Constructing up to G 2 continuous curve on freeform surface

Abstract

This paper presents new methods for G 1 and G 2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and both tangent direction and curvature vector, respectively, at every point. We design a G 1 or G 2 continuous curve in three-dimensional space, construct a so-called directrix vector field using the space curve and then project a special straight line segment onto the given surface along the directrix vector field. With the techniques in classical differential geometry, we derive a system of differential equations for the projection curve. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for the parametric case or in three-dimensional space for the implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in patterns design on surface.