrary topology have won common attractiveness in special effects and are being evaluated in engineering purposes. This improvement used to be complemented by way of ongoing efforts to boost acceptable mathematical instruments for a radical research, and at the present time, some of the attention-grabbing homes of subdivision are good understood.

This e-book summarizes the present wisdom at the topic. It includes either in the meantime classical effects in addition to brand-new, unpublished fabric, akin to a brand new framework for developing C^2-algorithms.

The concentration of the booklet is at the improvement of a finished mathematical conception, and not more on algorithmic points. it's meant to serve researchers and engineers - either new to the great thing about the topic - in addition to specialists, educational academics and graduate scholars or, briefly, anyone who's attracted to the rules of this flourishing department of utilized geometry.

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The abscissae separating its segments are called knots. Between two consecutive knots, the spline curve coincides with a function of a fixed type, say a polynomial of a certain degree. At knots, the spline has to satisfy smoothness conditions. To prepare our analysis of bivariate splines, and to motivate the setting to be developed then, we take an alternative view. 1 Further, I := Z is an index set enumerating segments xi : U → Rd , i ∈ I, of the spline curve x. The spline domain S := U × I consists of indexed copies (U, i) := U × {i} of the unit interval (Fig.

19 (Injectivity of an almost regular function). Let f ∈ C01 (C, C) be an almost regular function, and let z : U → C be a piecewise differentiable Jordan curve with ν(z, 0) = 1. • If |ν(f, z, 0)| = 1 then the restriction of f to a sufficiently small neighborhood Γ (0) of the origin is injective. • If |ν(f, z, 0)| = 1 then the restriction of f to any neighborhood of the origin is not injective. 5 1 Fig. 18/34: The given curve z = z 0 is transformed into the target curve z 1 , which separates z0 from z1 , .

In Sect. 3/44, C k -splines are defined in terms of standard differentiability properties of joint patches, which are obtained by embedding neighboring cells of the domain into R2 in a specific way. This approach turns out to be equivalent to the familiar characterization of C k -splines via the agreement of transversal derivatives along patch boundaries. Of course, in the case of spline surfaces, the analytical definition fails to yield geometric information at points where the parametrization is singular.

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Subdivision surfaces by Jörg Peters

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