Jüttler, Bert; Wang, Wenping The shape of spherical quartics. (English) Zbl 1069.41505 Comput. Aided Geom. Des. 20, No. 8-9, 621-636 (2003). Summary: We discuss the problem of interpolating \(C^{1}\) Hermite data on the sphere (two points with associated first derivative vectors) by spherical rational curves. With the help of the generalized stereographic projection (Dietz et al., 1993), we construct a two-parameter family of spherical quartics solving this problem. We study the shape of these solutions and derive criteria which guarantee solutions without cusps or self-intersections. Cited in 1 Document MSC: 41A10 Approximation by polynomials 65D05 Numerical interpolation 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry Keywords:Hermite interpolation; spherical rational curves; generalized stereographic projection PDFBibTeX XMLCite \textit{B. Jüttler} and \textit{W. Wang}, Comput. Aided Geom. Des. 20, No. 8--9, 621--636 (2003; Zbl 1069.41505) Full Text: DOI References: [1] Choi, H. I.; Lee, D. S.; Moon, H. P., Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comput. Math., 17, 5-48 (2002) · Zbl 0998.65024 [2] Dickson, L. E., History of the Theory of Numbers, Vol. II (1952), Chelsea: Chelsea New York [3] Dietz, R.; Hoschek, J.; Jüttler, B., An algebraic approach to curves and surfaces on the sphere and on other quadrics, Computer Aided Geometry Design, 10, 211-229 (1993) · Zbl 0781.65009 [4] Farouki, R. T., Pythagorean-hodograph curves, (Farin, G.; Hoschek, J.; Kim, M.-S., Handbook of Computer Aided Geometric Design (2002), Elsevier: Elsevier Amsterdam), 405-428 [5] Farouki, R. T.; al-Kandari, M.; Sakkalis, T., Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves, Adv. Comput. Math., 17, 369-383 (2002) · Zbl 1001.41003 [6] Gfrerrer, A., Rational interpolation on a hypersphere, Computer Aided Geometric Design, 16, 21-37 (1999) · Zbl 0908.68177 [7] Hoschek, J.; Seemann, G., Spherical splines, RAIRO Modél. Math. Anal. Numér., 26, 1-22 (1992) · Zbl 0755.41011 [8] Jupp, P. E.; Kent, J. T., Fitting smooth paths to spherical data, J. Roy. Stat. Soc. Ser. C, 36, 34-46 (1987) · Zbl 0613.62086 [9] Jüttler, B.; Wagner, M. G., Kinematics and animation, (Farin, G.; Hoschek, J.; Kim, M.-S., Handbook of Computer Aided Geometric Design (2002), Elsevier: Elsevier Amsterdam), 723-748 [10] Kim, M.-S.; Nam, K.-W., Interpolating solid orientations with circular blending quaternion curves, Computer-Aided Design, 27, 385-398 (1995) · Zbl 0836.65158 [11] Kim, M.-J.; Kim, M.-S.; Shin, S., A general construction scheme for unit quaternion curves with simple high order derivatives, Computer Graphics (SIGGRAPH), 29, 369-376 (1995) [12] Nielson, G. M.; Heiland, R. W., Animated rotations using quaternions and splines on a 4D sphere, Program. Comput. Softw., 18, 145-154 (1992) · Zbl 0875.68913 [13] Noakes, L., Non-linear corner cutting, Adv. Comput. Math., 8, 165-177 (1998) · Zbl 0908.65008 [14] Pletinckx, D., Quaternion calculus as a basic tool in computer graphics, The Visual Comput., 5, 2-13 (1989) · Zbl 0668.65012 [15] Shoemake, K., Animating rotations with quaternion curves, Computer Graphics (SIGGRAPH), 19, 245-254 (1985) [16] Pottmann, H.; Wallner, J., Computational Line Geometry (2001), Springer: Springer Berlin · Zbl 1006.51015 [17] Röschel, O., Rational motion design—a survey, Computer-Aided Design, 30, 169-178 (1998) · Zbl 0906.68175 [18] Wang, W.; Joe, B., Orientation interpolation in quaternion space using spherical biarcs, (Proceedings of Graphics Interface ’93 (1993)), 24-32 [19] Wang, W.; Joe, B., Interpolation on quadric surfaces with rational quadratic spline curves, Computer Aided Geometric Design, 14, 207-230 (1997) · Zbl 0906.68160 [20] Wang, W.; Qin, K., Existence and computation of spherical rational quartic curves for Hermite interpolation, The Visual Comput., 16, 187-196 (2000) 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.