Hostname: page-component-5cf477f64f-54txb Total loading time: 0 Render date: 2025-04-06T08:16:44.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized barycentric coordinates and applications*

Published online by Cambridge University Press:  27 April 2015

Michael S. Floater
Michael S. Floater*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway E-mail: michaelf@ifi.uio.no
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra. Applications include: surface mesh parametrization in geometric modelling; image, curve, and surface deformation in computer graphics; and polygonal and polyhedral finite element methods.

Type
Research Article
Information
Acta Numerica , Volume 24 , 01 May 2015 , pp. 161 - 214
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alfeld, P., Neamtu, M. and Schumaker, L. L. (1996), ‘Bernstein–Bézier polynomials on spheres and sphere-like surfaces’, Comput. Aided Geom. Design 13, 333–349.Google Scholar
Belyaev, A. (2006), On transfinite barycentric coordinates. In Symposium on Geometry Processing 2006, Eurographics Association, pp. 89–99.Google Scholar
Bishop, J. E. (2014), ‘A displacement-based finite element formulation for general polyhedra using harmonic shape functions’, Internat. J. Numer. Meth. Engng 97, 1–31.Google Scholar
Bruvoll, S. and Floater, M. S. (2010), ‘Transfinite mean value interpolation in general dimension’, J. Comput. Appl. Math. 233, 1631–1639.Google Scholar
Charrot, P. and Gregory, J. (1984), ‘A pentagonal surface patch for computer aided geometric design’, Comput. Aided Geom. Design 1, 87–94.Google Scholar
Coxeter, H. S. M. (1969), Introduction to Geometry, second edition, Wiley.Google Scholar
Dasgupta, G. and Wachspress, E. (2008), ‘The adjoint for an algebraic finite element’, Comput. Math. Appl. 55, 1988–1997.Google Scholar
Dyken, C. and Floater, M. S. (2009), ‘Transfinite mean value interpolation’, Comput. Aided Geom. Design 26, 117–134.Google Scholar
Farouki, R. T. (2012), ‘The Bernstein polynomial basis: A centennial retrospective’, Comput. Aided Geom. Design 29, 379–419.Google Scholar
Floater, M. S. (1997), ‘Parameterization and smooth approximation of surface triangulations’, Comput. Aided Geom. Design 14, 231–250.Google Scholar
Floater, M. S. (2000), Meshless parameterization and B-spline surface approximation. In The Mathematics of Surfaces IX, Springer, pp. 1–18.Google Scholar
Floater, M. S. (2003a), ‘Mean value coordinates’, Comput. Aided Geom. Design 20, 19–27.Google Scholar
Floater, M. S. (2003b), ‘One-to-one piecewise linear mappings over triangulations’, Math. Comp. 72, 685–696.Google Scholar
Floater, M. S. (2014), Wachspress and mean value coordinates. In Approximation Theory XIV: San Antonio 2013, Springer, pp. 81–102.Google Scholar
Floater, M. S. and Kosinka, J. (2010a), Barycentric interpolation and mappings on smooth convex domains. In Proc. 14th ACM Symposium on Solid and Physical Modeling, ACM, pp. 111–116.Google Scholar
Floater, M. S. and Kosinka, J. (2010b), ‘On the injectivity of Wachspress and mean value mappings between convex polygons’, Adv. Comput. Math. 32, 163–174.CrossRefGoogle Scholar
Floater, M. S. and Schulz, C. (2008), ‘Pointwise radial minimization: Hermite interpolation on arbitrary domains’, Comput. Graph. Forum 27, 1505–1512.CrossRefGoogle Scholar
