Constraints for geodesic network interpolation at a vertex

Abstract

Consider n space curves c˜i(t)i∈Zn, meeting at a vertex. The geodesic network interpolation problem is to determine n surface pieces (patches) xi(u,v) that are each internally C2, surround the vertex and each interpolate curves c˜i and c˜i+1 so that the curves are geodesics of the resulting surface. This paper proves that together three local constraints on the curves are necessary and sufficient for the existence of the surface patches xi: the binormal constraint, the geodesic crossing constraint and the vertex enclosure constraint. Additionally, the paper exhibits stronger geometric constraints, in terms of curvature and torsion, that imply the existence of a solution to the geodesic network interpolation problem.