Projecting curves onto free-form surfaces

This paper presents simple methods of projecting a space curve onto a surface. Here, the parallel projection and the central projection are particularly considered. We derive the differential equations of the projection curve on both parametrically and implicitly defined surface. The projection curve is obtained by numerically solving the initial-value problem for a system of first-order Ordinary Differential Equations (ODEs) in the parametric domain associated with the surface representation for parametric case or in 3D space for implicit case. Some examples are also given to demonstrate that the presented methods are effective and potentially useful in computer-aided design.