Floater, M. S., Gillette, A. and Sukumar, N. (2014), ‘Gradient bounds for Wachspress coordinates on polytopes’, SIAM J. Numer. Anal. 52, 515–532.Google Scholar
Floater, M. S., Hormann, K. and Kós, G. (2006), ‘A general construction of barycentric coordinates over convex polygons’, Adv. Comput. Math. 24, 311–331.Google Scholar
Floater, M. S., Kos, G. and Reimers, M. (2005), ‘Mean value coordinates in 3D’, Comput. Aided Geom. Design 22, 623–631.Google Scholar
Gillette, A., Rand, A. and Bajaj, C. (2012), ‘Error estimates for generalized barycentric interpolation’, Adv. Comput. Math. 37, 417–439.Google Scholar
Gordon, W. J. and Wixom, J. A. (1974), ‘Pseudo-harmonic interpolation on convex domains’, SIAM J. Numer. Anal. 11, 909–933.CrossRefGoogle Scholar
Goyal, S. and Goyal, V. (2012), ‘Mean value results for second and higher order partial differential equations’, Appl. Math. Sci. 6, 3941–3957.Google Scholar
Hormann, K. and Floater, M. S. (2006), ‘Mean value coordinates for arbitrary planar polygons’, ACM Trans. on Graphics 25, 1424–1441.Google Scholar
Hormann, K. and Sukumar, N. (2008), Maximum entropy coordinates for arbitrary polytopes. In Symposium on Geometry Processing 2008, Eurographics Association, pp. 1513–1520.Google Scholar
Iserles, A. (1996), A First Course in Numerical Analysis of Differential Equations, Cambridge University Press.Google Scholar
Ju, T., Schaefer, S. and Warren, J. (2005a), ‘Mean value coordinates for closed triangular meshes’, ACM Trans. on Graphics 24, 561–566.Google Scholar
Ju, T., Schaefer, S., Warren, J. and Desbrun, M. (2005b), A geometric construction of coordinates for convex polyhedra using polar duals. In Symposium on Geometry Processing 2005, Eurographics Association, pp. 181–186.Google Scholar
Kosinka, J. and Bartoň, M. (2014), ‘Convergence of Wachspress coordinates: From polygons to curved domains’, Adv. Comput. Math. doi:10.1007/s10444-014-9370-3 Google Scholar
Lai, M.-J. and Schumaker, L. L. (2007), Spline Functions on Triangulations, Cambridge University Press.Google Scholar
Langer, T., Belyaev, A. and Seidel, H.-P. (2006), Spherical barycentric coordinates. In Symposium on Geometry Processing 2006, Eurographics Association, pp. 81–88.Google Scholar
Lee, C. W. (1990), ‘Some recent results on convex polytopes’, Contemp. Math. 114, 3–19.Google Scholar
Li, X.-Y. and Hu, S.-M. (2013), ‘Poisson coordinates’, IEEE Trans. Vis. Comput. Graph. 19, 344–352.Google Scholar
Li, X.-Y., Ju, T. and Hu, S.-M. (2013), ‘Cubic mean value coordinates’, ACM Trans. on Graphics 32, #126.Google Scholar
Lipman, Y., Kopf, J., Cohen-Or, D. and Levin, D. (2007), GPU-assisted positive mean value coordinates for mesh deformation. In Symposium on Geometry Processing 2007, Eurographics Association, pp. 117–123.Google Scholar
Lipman, Y., Levin, D. and Cohen-Or, D. (2008), ‘Green coordinates’, ACM Trans. on Graphics 27, #78.CrossRefGoogle Scholar
Loop, C. T. and DeRose, T. D. (1989), ‘A multisided generalization of Bézier surfaces’, ACM Trans. on Graphics 8, 204–234.Google Scholar
Manson, J. and Schaefer, S. (2010), ‘Moving least squares coordinates’, Computer Graphics Forum 29, 1517–1524.Google Scholar
Manson, J., Li, K. and Schaefer, S. (2011), ‘Positive Gordon–Wixom coordinates’, Computer Aided Design 43, 1422–1426.Google Scholar
Manzini, G., Russo, A. and Sukumar, N. (2014), ‘New perspectives on polygonal and polyhedral finite element methods’, Math. Models Methods Appl. Sci. 24, 1665–1699.Google Scholar
