Approximately Arc-Length Parametrized C3 Quintic Interpolatory Splines

Abstract

A quasi-global interpolation method that fits a quintic spline curve to a set of designated data points is described in this paper. The resultant curve has several important features. First, the curve is smooth with C3 continuity and has no unwanted oscillations. Second, the generated quintic spline is “optimally” parametrized; that is, the curve is parametrized very closely to its arc length. In addition, with the interpolation method, straight line segments can be preserved to generate a quintic spline of hybrid curve segments. The properties of C3 continuity and the “near arc length” parametrization have direct applications to trajectory planning in robotics and the development of new types of machine tool controllers for high speed and precision machining. The encapsulation of straight line segments enhances the capability for the shape designers to design more complicated shapes, including free form curves and straight line segments in a uniform way.