Constructing G¹ continuous curve on a free-form surface with normal projection

Abstract

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is 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 a parametric case or in three-dimensional space for an 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 surface trim, robot, patterns design on surface and other industrial and research fields.

Keywords:

• Hermite interpolation
• normal projection
• surface
• G¹ continuity
• ordinary differential equation

2000 AMS Subject Classification :

• 68U05
• 68U07
• 65D17
• 65D18
• 65L05

CCS Category :

• I.3.5
• G.1.7

Acknowledgements

The work reported in this paper was supported by National Natural Science Foundation of China under a grant (nos. 60673026 and 50875130) and Huo Ying-Dong Educational Foundation (no. 03-91053).