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Polynomial Surfaces Interpolating Arbitrary Triangulations
January-March 2003 (vol. 9 no. 1)
pp. 99-109

Abstract—Triangular Bézier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bézier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.

[1] R. Barnhill, J. Brown, and I. Klucewicz, “A New Twist in CAGD,” Computer Graphics and Image Processing, 1978.
[2] C. Bajaj, “Smoothing Polyhedra Using Implicit Algebraic Splines,” Computer Graphics, vol. 26, no. 2, pp. 79-88, 1992.
[3] P. Brunet, “Increasing the Smoothness of Bicubic Spline Surfaces,” Proc. Surfaces in CAGD '84, R. Barnhill and W. Böhm, eds., 1985.
[4] H.W. Du and F. Schmitt, CAD, vol. 22, no. 9, pp. 556-573, 1990.
[5] N. Dyn, D.F. Levin, and J.A. Gregory, “A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control,” ACM Trans. Graphics, vol. 9, pp. 160-169, 1990.
[6] G. Farin, “A Construction for Visual $\big. {\rm C}^1\bigr.$ Continuity of Polynomial Surface Patches,” Computer Graphics and Image Processing, vol. 20, pp. 272-282, 1982.
[7] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, fourth ed. New York: Academic Press, 1996.
[8] J.A. Gregory, “N-Sided Surface Patches,” The Math. of Surfaces, J. Gregory, ed., pp. 217-232, Oxford: Clarendon Press, 1986.
[9] H. Hagen, “Geometric Surface Patches without Twist Constraints,” Computer Aided Geometric Design, vol. 3, pp. 179-184, 1986.
[10] H. Hagen and G. Schulze, “Automatis Smoothing with Geometric Surface Patches,” Computer Aided Geometric Design, vol. 4, pp. 231-236, 1987.
[11] H. Hagen and H. Pottmann, “Curvature Continuous Triangular Interpolants,” Math. Methods in Computer Aided Geometric Design, T. Lyche and L.L. Schumaker, eds., pp. 373-384, New York: Academic Press, 1989.
[12] S. Hahmann and G.-P. Bonneau, “Triangular $\big. {\rm G}^1\bigr.$ Interpolation by 4-Splitting Domain Triangles,” Computer Aided Geometric Design, vol. 17, pp. 731-757, 2000.
[13] S. Hahmann, G.-P. Bonneau, and R. Taleb, “Smooth Irregular Mesh Interpolation,” Curve and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L.L. Schumaker, eds., pp. 237-246, Nashville: Vanderbilt Univ. Press, 2000.
[14] T. Jensen, “Assembling Triangular and Rectangular Patches and Multivariate Splines,” Geometric Modeling: Algorithms and New Trends, G. Farin, ed., pp. 203-220, SIAM, 1987.
[15] C. Loop, “A $\big. {\rm G}^1\bigr.$ Triangular Spline Surface of Arbitrary Topological Type,” Computer Aided Geometric Design, vol. 11, pp. 303-330, 1994.
[16] S. Mann, “Surface Approximation Using Geometric Hermite Patches,” PhD dissertation, Univ. of Washington, 1992.
[17] S. Mann, C. Loop, M. Lounsbery, D. Meyers, J. Painter, T. DeRose, and K. Sloan, “A Survey of Parametric Scattered Data Fitting Using Triangular Interpolants,” Curve and Surface Design, H. Hagen, ed., pp. 145-172, SIAM, 1992.
[18] M. Neamtu and P. Pluger, “Degenerate Polynomial Patches of Degree 4 and 5 Used for Geometrically Smooth Interpolation in $\big. {\rlap{I}\kern 2.0pt{R}}^3\bigr.$,” Computer Aided Geometric Design, vol. 11, pp. 451-474, 1994.
[19] G. Nielson, “A Transfinite, Visually Continuous, Triangular Interpolant,” Geometric Modeling: Algorithms and New Trends, G. Farin, ed., pp. 235-246, SIAM, 1987.
[20] H. Nowacki and D. Reese, “Design and Fairing of Ship Surfaces,” Computer Aided Geometric Design, R. Barnhill and W. Böhm, eds., pp. 121-134, North-Holland, 1982.
[21] J. Peters, “Smooth Interpolation of a Mesh of Curves,” Constructive Approximation, vol. 7, pp. 221-246, 1991.
[22] B.R. Piper, “Visually Smooth Interpolation with Triangular Bézier Patches,” Geometric Modeling: Algorithms and New Trends, G. Farin, ed., pp. 221-233, SIAM, 1987.
[23] S. Selesnick, “Local Invariants and Twist Vectors in CAGD,” Computer Graphics and Image Processing, vol. 17, pp. 145-160, 1981.
[24] L.A. Shirman and C.H. Séquin, “Local Surface Interpolation with Bézier Patches,” Computer Aided Geometric Design, vol. 4, pp. 279-295, 1987.
[25] J.J. Van Wijk, “Bicubic Patches for Approximating Non-Rectangular Control Meshes,” Computer Aided Geometric Design, vol. 3, pp. 1-13, 1986.

Index Terms:
Triangulation, irregular 3D meshes, arbitrary topology, modeling, surfaces, triangular patches, piecewise polynomial patches, interpolation, arbitrary tangent vectors, reconstruction.
Stefanie Hahmann, Georges-Pierre Bonneau, "Polynomial Surfaces Interpolating Arbitrary Triangulations," IEEE Transactions on Visualization and Computer Graphics, vol. 9, no. 1, pp. 99-109, Jan.-March 2003, doi:10.1109/TVCG.2003.1175100
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