On transfinite interpolations with respect to convex domains

We propose an approach for constructing a transfinite interpolation where the domain of definition is a convex polytope. For multifaceted domains which are not of tensor product type, it is difficult to directly generalize the usual approach of transfinite interpolation which blends opposite faces and which then substracts that by a mixed term. Therefore, we suggest a short formula which uses topologic entities of the convex domain. Our formula uses some projection operator onto the faces of the polytope. Both representations (VV-setting and HH-setting) of a polytope are used. We show also that the transfinite interpolation is stable under affine transformations. As a supplement to the theoretical demonstrations, we show some interesting practical illustrations.

Research highlights
► We generalize the Coons transfinite interpolation for polytopes.
► Hyperplane arrangements are used to formulate the interpolation.
► Combining opposite faces is unnecessary even in higher dimension.
► We use a succinct formula by combining the polytope faces.
► Being affinely stable is an important property of the interpolation.