Existence conditions for Coons patches interpolating geodesic boundary curves

Given two pairs of regular space curves r 1 ( u ) , r 3 ( u ) and r 2 ( v ) , r 4 ( v ) that define a curvilinear rectangle, we consider the problem of constructing a C 2 surface patch R ( u , v ) for which these four boundary curves correspond to geodesics of the surface. The possibility of constructing such a surface patch is shown to depend on the given boundary curves satisfying two types of consistency constraints. The first constraint is global in nature, and is concerned with compatibility of the variation of the principal normals along the four curves with the normal to an oriented surface. The second constraint is a local differential condition, relating the curvatures and torsions of the curves meeting at each of the four patch corners to the angle between those curves. For curves satisfying these constraints, the surface patch is constructed using a bicubically-blended Coons interpolation process.