Pythagorean-Hodograph B-Spline Curves

TL;DR

This paper introduces a new class of planar Pythagorean-Hodograph B-Spline curves that generalize existing PH curves, offering exact arc-length and offset computations, with applications in CAD, robotics, and computer graphics.

Contribution

The paper defines Pythagorean-Hodograph B-Spline curves, extending PH curves to B-Spline form, and provides formulas, properties, and practical examples for their use.

Findings

Arc-length of these curves is a B-Spline function.

Offsets of these curves are NURBS curves.

They have advantages in CAD and motion control applications.

Abstract

We introduce the new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) B\'ezier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B-Spline curves are non-uniform parametric B-Spline curves whose arc-length is a B-Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B-Spline curves are non-uniform rational B-Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B-Spline curves have fewer degrees of freedom than general B-Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we present useful formulae for their construction, discuss their remarkable attractive properties and give some examples of their practical use.