Transfinite mean value interpolation

Abstract

Transfinite mean value interpolation has recently emerged as a simple and robust way to interpolate a function f defined on the boundary of a planar domain. In this paper we study basic properties of the interpolant, including sufficient conditions on the boundary of the domain to guarantee interpolation when f is continuous. Then, by deriving the normal derivative of the interpolant and of a mean value weight function, we construct a transfinite Hermite interpolant, and discuss various applications.

On the injectivity of Wachspress and mean value mappings between convex polygons

“…To simplify this task we again appeal to Lemma 2 but this time must use the polynomial w i . We will show that D(w −1 , w , w +1 )(x) > 0 with w i as in (5). From (16), we can write the determinant as…”

“…For example, one way of deforming a curve is to first enclose it in a polygon, and then to use the vertices of the polygon as control points: moving the vertices will change the shape of the curve. Most of these methods focus on two kinds of barycentric coordinates: Wachspress coordinates [13,20,21] and mean value coordinates [5,7,9,10,14]. Further work on generalized barycentric coordinates can be found in [2,8,11,[15][16][17]19].…”

Convergence of Wachspress coordinates: from polygons to curved domains

“…Examples of such kernels are the kernel defined by Warren et al [15], which we refer to as the Wachspress kernel, and the mean value kernel [1].…”

Poisson Coordinates

“…Therefore, (6) is a transfinite interpolation. Properties of MVCs are further discussed in [22] and [23]; it has been proved that MVCs are barycentric and smooth in 2D. In higher dimensions, similar properties have also been considered in [26].…”

“…These techniques provide explicit expressions of harmonic-like functions with desired boundary values. As a representative instance, Floater's Mean Value coordinates (MVCs) [19], [20], [21], [22], [23] always produce smooth interpolations in both continuous and discrete cases. However, a crucial problem of MVCs is that they do not produce rigorous harmonic functions on the most ordinary regions: n-dimensional balls.…”