Meyer, M., Barr, A., Lee, H. and Desbrun, M. (2002), ‘Generalized barycentric coordinates for irregular polygons’, J. Graph. Tools 7, 13–22.Google Scholar
Möbius, A. F. (1827), Der Barycentrische Calcul, Johann Ambrosius Barth, Leipzig.Google Scholar
Möbius, A. F. (1846), Ueber eine neue Behandlungsweise der analytischen Sphärik. In Abhandlungen bei Begründung der Königl. Sächs. Gesellschaft der Wissenschaften, Fürstliche Jablonowski’schen Gesellschaft, Leipzig, pp. 45–86.Google Scholar
Pinkall, U. and Polthier, K. (1993), ‘Computing discrete minimal surfaces and their conjugates’, Experimental Mathematics 2, 15–36.CrossRefGoogle Scholar
Polyanin, A. D. (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall.Google Scholar
Prautzsch, H., Boehm, W. and Paluszny, M. (2002), Bézier and B-Spline Techniques, Springer.CrossRefGoogle Scholar
Rand, A., Gillette, A. and Bajaj, C. (2013), ‘Interpolation error estimates for mean value coordinates over convex polygons’, Adv. Comput. Math. 39, 327–347.Google Scholar
Rand, A., Gillette, A. and Bajaj, C. (2014), ‘Quadratic serendipity finite elements on polygons using generalized barycentric coordinates’, Math. Comp. 83, 2691–2716.Google Scholar
Schneider, T., Hormann, K. and Floater, M. S. (2013), Bijective composite mean value mappings. In Symposium on Geometry Processing 2013, Eurographics Association, pp. 137–146.Google Scholar
Sukumar, N. (2004), ‘Construction of polygonal interpolants: A maximum entropy approach’, Internat. J. Numer. Meth. Engng 61, 2159–2181.Google Scholar
Sukumar, N. (2013), ‘Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons’, Comput. Meth. Appl. Mech. Engrg. 263, 27–41.Google Scholar
Sukumar, N. and Tabarraei, A. (2004), ‘Conforming polygonal finite elements’, Internat. J. Numer. Meth. Engng 61, 2045–2066.Google Scholar
Talischi, C., Paulino, G. H. and Le, C. H. (2009), ‘Honeycomb Wachspress finite elements for structural topology optimization’, Struct. Multidisc. Optim. 37, 569–583.Google Scholar
Thiery, J.-M., Tierny, J. and Boubekeur, T. (2013), ‘Jacobians and Hessians of mean value coordinates for closed triangular meshes’, The Visual Computer.Google Scholar
Tutte, W. T. (1963), ‘How to draw a graph’, Proc. London Math. Soc. 13, 743–768.Google Scholar
Wachspress, E. L. (1975), A Rational Finite Element Basis, Academic Press.Google Scholar
Wachspress, E. L. (2011), ‘Barycentric coordinates for polytopes’, Comput. Math. Appl. 61, 3319–3321.Google Scholar
Warren, J. (1996), ‘Barycentric coordinates for convex polytopes’, Adv. Comput. Math. 6, 97–108.CrossRefGoogle Scholar
Warren, J., Schaefer, S., Hirani, A. N. and Desbrun, M. (2007), ‘Barycentric coordinates for convex sets’, Adv. Comput. Math. 27, 319–338.Google Scholar
Weber, O., Ben-Chen, M. and Gotsman, C. (2009), ‘Complex barycentric coordinates with applications to planar shape deformation’, Comput. Graph. Forum 28, 587–597.Google Scholar
Weber, O., Ben-Chen, M., Gotsman, C. and Hormann, K. (2011), ‘A complex view of barycentric mappings’, Comput. Graph. Forum 30, 1533–1542.Google Scholar
Wicke, M., Botsch, M. and Gross, M. (2007), A finite element method on convex polyhedra. In Proceedings of Eurographics 2007, pp. 255–364.Google Scholar
Wolberg, G. (1990), Digital Image Warping, IEEE Computer Society Press.Google Scholar