Euler–Rodrigues frames on spatial Pythagorean-hodograph curves

✍ Scribed by Hyeong In Choi; Chang Yong Han

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 166 KB
• Volume: 19
• Category: Article
• ISSN: 0167-8396
• DOI: 10.1016/s0167-8396(02)00165-6

✦ Synopsis

We investigate the properties of a special kind of frame, which we call the Euler-Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a characterization of PH curves whose ERF achieves rotationminimizing property. This computation leads into a new fact that this ERF is equivalent to the Frenet frame on cubic PH curves. Furthermore, we prove that the minimum degree of non-planar PH curves whose ERF is an rotation-minimizing frame is seven, and provide a parameterization of the coefficients of those curves.