Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling

Abstract

This article is devoted to cubic Pythagorean hodograph (PH) curves which enjoy a number of remarkable properties, such as polynomial arc-length function and existence of associated rational frames. First we derive a construction of such curves via interpolation of G 1 Hermite boundary data with Pythagorean hodograph cubics. Based on a thorough discussion of the existence of solutions we formulate an algorithm for approximately converting arbitrary space curves into cubic PH splines, with any desired accuracy. In the second part of the article we discuss applications to sweep surface modeling. With the help of the associated rational frames of PH cubics we construct rational representations of sweeping surfaces. We present sufficient criteria ensuring G 1 continuity of the sweeping surfaces. This article concludes with some remarks on offset surfaces and rotation minimizing frames.