Polynomial Patches through Geodesics

The main goal of this paper is to exhibit a simple method to create surface patches that contain given curves as geodesics, when reparameterized by arclength. The method is based on the observation that a curve which lies on a surface is a geodesic (up to parametrization) if and only if its tangent and acceleration vectors are coplanar with the surface normal along the curve. This observation rephrases the statement of B. O’Neill in his book Elementary Differential Geometry [1, p. 330]: a curve has geodesic curvature zero if and only if its tangent vector and the surface tangential component of the acceleration are collinear. The problem of finding surface patches that contain a given curve as a geodesic is considered in [2]. Their main emphasis in applications is in the shoe industry: finding surfaces that could model the shoe piece with a prescribed girth. The mathematical connection between girth and geodesics lies in the fact that the shoe piece is fabricated from an approximately flat sheet with a minimum of stretching, so it is nearly isometric to the plane, and hence geodesics correspond to straight lines. Further applications are in textile manufacturing. This is discussed in [2] and the references therein. 2. GEODESICS