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  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; see . For arbitrary polygons, a specific proof of interpolation for MV coordinates was derived in .
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 [7, 1]. 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  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 interpolates any continuous data on the boundary of an arbitrary polygon, and this is what we establish in this paper. The proof parallels that of  in that we treat interpolation at edge points and vertices separately: in Theorems 1 and 2 respectively. At the end of the paper we give two examples that numerically confirm the interpolation property.
In the future we would like to extend the proof of interpolation to 3D geometry such as volumes enclosed by triangular meshes [5, 7] but there does not seem to be any straightforward generalization of the proof in the 2D case, not even for piecewise-linear boundary data. It would also be interesting to establish transfinite interpolation over more general domains with weaker conditions on the shape of the boundary than those used in .
Let be a polygon with vertices and edges . Suppose that is a continuous function on the boundary . We define a function as follows. For each edge , let denote the outward unit normal to with respect to , and for each point , let be its signed distance to ,
for any . We let be the sign of the distance,
Let denote the unit circle in . For , let denote the circular arc on formed by projecting onto the unit circle centred at ,
with the Euclidean norm. This arc is just a point in the case that . Suppose
. Then for each unit vector, let be the unique point of such that
In the case that , we define .
3 Interpolation on an edge
Let be an interior point of some edge of . Then as for .
Proof. From the form of (1),
where and therefore
Let be the edge containing , as in Figure 1.
Let . By the continuity of , there is some , where
such that if and then . Let , , be the point such that , and let . Then,
and it follows that , where
Similar to , we can express as
For close enough to , , and then
As , , and since for all ,
Therefore, by (3), as ,
Thus as . Hence,
for any which shows that as .
4 Interpolation at a vertex
For , as for .
Figure 3: Interpolation at a concave vertex .
Let . By the continuity of , there is some , where
such that if is in or and then . Let , , be the point such that , and define and . Then,
It follows that , where
and is as in (5). We can similarly express as
Then using (3), and multiplying both and by , we have
where as . Letting and , , and using the fact that for , we can rewrite this as
Next, using the identity
and the fact that , it follows that
and so also as .
Finally, we consider the two cases (i) is a convex vertex and (ii) is a concave vertex. In case (i), referring to Figure 2 we see that for close enough to , and so
In case (ii), the values of and depend on the location of , even when is close to . However, for any that is close enough to , we have the identity (observed in )
This can be verified in the three cases illustrated in Figure 3. In the three configurations, from left to right, we have, respectively,
Since , it follows that in case (ii),
5 Numerical examples
In this section we present two examples of transfinite mean value interpolants of different functions over a polygonal-shaped domain in order to confirm the theoretical interpolation property proven in Sections 3 and 4. For the implementation we have evaluated the mean value interpolant using the boundary integral formula of . This is more efficient than applying the definition, equation (1), which would require computing intersection points.
The first function we consider is
defined on the non-convex polygon in Figure 4a. Figures 4a and 4b illustrate the exact surface and Figures 4c and 4d the corresponding interpolant . Figure 4e shows the absolute error . The darker the colour the smaller the error and, as expected, the error vanishes as we get close to the boundary.
For our second example we chose the function
Acknowledgement. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675789.
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