Download Computer Graphics and Geometric Modeling Using Beta-splines by Brian A. Barsky PDF

By Brian A. Barsky

Special effects and Geometric Modeling utilizing Beta-splines (Computer technological know-how Workbench) [Hardcover] [May 03, 1988]

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Observe that a more efficient algorithm would be possible if all curve segments were evaluated at the same set of values of the domain parameter. Note that this does not impose any restriction on the selection of a particular set of values of the domain parameter. The following algorithm requires 7 + (p + 1)(9 + 4dm) multiplications, 14 + 3(p + 1)(3 + dm) additionsjsubtractions, and 8 divisions to evaluate the curve at p + 1 values of the domain parameter on each of the m segments. compute_d (beta1, beta2, d); compute_delta (d, delta); compute_c (delta, d, c); for each u in {u 0 , u 1 , ...

3) 6 IX1 2 Q~l(1) = _ _ m[Vm- Vm- 1] Ym An important property of an end condition is the curvature at the endpoints of the curve. It is sometimes desirable to have zero curvature at an endpoint to join the curve to a straight line, while other applications require nonzero curvature. 4, and required the evaluation of the first and second derivative vectors.

12) Hence the segments join with a continuous parametric second derivative vector even though the curvature vector is discontinuous since it jumps from [ -1/a, 0] for Q7 (u) to [1/b, 0] for Q8 (u). 1 Unit Tangent Vector Given the two curves Q1 (u) and Q 2(u), consider the joint Q1 (u). 3). 4) It is easy to see that a sufficient solution is Q~2l(O) = p12Q\2>(1) . 5) More generally, Q~2 >(Q) can also include an additional term which is a multiple of 6 Geometrie Continuity and Shape Parameters 28 Q\1)(1): Q~2l(O) = ß12Q\2>(1) + ß2Q\1>(1) .

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