Transfinite mean value interpolation over polygons
- PDF / 623,848 Bytes
- 9 Pages / 439.642 x 666.49 pts Page_size
- 77 Downloads / 158 Views
Transfinite mean value interpolation over polygons Michael S. Floater1 · Francesco Patrizi2 Received: 18 June 2019 / Accepted: 7 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications in computer graphics and curve and surface modelling. The method generalizes to transfinite interpolation, i.e., to any continuous data on the boundary but a mathematical proof that interpolation always holds has so far been missing. The purpose of this note is to complete this gap in the theory. Keywords Mean value coordinates · Mean value interpolation · Transfinite interpolation
1 Introduction One of the main uses of generalized barycentric coordinates (GBCs) is to interpolate piecewise-linear data prescribed on the boundary of a polygon with a smooth function. This kind of barycentric interpolation has been used, for example, in computer graphics, as the basis for image warping, and in higher dimension, for mesh deformation. One type of GBC that is frequently used for this is mean value (MV) coordinates due to a simple closed formula. MV coordinates have been studied extensively in various papers [3] but while they are simple to implement, a mathematical proof of interpolation seems surprisingly difficult. A proof for convex polygons is relatively simple and follows from the fact that MV coordinates are positive in this case. Interpolation for a convex polygon holds in fact for any positive barycentric coordinates
Michael S. Floater
[email protected] Francesco Patrizi [email protected] 1
Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 Oslo, Norway
2
SINTEF, PO Box 124 Blindern, 0314 Oslo, Norway
Numerical Algorithms
(see [4]). For arbitrary polygons, a specific proof of interpolation for MV coordinates was derived in [6]. The MV interpolant to piecewise-linear boundary data is based on integration with respect to angles around each chosen point inside the polygon. This construction extends in a natural way to any continuous boundary data thus providing a transfinite interpolant [1, 7]. Such interpolation could have various applications, one of which is its use as a building block for interpolants of higher order that also match derivative data on the boundary. However, there is currently no mathematical proof of interpolation in the transfinite setting in all cases, only numerical evidence. Like in the piecewise-linear case, when the polygon is convex, interpolation is easier to establish. In fact it was shown in [1] for more general domains, convex or otherwise, under the condition that the distance between the external medial axis of the domain and the domain boundary is strictly positive. This latter condition trivially holds for convex domains since there is no external medial axis in this case. This still leaves open the question of whether MV interpolation really interp
Data Loading...
