Planar Pythagorean-hodograph B-spline curves. (English) Zbl 1379.65011
Summary: We introduce a new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bézier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B-Spline curves are non-uniform parametric B-Spline curves whose arc length is a B-Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B-Spline curves are non-uniform rational B-Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B-Spline curves have fewer degrees of freedom than general B-Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we present useful formulae for their construction and discuss their remarkable attractive properties. Then we solve the reverse engineering problem consisting of determining the complex pre-image spline of a given PH B-Spline, and we also provide a method to determine within the set of all PH B-Splines the one that is closest to a given reference spline having the same degree and knot partition.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 65D07 | Numerical computation using splines |
Keywords:
plane curve; non-uniform B-spline; Pythagorean-hodograph; arc-length; offset; reverse engineering; non-uniform rational B-spline curves; computer-aided design; computer graphicsReferences:
| [1] | Albrecht, G.; Beccari, C. V.; Canonne, J.-Ch.; Romani, L., Pythagorean hodograph B-spline curves (2016) |
| [2] | Albrecht, G.; Farouki, R. T., Construction of \(C^2\) Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math., 5, 417-442 (1996) · Zbl 0866.65008 |
| [3] | Che, X.; Farin, G.; Gao, Z.; Hansford, D., The product of two B-spline functions, (Adv. Mater. Res., vol. 186 (2011)), 445-448 |
| [4] | de Boor, C., A Practical Guide to Splines, Appl. Math. Sci., vol. 27 (2001), Springer: Springer New York, Berlin · Zbl 0987.65015 |
| [5] | Farouki, R. T., The conformal map \(z \to z^2\) of the hodograph plane, Comput. Aided Geom. Des., 11, 363-390 (1994) · Zbl 0806.65005 |
| [6] | Farouki, R. T.; Giannelli, C.; Sestini, A., Identification and “reverse engineering” of Pythagorean-hodograph curves, Comput. Aided Geom. Des., 34, 21-36 (2015) · Zbl 1375.65022 |
| [7] | Farouki, R. T.; Giannelli, C.; Sestini, A., Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form, Adv. Comput. Math., 42, 1, 199-225 (2016) · Zbl 1334.65034 |
| [8] | Farouki, R. T.; Kuspa, B. K.; Manni, C.; Sestini, A., Efficient solution of the complex quadratic tridiagonal system for \(C^2\) PH quintic splines, Numer. Algorithms, 27, 35-60 (2001) · Zbl 0987.65016 |
| [9] | Farouki, R. T.; Neff, C. A., Hermite interpolation by Pythagorean hodograph quintics, Math. Comput., 64, 212, 1589-1609 (1995) · Zbl 0847.68125 |
| [10] | Farouki, R. T.; Sakkalis, T., Pythagorean hodographs, IBM J. Res. Dev., 34, 736-752 (1990) |
| [11] | Gallier, J., Curves and Surfaces in Geometric Modeling (2000), Morgan Kaufmann Publishers: Morgan Kaufmann Publishers San Francisco, CA |
| [12] | Hoschek, J.; Lasser, D., Fundamentals of Computer Aided Geometric Design (1996), A K Peters: A K Peters Wellesley, MA |
| [13] | Joyce, D., Introduction to Modern Algebra (2008), Clark University, Version 0.0.6, 3 Oct 2008 |
| [14] | Kubota, K. K., Pythagorean triples in unique factorization domains, Math. Mon., 79, 503-505 (1972) · Zbl 0242.10008 |
| [15] | Martin, R. S.; Wilkinson, J. H., Symmetric decomposition of positive definite band matrices, Numer. Math., 7, 355-361 (1965) · Zbl 0137.32806 |
| [16] | Mørken, K., Some identities for products and degree raising of splines, Constr. Approx., 7, 195-208 (1991) · Zbl 0732.41004 |
| [17] | Pelosi, F.; Sampoli, M. L.; Farouki, R. T.; Manni, C., A control polygon scheme for design of planar \(C^2\) PH quintic spline curves, Comput. Aided Geom. Des., 24, 28-52 (2007) · Zbl 1171.65347 |
| [18] | Romani, L.; Saini, L.; Albrecht, G., Algebraic-trigonometric Pythagorean-hodograph curves and their use for Hermite interpolation, Adv. Comput. Math., 40, 5-6, 977-1010 (2014) · Zbl 1310.65020 |
| [19] | Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions, Q. Appl. Math., 4, 45-99 (1946), and 112-141 · Zbl 0061.28804 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
