Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves

Abstract

An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. The result is an angular function θ( t) of the curve parameter, comprising in general both rational and logarithmic terms, that specifies the orientation of the rotation-minimizing frame relative to the Frenet frame. For PH cubics and quintics, the solution employs only arithmetic operations on the curve coefficients and some complex square and cube root extractions. Moreover, the generalization to PH curves of arbitrary order entails only standard polynomial algorithms (i.e., arithmetic, greatest common divisors, and resultants), solution of a linear system, and a minimal element of polynomial root-solving. Rotation-minimizing frames are employed in computer animation, the construction of swept surfaces, and in robotics applications where the axis of a tool or probe should remain tangential to a given spatial path while minimizing changes of orientation about this axis.