Shape-preserving interpolation on surfaces via variable-degree splines

This paper proposes two, geodesic-curvature based, criteria for shape-preserving interpolation on smooth surfaces, the first criterion being of non-local nature, while the second criterion is a local (weaker) version of the first one. These criteria are tested against a family of on-surface $C^2$ splines obtained by composing the parametric representation of the supporting surface with variable-degree (≥3) splines amended with the preimages of the shortest-path geodesic arcs connecting each pair of consecutive interpolation points. After securing that the interpolation problem is well posed, we proceed to investigate the asymptotic behaviour of the proposed on-surface splines as degrees increase. Firstly, it is shown that the local-convexity sub-criterion of the local criterion is satisfied. Second, moving to non-local asymptotics, we prove that, as degrees increase, the interpolant tends uniformly to the spline curve consisting of the shortest-path geodesic arcs. Then, focusing on isometrically parametrized developable surfaces, sufficient conditions are derived, which secure that all criteria of the first (strong) criterion for shape-preserving interpolation are met. Finally, it is proved that, for adequately large degrees, the aforementioned sufficient conditions are satisfied. This permits to build an algorithm that, after a finite number of iterations, provides a $C^2$ shape-preserving interpolant for a given data set on a developable surface.