A Laguerre geometric approach to rational offsets. (English) Zbl 0903.68190

Summary: Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.


68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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[1] Ait Haddou, R.; Biard, L., \(G^2\) approximation of an offset curve by Tschirnhausen quartics, (Daehlen, M.; Lyche, T.; Schumaker, L. L., Mathematical Methods for Curves and Surfaces (1995), Vanderbilt University Press), 1-10 · Zbl 0835.65016
[2] Albrecht, G.; Farouki, R. T., Construction of \(C^2\) Pythagorean-hodograph interpolating splines by the homotopy method, (Advances in Comp. Math. (1996)), to appear · Zbl 0866.65008
[3] Aumann, G., Curvature continuous connections of cones and cylinders, Computer-Aided Design, 27, 293-310 (1995) · Zbl 0830.65008
[4] Blaschke, W., Untersuchungen über die Geometrie der Speere in der Euklidischen Ebene, Monatshefte für Mathematik und Physik, 21, 3-60 (1910)
[5] Blaschke, W., Vorlesungen über differentialgeometrie III (1929), Springer: Springer Berlin
[6] Brezinski, C., An Introduction to Padé Approximations, (Laurent, P. J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces II (1993), A K Peters: A K Peters Boston), 1-10 · Zbl 0835.41021
[7] Cecil, T. E., Lie Sphere Geometry (1992), Springer: Springer Berlin · Zbl 0752.53003
[8] DeRose, T. D., Rational Bézier curves and surfaces on projective domains, (Farin, G., NURBS for Curve and Surface Design (1991), SIAM: SIAM Philadelphia), 35-45 · Zbl 0760.68088
[9] Dietz, R., Rationale Bézier-Kurven und Bézier-Flächenstücke auf Quadriken, dissertation (1995), TH Darmstadt
[10] Dietz, R.; Hoschek, J.; Jüttler, B., An algebraic approach to curves and surfaces on the sphere and on other quadrics, Computer Aided Geometric Design, 10, 211-229 (1993) · Zbl 0781.65009
[11] Farouki, R. T., Pythagorean-hodograph curves in practical use, (Barnhill, R. E., Geometry Processing for Design and Manufacturing (1992), SIAM: SIAM Philadelphia), 3-33 · Zbl 0770.41017
[12] Farouki, R. T., The conformal map \(z\) ↦ \(z^2\) of the hodograph plane, Computer Aided Geometric Design, 11, 363-390 (1994) · Zbl 0806.65005
[13] Farouki, R. T.; Neff, C. A., Hermite interpolation by Pythagorean-hodograph quintics, Math. Comp., 64, 1589-1609 (1995) · Zbl 0847.68125
[14] Farouki, R. T.; Sakkalis, T., Pythagorean hodographs, IBM J. Res. Develop., 34, 736-752 (1990)
[15] Farouki, R. T.; Sakkalis, T., Pythagorean-hodograph space curves, Advances in Comp. Math., 2, 41-66 (1994) · Zbl 0829.65011
[16] Farouki, R. T., Offset curves in layered manufacturing, ASME PED, Vol. 68-2, 557-568 (1994), Manufacturing Science and Engineering Book No. G0930B
[17] Fiorot, J. C.; Gensane, T., Characterization of the set of rational parametric curves with rational offsets, (Laurent, P. J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces in Geometric Design (1994), A K Peters: A K Peters Wellesley, MA), 153-160 · Zbl 0813.65032
[18] Hoffmann, Ch. M.; Peters, J., Geometric Constraints for CAGD, (Daehlen, M.; Lyche, T.; Schumaker, L. L., Mathematical Methods in CAGD III (1995)), 1-16 · Zbl 0835.65028
[19] Jüttler, B.; Wagner, M., Computer aided design with spatial rational B-spline motions, ASME J. of Mechanical Design, 118, 193-201 (1996)
[20] Lü, W., Rationality of the offsets to algebraic curves and surfaces, Applied Mathematics, 9, 265-278 (1994), (Ser. B) · Zbl 0814.14048
[21] Lü, W., Offset-rational parametric plane curves, Computer Aided geometric Design, 12, 601-616 (1995) · Zbl 0875.68853
[22] Lü, W., Rational parameterization of quadrics and their offsets, Computing, 57, 135-147 (1996) · Zbl 0855.68113
[23] Lü, W.; Pottmann, H., Pipe surfaces with rational spine curve are rational, Computer Aided Geometric Design, 13, 621-628 (1996) · Zbl 0900.68410
[24] Martin, R. R., Principal patches for computational geometry, (PhD Thesis (1982), Cambridge University)
[25] Peternell, M.; Pottmann, H., Designing rational surfaces with rational offsets, (Fontanella, F.; Jetter, K.; Laurent, P. J., Advanced Topics in Multivariate Approximation (1996)), 275-286, World Scientific · Zbl 1273.65028
[26] Peternell, M.; Pottmann, H., Computing rational parametrizations of canal surfaces, J. Symbolic Computation, 23, 255-266 (1997) · Zbl 0877.68116
[27] Pottmann, H., Applications of the dual Bézier representation of rational curves and surfaces, (Laurent, P. J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces in Geometric Design (1994), A K Peters: A K Peters Wellesley, MA), 377-384 · Zbl 0813.65047
[28] Pottmann, H., Rational curves and surfaces with rational offsets, Computer Aided Geometric Design, 12, 175-192 (1995) · Zbl 0872.65011
[29] Pottmann, H., Curve design with rational Pythagorean-hodograph curves, Advances in Comp. Math., 3, 147-170 (1995) · Zbl 0831.65013
[30] Pottmann, H.; Farin, G., Developable rational Bézier and B-spline surfaces, Computer Aided Geometric Design, 12, 513-531 (1995) · Zbl 0875.68847
[31] Pottmann, H.; Lü, W.; Ravani, B., Rational ruled surfaces and their offsets, Graphical Models and Image Processing, 58, 544-552 (1996)
[32] Pottmann, H.; Peternell, M., Applications of Laguerre geometry in CAGD, Computer Aided Geometric Design (1997), to appear
[33] Pottmann, H.; Wagner, M., Principal Surfaces, (Goodman, T. N.T, The Mathematics of Surfaces VII (1997), Information Geometers Ltd), to appear · Zbl 0982.53005
[34] Tschirnhaus, W.von, Methodus curvas determinandi, quae formantur a radiis reflectis, quorum incidentes ut paralleli considerantur, (Acta Eruditorum (1690)), 68-75, Febr. 1690
[35] Wagner, M.; Ravani, B., Computer aided design of robot trajectories using rational Frenet-Serret motions, (Lenarcic, J.; Parenti-Castelli, V., Recent Advances in Robot Kinematics (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 151-158
[36] Wallner, J.; Pottmann, H., Rational blending surfaces between quadrics, Computer Aided Geometric Design (1997), to appear · Zbl 0896.65010
[37] Wunderlich, W., Spiegelung am elliptischen Paraboloid, Monatsh. Math., 52, 13-37 (1948) · Zbl 0031.07101
[38] Wunderlich, W., Algebraische Böschungslinien dritter und vierter Ordnung, Sitzungsberichte der Österreichischen Akademie der Wissenschaften, 181, 353-376 (1973) · Zbl 0273.53003
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