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Linear precision for parametric patches. (English) Zbl 1193.65018
Summary: We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for {\it R. Krasauskas}’ toric patches [Adv. Comput. Math. 17, No. 1--2, 89--113 (2002; Zbl 0997.65027)], which we show is equivalent to a certain rational map on ${\mathbb C}{\mathbb P}^d$ being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.

65D17Computer aided design (modeling of curves and surfaces)
Full Text: DOI arXiv
[1] Catanese, F., Hoşten, S., Khetan, A., Sturmfels, B.: The maximum likelihood degree. Amer. J. Math. 128(3), 671--697 (2006) · Zbl 1123.13019 · doi:10.1353/ajm.2006.0019
[2] Clemens, C.H., Griffiths, P.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95(2), 281--356 (1972) · Zbl 0231.14004 · doi:10.2307/1970801
[3] Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. UTM. Springer, New York (1992) · Zbl 0756.13017
[4] Cox, D.: What is a toric variety? In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. Math., vol. 334, pp. 203--223. American Mathematical Society, Providence (2003) · Zbl 1038.14021
[5] Darroch, J.N., Ratcliff, D.: Generalized iterative scaling for log-linear models. Ann. Math. Stat. 43, 1470--1480 (1972) · Zbl 0251.62020 · doi:10.1214/aoms/1177692379
[6] DeRose, T., Goldman, R., Hagen, H., Mann, S.: Functional composition algorithms via blossoming. ACM Trans. Graph. 12, 113--135 (1993) · Zbl 0771.68100 · doi:10.1145/151280.151290
[7] DeRose, T.D.: Rational Bézier curves and surfaces on projective domains. In: NURBS for Curve and Surface Design (Tempe, AZ, 1990), pp. 35--45. SIAM, Philadelphia (1991) · Zbl 0760.68088
[8] Farin, G.: Curves and surfaces for computer-aided geometric design. In: Computer Science and Scientific Computing. Academic, San Diego (1997) · Zbl 0919.68120
[9] Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Design 20(1), 19--27 (2003) · Zbl 1069.65553 · doi:10.1016/S0167-8396(03)00002-5
[10] Fulton, W.: Introduction to toric varieties. In: Annals of Mathematics Studies. The William H. Roever Lectures in Geometry, vol. 131. Princeton University Press, Princeton (1993) · Zbl 0813.14039
[11] Karčiauskas, K.: Rational M-patches and tensor-border patches. In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. Math., vol. 334, pp. 101--128. American Mathematical Society, Providence (2003) · Zbl 1045.65013
[12] Karčiauskas, K., Krasauskas, R.: Comparison of different multisided patches using algebraic geometry. In: Laurent, P.-J., Sablonniere, P. Schumaker, L.L. (eds.) Curve and Surface Design: Saint-Malo 1999, pp. 163--172. Vanderbilt University Press, Nashville (2000)
[13] Krasauskas, R.: Toric surface patches. Advances in geometrical algorithms and representations. Adv. Comput. Math. 17(1--2), 89--133 (2002) · Zbl 0997.65027 · doi:10.1023/A:1015289823859
[14] Krasauskas, R.: Bézier patches on almost toric surfaces. In: Algebraic Geometry and Geometric Modeling. Math. Vis., pp. 135--150. Springer, Berlin (2006) · Zbl 1110.14307
[15] Lauritzen, S.L.: Graphical models. Oxford Statistical Science Series, vol. 17. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1996)
[16] Loop, C.T., DeRose, T.D.: A multisided generalization of Bézier surfaces. ACM Trans. Graph. 8(3), 204--234 (1989) · Zbl 0746.68097 · doi:10.1145/77055.77059
[17] Pachter, L., Sturmfels, B. (eds.): Algebraic Statistics for Computational Biology. Cambridge University Press, New York (2005) · Zbl 1108.62118
[18] Sottile, F.: Toric ideals, real toric varieties, and the moment map. In: Topics in Algebraic Geometry and Geometric Modeling. Contemp. Math., vol. 334, pp. 225--240. American Mathematical Society, Providence (2003) · Zbl 1051.14059
[19] Sturmfels, B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence (1996) · Zbl 0856.13020
[20] von Bothmer, H.-C.G., Ranestad, K., Sottile, F.: Linear precision for toric surface patches, 25 pp. ArXiv.org/0806.3230 (submitted)
[21] Wachspress, E.L.: A rational finite element basis. In: Mathematics in Science and Engineering, vol. 114. Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975) · Zbl 0322.65001
[22] Warren, J.: Creating multisided rational Bézier surfaces using base points. ACM Trans. Graph. 11(2), 127--139 (1992) · Zbl 0760.65019 · doi:10.1145/130826.130828
[23] Warren, J.: Barycentric coordinates for convex polytopes. Adv. Comput. Math. 6(2), 97--108 (1996, 1997) · Zbl 0873.52013 · doi:10.1007/BF02127699