Hermite interpolation by Pythagorean hodograph curves of degree seven

Jüttler, B.

doi:10.1090/S0025-5718-00-01288-6

Hermite interpolation by Pythagorean hodograph curves of degree seven
HTML articles powered by AMS MathViewer

by B. Jüttler PDF

Math. Comp. 70 (2001), 1089-1111 Request permission

Abstract:

Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on $C^1$ Hermite interpolation with PH quintics. We extend the $C^1$ Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of $G^2[C^1]$ boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials. With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize $\Delta$. It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize $\Delta$, provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be $G^2$ (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bézier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations. It produces $G^2$ polynomial PH spline curves of degree 7.

References

Gudrun Albrecht and Rida T. Farouki, Construction of $C^2$ Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math. 5 (1996), no. 4, 417–442. MR 1414289, DOI 10.1007/BF02124754
Carl de Boor, Klaus Höllig, and Malcolm Sabin, High accuracy geometric Hermite interpolation, Comput. Aided Geom. Design 4 (1987), no. 4, 269–278. MR 937366, DOI 10.1016/0167-8396(87)90002-1
I. N. Bronshtein and K. A. Semendyayev, Handbook of mathematics, Reprint of the third (1985) English edition, Springer-Verlag, Berlin, 1997. Translated from the German; Translation edited by K. A. Hirsch. MR 1450092
G. Elber, I.-K. Lee, and M.-S. Kim, Comparing Offset Curve Approximation Methods. IEEE Comp. Graphics and Appl. 17 (1998), 62–71.
Gerald Farin, Curves and surfaces for computer aided geometric design, 3rd ed., Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1993. A practical guide; With 1 IBM-PC floppy disk (5.25 inch; DD). MR 1201325
R. T. Farouki and C. A. Neff, Hermite interpolation by Pythagorean hodograph quintics, Math. Comp. 64 (1995), no. 212, 1589–1609. MR 1308452, DOI 10.1090/S0025-5718-1995-1308452-6
Rida T. Farouki and Takis Sakkalis, Real rational curves are not “unit speed”, Comput. Aided Geom. Design 8 (1991), no. 2, 151–157. MR 1107850, DOI 10.1016/0167-8396(91)90040-I
R.T. Farouki and S. Shah, Real-time CNC interpolators for Pythagorean-hodograph curves. Comput. Aided Geom. Des. 13 (1996), 583-600.
R.T. Farouki, Y.-F. Tsai and G.-F. Yuan, Contour machining of free form surfaces with real–time PH curve CNC interpolators. Comput. Aided Geom. Des. 16 (1999), 61-76.
Josef Hoschek and Dieter Lasser, Fundamentals of computer aided geometric design, A K Peters, Ltd., Wellesley, MA, 1993. Translated from the 1992 German edition by Larry L. Schumaker. MR 1258308
Erwin Kreyszig, Differential geometry, Dover Publications, Inc., New York, 1991. Reprint of the 1963 edition. MR 1118149
K. K. Kubota, Pythagorean triples in unique factorization domains, Amer. Math. Monthly 79 (1972), 503–505. MR 297690, DOI 10.2307/2317570
D. S. Meek and D. J. Walton, Geometric Hermite interpolation with Tschirnhausen cubics, J. Comput. Appl. Math. 81 (1997), no. 2, 299–309. MR 1459031, DOI 10.1016/S0377-0427(97)00066-6
Knut Mørken and Karl Scherer, A general framework for high-accuracy parametric interpolation, Math. Comp. 66 (1997), no. 217, 237–260. MR 1372007, DOI 10.1090/S0025-5718-97-00796-5
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
Helmut Pottmann, Curve design with rational Pythagorean-hodograph curves, Adv. Comput. Math. 3 (1995), no. 1-2, 147–170. MR 1314907, DOI 10.1007/BF03028365
D. J. Walton and D. S. Meek, $G^2$ curves composed of planar cubic and Pythagorean hodograph quintic spirals, Comput. Aided Geom. Design 15 (1998), no. 6, 547–566. MR 1629818, DOI 10.1016/S0167-8396(97)00028-9