Barycentric Coordinates for Arbitrary Polygons in the Plane

More documents

Recommendations

Info

BARYCENTRIC COORDINATES FOR ARBITRARY POLYGONS 19 Meyer, M., Lee, H., Barr, A. H., and Desbrun, M. (2002). Generalized barycentric coordinates on irregular polygons. Journal of Graphics Tools, 7(1):13–22. Milliron, T., Jensen, R. J., Barzel, R., and Finkelstein, A. (2002). A framework for geometric warps and deformations. ACM Transactions on Graphics, 21(1):20–51. Möbius, A. F. (1827). Der barycentrische Calcul. Johann Ambrosius Barth, Leipzig. Nürnberger, G. and Zeilfelder, F. (2000). Developments in bivariate spline interpolation. Journal of Computational and Applied Mathematics, 121(1-2):125–152. Pinkall, U. and Polthier, K. (1993). Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 2(1):15–36. Ruprecht, D. and Müller, H. (1995). Image warping with scattered data interpolation. IEEE Computer Graphics and Applications, 15(2):37–43. Shewchuk, J. R. (2002). Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications, 22(1-3):21–74. Sibson, R. (1980). A vector identity for the Dirichlet tesselation. Mathematical Proceedings of the Cambridge Philosophical Society, 87:151–155. Sibson, R. (1981). A brief description of natural neighbour interpolation. In Barnett, V., editor, Interpolating Multivariate Data, pages 21–36. Wiley, New York. Wachspress, E. L. (1975). A Rational Finite Element Basis. Academic Press, New York. Warren, J. (1996). Barycentric coordinates for convex polytopes. Advances in Computational Mathematics, 6(2):97–108. Warren, J. (2003). On the uniqueness of barycentric coordinates. In Contemporary Mathematics, Proceedings of AGGM ’02, pages 93–99. Warren, J., Schaefer, S., Hirani, A. N., and Desbrun, M. (2003). Barycentric coordinates for convex sets. Technical report, Rice University. Wolberg, G. (1990). Digital Image Warping. IEEE Computer Society Press, Los Alamitos. Technical Report IfI-05-05
20 K. Hormann v j+2 v Ψ^ r j+1 α j+1 ^ α α + r^ j ^ α j–1 α – r j v j–1 v^ v j v j+1 Figure 13: Notation for refining a polygon by adding a vertex. A Refinability If we refine the polygon Ψ to ̂Ψ by adding a vertex ˆv between v j and v j+1 as in Figure 13, then it follows from the locality of the homogeneous coordinates that ŵ i = w i for i ≠ j, j + 1. According to Equation (11), the other three homogeneous coordinates are ŵ j (v) = 2 ( tan(α j−1 (v)/2) + tan(̂α − (v)/2) ) /r j (v), ŵ(v) = 2 ( tan(̂α − (v)/2) + tan(̂α + (v)/2) ) /ˆr(v), ŵ j (v) = 2 ( tan(̂α + (v)/2) + tan(α j+1 (v)/2) ) /r j+1 (v). If we now write ˆv as an affine combination of v, v j , and v j+1 , i.e., with barycentric coordinates and ˆv = ρ(v)v + σ(v)v j + τ(v)v j+1 σ(v) = ˆr(v) sin ̂α+ (v) r j (v) sin α j (v) , τ(v) = ˆr(v) sin ̂α− (v) r j+1 (v) sin α j (v) , ρ(v) = 1 − σ(v) − τ(v), then some elementary transformations show that the homogeneous coordinates w j and w j+1 can be expressed as w j (v) = ŵ j (v) + σ(v)ŵ(v), w j+1 (v) = ŵ j+1 (v) + τ(v)ŵ(v), and that the sum of the refined homogeneous coordinates satisfies Ŵ (v) = W (v) + ρ(v)ŵ(v). It further follows that Ŵ (v) = W (v) in the special case that ˆv lies on the edge e j = (v j , v j+1 ) since then ρ(v) = 0. INSTITUT FÜR INFORMATIK
Page 1 and 2: IFI TECHNICAL REPORTS Institute of
Page 3 and 4: 2 K. Hormann In many applications i
Page 5 and 6: 4 K. Hormann (a) (b) (c) (d) (e) (f
Page 7 and 8: 6 K. Hormann Figure 3: Zero-set of
Page 9 and 10: 8 K. Hormann Figure 4: In this exam
Page 11 and 12: 10 K. Hormann (a) (b) (c) (d) Figur
Page 13 and 14: 12 K. Hormann (b) (a) (c) (d) Figur
Page 15 and 16: 14 K. Hormann This is due to the la
Page 17 and 18: 16 K. Hormann (a) (b) (c) (d) Figur
Page 19: 18 K. Hormann Coxeter, H. S. M. (19
Page 23 and 24: 22 K. Hormann twice) and also in th

Barycentric Coordinates for Arbitrary Polygons in the Plane

Create successful ePaper yourself

Delete template?

Save as template?