Abstract: Since the curvature vector of a curve on a surface can be decomposed into normal curvature vector and geodesic curvature vector, a surface curve can be designed first on a plane and then mapped onto the surface. The desired surface curve is first constructed as a plane curve with the given geodesic curvature. After assigning a start point and initial direction on the surface, the whole plane curve is traced onto the surface through point by point mapping. This method can be used for pattern design as well as for geodesic curve computation on smooth surfaces.