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Problems with defining barycentric coordinates for the sphere. (English) Zbl 0748.65014

Barycentric coordinates \((b_ 1,b_ 2,b_ 3)\) in a triangle of the plane have the following properties: 1) the coordinates add up to one, 2) on the edges of the triangle the coordinate is the ratio of geodesic lengths. Coordinates are consistent in the sense that the coordinates with respect to a subtriangle \(T_ 1\subset T\) can be calculated from the coordinates with respect to \(T\) and the coordinates of the vertices of \(T_ 1\) with respect to \(T\). The authors show that it is impossible to define on all geodesic triangles in an open subset of the sphere barycentric coordinates which are consistent and have properties 1) and 2).
As an alternative one may use a triangulation which is obtained by an area preserving mapping from the plane to the sphere. The triangles so obtained are not geodesic. It is shown by examples that the distortion may become unacceptably large.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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References:

[1] R. E. BARNHILL (1985), Surfaces in computer aided Geometrie design : A survey with new results, Comput. Aided Geom. Design, 2, pp. 1-17. Zbl0597.65001 MR828527 · Zbl 0597.65001 · doi:10.1016/0167-8396(85)90002-0
[2] R. E. BARNHILL and H. S. OU (1990), Surfaces defined on surfaces, Comput. Aided Geom. Design, 7, pp. 323-336. Zbl0726.65013 MR1074618 · Zbl 0726.65013 · doi:10.1016/0167-8396(90)90040-X
[3] M. BERGER (1987), Geometry II, Springer-Verlag. Zbl0606.51001 MR882916 · Zbl 0606.51001
[4] W. BOEHM, G. FARIN and J. KAHMANN (1984), A survey of curve and surface methods in CAGD, Comput. Aided Geom. Design 1, pp. 1-60. Zbl0604.65005 · Zbl 0604.65005 · doi:10.1016/0167-8396(84)90003-7
[5] W. BOEHM (1980), Inserting new knots into B-spline curves, Comput. Aided Design 12, 199-201.
[6] G. FARIN (1986), Triangular Bernstein-Bézier patches, Comput. Aided Geom.Design 3, 83-128. MR867116
[7] G. FARIN (1989), Curves and Surfaces for Computer Aided Geometrie Design, Academic Press. Zbl0694.68004 · Zbl 0694.68004
[8] T. A. FOLEY (1990a), Surfaces on surfaces, presentation at the Geometrie design meeting, Ettore Majorana, Erice, Sicily, May 1990.
[9] T. A. FOLEY (1990b), Interpolation to scattered data on a spherical domain, in : M. Cox and J. Mason, eds., Algorithms for Approximation II, Chapman & Hall, London, pp. 303-310. Zbl0749.41003 MR1071988 · Zbl 0749.41003
[10] C. L. LAWSON (1984), C1 Surface interpolation for scattered data on a sphere, Rocky Mountain J. Math. 14, pp. 177-202. Zbl0579.65008 MR736173 · Zbl 0579.65008 · doi:10.1216/RMJ-1984-14-1-177
[11] G. M. NIELSON and R. RAMARAJ (1987), Interpolation over a sphere based upon a minimum norm network, Comput. Aided Geom. Design, 4, pp. 41-57. Zbl0632.65010 MR898022 · Zbl 0632.65010 · doi:10.1016/0167-8396(87)90023-9
[12] G. M. NIELSON (1989), Interpolation on the sphere, presentation at the Computer Aided Geometrie Design meeting, Oberwolfach, West Germany, April 1989.
[13] H. POTTMANN and M. ECK (1990), Modified multiquadric methods for scattered data interpolation over a sphere, Comput. Aided Geom. Design, 7, pp. 313-322. Zbl0714.65010 MR1074617 · Zbl 0714.65010 · doi:10.1016/0167-8396(90)90039-T
[14] R. L. RENKA (1984), Interpolation of data on the surface of a sphere, ACM Trans. Math. Software, pp. 417-436. Zbl0548.65001 MR792004 · Zbl 0548.65001 · doi:10.1145/2701.2703
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