Hardware Assist For Constrained Circle Constructions I: Sequential Problems
Hardware Assist For Constrained Circle Constructions I: Sequential Problems |
Abstract
In geometric constraint solving, constructing circles with indeterminate radius is an important sub problem. Such constructions are both sequential, meaning that we seek a circle tangent to three known geometric entities, as well as simultaneous, when several sets of entities, among them variable-radius circles, must be determined together. In Part I, we investigate techniques to solve sequential construction problems of variable-radius circles, analyzing the case when at least one of the constraining entities is a B?©zier curve. We consider first an algebraic solution in which we restrict to Pythagorean hodographs (PH). However, the polynomial degrees become very large, rendering this approach impractical. So, we develop an approach in which the needed computations are assisted by graphics hardware commonly available on PCs and laptops. Here we achieve greater generality, allowing arbitrary curves, greater numerical stability, and extreme speed-ups.
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Citation
Ching-Shoei Chiang, C Hoffmann, and Paul Rosen. Hardware Assist For Constrained Circle Constructions I: Sequential Problems. Computer-Aided Design and Applications, 2010.
Bibtex
@article{chiang2010hardware1, title = {Hardware Assist for Constrained Circle Constructions I: Sequential Problems}, author = {Chiang, Ching-Shoei and Hoffmann, C and Rosen, Paul}, journal = {Computer-Aided Design and Applications}, volume = {7}, pages = {17--33}, year = {2010}, keywords = {geometric constraint solving, variable-radius circles, cyclographic maps, PH curves, equation solving, hardware acceleration, GPU}, abstract = {In geometric constraint solving, constructing circles with indeterminate radius is an important sub problem. Such constructions are both sequential, meaning that we seek a circle tangent to three known geometric entities, as well as simultaneous, when several sets of entities, among them variable-radius circles, must be determined together. In Part I, we investigate techniques to solve sequential construction problems of variable-radius circles, analyzing the case when at least one of the constraining entities is a B?©zier curve. We consider first an algebraic solution in which we restrict to Pythagorean hodographs (PH). However, the polynomial degrees become very large, rendering this approach impractical. So, we develop an approach in which the needed computations are assisted by graphics hardware commonly available on PCs and laptops. Here we achieve greater generality, allowing arbitrary curves, greater numerical stability, and extreme speed-ups. } }