The hdograph of a plane parametric curve r(t) = (x(t), y(t)f is the locus described by the first parametric derivative r' (t) = (x ' (t), y ' (t)) of that curve. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial a(t) such that x t 2 (t) + y " (t) "Pythagorean triple." Although Pythagoreanhodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkably attractive properties for practical use. For example, their arc length is expressible as a polynomial function of the parameter, and their offsets are rational curves. We present a sufficient-andnecessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein-Bezier forms, and survey the useful attributes it entails in various applications. ~' (t) , i.e., (x' (t), y ' (t) , a (t)) form a

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A characterization for spatial Pythagorean-hodograph (PH) curves of degree 7 with rotation-minimizing Euler-Rodrigues frames (ERFs) is determined, in terms of one real and two complex constraints on the curve coefficients. These curves can interpolate initial/final positions p i and p f and orientational frames (t i , u i , v i) and (t f , u f , v f) so as to define a rational rotation-minimizing rigid body motion. Two residual free parameters, that determine the magnitudes of the end derivatives, are available for optimizing shape properties of the interpolant. This improves upon existing algorithms for quintic PH curves with rational rotation-minimizing frames (RRMF quintics), which offer no residual freedoms. Moreover, the degree 7 PH curves with rotation-minimizing ERFs are capable of interpolating motion data for which the RRMF quintics do not admit real solutions. Although these interpolants are of higher degree than the RRMF quintics, their rotation-minimizing frames are actually of lower degree (6 versus 8), since they coincide with the ERF. This novel construction of rational rotation-minimizing motions may prove useful in applications such as computer animation, geometric sweep operations, and robot trajectory planning.