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On Approximating Contours of the Piecewise Trilinear Interpolant Using Triangular Rational-Quadratic Bézier Patches
July-September 1997 (vol. 3 no. 3)
pp. 215-227

Abstract—Given a three-dimensional (3D) array of function values Fi, j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation ${\cal T}$ approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C0-continuous surface consisting of rational-quadratic surface patches interpolating the triangles in ${\cal T}.$ We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.

[1] C.L. Bajaj, J. Chen, and G. Xu, "Modeling with Cubic A-Patches," ACM Trans. Graphics, vol. 14, no. 2, pp. 103-133, 1995.
[2] W. Boehm and D.C. Hansford, "Bézier Patches on Quadrics," NURBS for Curve and Surface Design, G. Farin, ed., pp. 1-14.Philadelphia, Penn: SIAM, 1991.
[3] W. Boehm and D.C. Hansford, "Parametric Representation of Quadric Surfaces, Modélisation Mathématique et Analysis Numérique, vol. 26, pp. 191-200, 1991.
[4] W. Boehm and H. Prautzsch, Geometric Concepts for Geometric Design.Wellesley, Mass: A.K. Peters, Ltd., 1994.
[5] B. K. Choi, H.Y. Shin, Y.I. Yoon, and J.W. Lee, "Triangulation of Scattered Data in 3D Space," Computer-Aided Design, vol. 20, pp. 239-248, 1988.
[6] J.H. Chuang and C.M. Hoffmann, "Curvature Computations on Surfaces in n-Space," Technical Report #90.2, The Leonardo Fibonacci Inst. for the Foundations of Computer Science, Trento, Italy, 1990.
[7] J.H. Chuang and C.M. Hoffmann, "Curvature Computations on Surfaces in n-Space," Modélisation Mathématique et Analysis Numérique, vol. 26, pp. 95-112, 1992.
[8] W. Dahmen, "Smooth Piecewise Quadric Surfaces," Mathematical Methods in Computer Aided Geometric Design, T. Lyche, and L.L. Schumaker, eds., pp. 181-193.Boston: Academic Press, 1989.
[9] G.E. Farin, “Triangular Bernstein-Bézier Patches,” Computer Aided Geometric Design, pp. 83-127, Mar. 1986.
[10] G. Farin, NURB Curves and Surfaces.Wellesley, Mass.: A.K. Peters, Ltd., 1995.
[11] G. Farin, Curves and Surfaces for CAGD, 4th ed. Boston: Academic Press, 1997.
[12] C.M. Grimm and J.F. Hughes, Smooth Isosurface Approximation, B. Wyvill, and M.P. Gascuel, eds. Implicit Surfaces '95, Eurographics workshop, pp. 57-76.U.K.: Blackwell Publishers, 1995.
[13] B. Hamann, "Modeling Contours of Trivariate Data, Modélisation Mathématique et Analysis Numérique, vol. 26, pp. 51-75, 1992.
[14] B. Hamann, "A Data Reduction Scheme for Triangulated Surfaces," Computer Aided Geometric Design, vol. 11, no. 2, pp. 197-214 1994.
[15] B. Hamann, G. Farin, and G.M. Nielson, "A Parametric Triangular Patch Based on Generalized Conics," NURBS for Curve and Surface Design, G. Farin, ed., pp. 75-85.Philadelphia, Penn.: SIAM, 1991.
[16] D.C. Hansford, "Boundary Curves with Quadric Precision for a Tangent Continuous Scattered Data Interpolant," PhD dissertation, Arizona State Univ., Tempe, AZ, 1991.
[17] D.C. Hansford, R.E. Barnhill, and G. Farin, "Curves with Quadric Boundary Precision," Computer Aided Geometric Design, vol. 11, pp. 519-531, 1994.
[18] C.L. Lawson, "Software for C1Surface Interpolation," Mathematical Software III, J.R. Rice, ed., pp. 161-194.San Diego, Calif: Academic Press, 1977.
[19] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Computer Graphics (SIGGRAPH '87 Proc.), vol. 21, pp. 163-169, 1987.
[20] J. Niebuhr, "B-Patches on Quadrics (in German)," PhD dissertation, Technische Universität Braunschweig, Germany, 1991.
[21] G.M. Nielson and B. Hamann, The Asymptotic Decider: Removing the Ambiguity in Marching Cubes Proc. Visualization '91, pp. 83-91, 1991.
[22] L. Piegl and W. Tiller, The NURBS Book.New York: Springer-Verlag, 1995.
[23] T. Schreiber, "Arithmetische Operationen auf Bézierflächen," Internal Report 224/92, Fachbereich Informatik, Technische Universität Kaiserslautern, Germany, 1992.
[24] L.L. Schumaker, "Computing Optimal Triangulations Using Simulated Annealing," Computer Aided Geometric Design, vol. 10, pp. 329-345, 1993.
[25] T.W. Sederberg, "Piecewise Algebraic Surface Patches," Computer Aided Geometric Design, vol. 2, nos. 1-3, pp. 53-59, 1985.

Index Terms:
Approximation, contour, isosurface, marching cubes, rational Bézier curve, rational Bézier surface, triangular patch, triangulation, trilinear interpolation, visualization.
Citation:
Bernd Hamann, Issac J. Trotts, Gerald E. Farin, "On Approximating Contours of the Piecewise Trilinear Interpolant Using Triangular Rational-Quadratic Bézier Patches," IEEE Transactions on Visualization and Computer Graphics, vol. 3, no. 3, pp. 215-227, July-Sept. 1997, doi:10.1109/2945.620489
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