Least-Squares Approximation by Pythagorean Hodograph Spline Curves Via an Evolution Process

摘要

The problem of approximating a given set of data points by splines composed of Pythagorean Hodograph (PH) curves is addressed. In order to solve this highly non-linear problem, we formulate an evolution process within the family of PH spline curves. This process generates a one-parameter family of curves which depends on a time-like parameter t. The best approximant is shown to be a stationary point of this evolution. The evolution process - which is shown to be related to the Gauss-Newton method - is described by a differential equation, which is solved by Euler's method.