دانلود رایگان مقاله یکنواختی مختصات گرانیگاهی تعمیم یافته در چند ضلعی محدب

Abstract

We show that four well-known kinds of generalized barycentric coordinates in convex polygons share a simple monotonicity property: the coordinate function associated with a vertex is increasing along any line from the polygon boundary to that vertex. This shows that the coordinate functions have no local extrema and that their contours are single curves connecting pairs of points on the two edges adjacent to the vertex.

6. Higher dimensions

As one might expect, the monotonicity property studied here extends in the obvious way to convex polyhedra and, more generally, convex polytopes in higher dimensions for the four coordinates considered here, if we make the right definitions. Considering first Wachspress coordinates, the proof of monotonicity of Section 2 generalizes at least to simple convex polytopes in Rd, i.e., convex polytopes in which every vertex has d adjacent (d − 1)-dimensional faces. Using the definitions derived in Warren et al. (2007), and applying the notation and gradient formulas of Sec. 2 of Floater et al. (2014), the argument is similar, and inequality (5) now follows from the fact that aii(x) = d and aij(x) < d, j = i. For the remaining three kinds of GBCs: harmonic, GW, and MV, we do not need to restrict to a simple polytope, and, moreover, we have a choice of how to generalize the boundary functions fi of (2). For each (d − 1)-dimensional face of the convex polytope we could choose any set of GBCs for that face that are themselves monotonic. These face GBCs then define the boundary functions fi for the vertices vi of the polytope. The proof of monotonicity of the harmonic coordinates is now as before, with ψi harmonic in P and ψi ≥ 0 on ∂ P due to fi being monotonic on the faces adjacent to vi . The proof of monotonicity for GW and MV coordinates is also similar to the 2-D case, with integration carried out over unit vectors μ on the unit sphere in Rd instead of angles around the unit circle. We would replace the ray Lθ of (6) by Lμ := {x + sμ : s ≥ 0}, and could denote by yμ its point of intersection with ∂ P . Then for any fixed μ, the five points vi , x, and x˜ of (3) and yμ and y˜μ would all lie in the same plane, and the steps used to prove (8) and (13) are the same.