Computer Aided Geometric Design 19 (2002) 603–620 www.elsevier.com/locate/cagd Euler–Rodrigues frames on spatial Pythagorean-hodograph curves ✩ Hyeong In Choi, Chang Yong Han ∗,1 Department of Mathematics, Seoul National University, Seoul 151-747, South Korea Received 28 August 2002; received in revised form 28 August 2002 Abstract 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 rotation- minimizing 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.  2002 Elsevier Science B.V. All rights reserved. Keywords: Euler–Rodrigues frame; Pythagorean-hodograph curve; Rotation-minimizing frame; Quaternion 1. Introduction A sweep surface (Jüttler and Mäurer, 1999; Pottmann and Wagner, 1998; Wang and Joe, 1997) is a preferred technique to model surfaces in the CAD systems. It is created by extruding a planar profile (or cross-section) curve along a spatial spine (or axial) curve. During the extrusion, the profile curve is always on the normal plane of the spine curve, which requires a local coordinate system, i.e., an ordered orthonormal basis on the normal plane. By selecting such ordered orthonormal basis on each normal plane, we build two vector fields on the spine curve. Together with the unit tangent vector field, those two vector fields constitute a frame (field) on the spine curve. ✩ Supported in part by KOSEF-102-07-2, KOSEF-R01-2001-00396, and BK21 program. * Corresponding author. E-mail addresses: hichoi@math.snu.ac.kr (H.I. Choi), cyinblue@hotmail.com (C.Y. Han). 1 Supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University. 0167-8396/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0167-8396(02)00165-6