Constructing G~2 Continuous Curve on Freeform Surface with Normal Projection

This article presents a new method for G² continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G² continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. 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 for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields.