Name
Abstract | ||
|---|---|---|
We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond
geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape
flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space
curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any
segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations. |
Year | DOI | Venue |
|---|---|---|
1994 | 10.1007/BF02519035 | Adv. Comput. Math. |
Keywords | Field | DocType |
Space curve,Pythagorean hodograph,helix,arc length,canal surface | Polygon,Family of curves,Polynomial,Mathematical analysis,Numerical integration,Arc length,PSPACE,Bézier curve,Hermite interpolation,Mathematics | Journal |
Volume | Issue | Citations |
2 | 1 | 55 |
PageRank | References | Authors |
5.36 | 7 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Rida T. Farouki | 1 | 1396 | 137.40 |
| Takis Sakkalis | 2 | 347 | 34.52 |