Rational framing motions and spatial rational Pythagorean hodograph curves

Abstract

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations.

Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions.

Examples illustrate our ideas and relate them to existing literature.

Keywords

• Rational curve
• Tangent indicatrix
• Euler-Rodrigues frame
• Rational motion
• Motion polynomial
• Spherical motion