Abstract
The problem of interpolation by a convex curve to the vertices of a convex polygon is considered. A natural 1-parameter family ofC ∞ algebraic curves solving this problem is presented. This is extended to a solution, of a general Hermite-type problem, in, which the curve also interpolates to one or two prescribedtangents at any desired vertices of the polygon. The construction of these curves is a generalization of well known methods for generatingconic sections. Several properties of this family of algebraic curves are discussed. In addition, the method is generalized to convexC ∞ interpolation of strictly convex data sets inR 3 by algebraicsurfaces.
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References
C.L. Bajaj, Geometric modelling with algebraic surfaces, inThe Mathematics of Surfaces III, D.C. Handscomb (ed.) (Oxford University Press, 1989) pp. 3–48.
C.L. Bajaj, Surface fitting using implicit algebraic surface patches, inTopics in Surface Modeling, H. Hagen (ed.) (SIAM, 1992) pp. 23–52.
C.L. Bajaj, and G. Xu, A-Splines: Local interpolation and approximation usingC k-continuous piecewise real algebraic curves, Purdue University, Computer Sciences Department, CSD-TR-92-095 (1992).
E. Brieskorn, and H. Knörrer,Plane Algebraic Curves (Birkhäuser, 1986).
B.E.J. Dahlberg, and B. Johansson, Shape preserving approximation, inThe Mathematics of Surfaces, R.R. Martin (ed.) (Oxford-Clarendon Press, 1987) pp. 418–426.
N. Dyn, D. Levin and D. Liu, Interpolatory convexity preserving subdivision schemes for curves and surfaces, CAD 24 (1992) 211–216.
F. Fontanella, Shape preserving surface interpolation, inTopics in Multivariate Approximation, C.K. Chui, L.L. Schumaker and F.I. Utreras (eds.) (Academic Press, 1987) pp. 63–78.
C. Hoffmann, and J. Hopcroft, The potential method for blending surfaces and corners, inGeometric Modeling: Algorithms and New Trends, G. Farin (ed.) (SIAM, 1987) pp. 347–365.
J.K. Johnstone, and C.L. Bajaj, Sorting points along an algebraic curve, SIAM J. Comput. 19 (1990) 925–967.
J.K. Johnstone, The sorting of points along an algebraic curve, Ph.D. Thesis, Cornell University (1987).
R.A. Liming,Practical Analytical Geometry with Applications to Aircraft (McMillan, 1944).
J. Li, J. Hoschek and E. Hartmann,G n-1-functional splines for interpolation and approximation of curves, surfaces and solids, Comp. Aided Geom. Design 7 (1990) 209–220.
A.E. Middleditch, and K.H. Sears, Blend surfaces for set theoretic volume modelling systems, Comp. Graphics 19 (1985) 161–170.
M. Paluszny, and R.R. Patterson, Curvature continuous cubic algebraic splines, inCurves and Surfaces in Computer Vision and Graphics III, Proceedings SPIE 1830 (1992) pp. 48–57.
R.R. Patterson, and J. Sarkar, private communication (April 1992).
T.W. Sederberg, Piecewise algebraic curves, Comp. Aided Geom. Design 1 (1984) 241–255.
T.W. Sederberg, Piecewise algebraic surface patches, Comp. Aided Geom. Design 2 (1985) 53–59.
T.W. Sederberg, Algebraic geometry for surface and solid modeling, inGeometric Modeling: Algorithms and New Trends, G. Farin (ed.) (SIAM, 1987) pp. 29–42.
R.J. Walker,Algebraic Curves (Princeton University Press, 1950).
J. Warren, Blending algebraic surfaces,. ACM Trans. Graphics 8 (1989) 263–278.
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Levin, D., Nadler, E. Convexity preserving interpolation by algebraic curves and surfaces. Numer Algor 9, 113–139 (1995). https://doi.org/10.1007/BF02143930
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DOI: https://doi.org/10.1007/BF02143930