Scalar-vector algorithm for the roots of quadratic quaternion polynomials, and the characterization of quintic rational rotation-minimizing frame curves

The scalar–vector representation is used to derive a simple algorithm to obtain the roots of a quadratic quaternion polynomial. Apart from the familiar vector dot and cross products, this algorithm requires only the determination of the unique positive real root of a cubic equation, and special cases (e.g., double roots) are easily identified through the satisfaction of algebraic constraints on the scalar/vector parts of the coefficients. The algorithm is illustrated by computed examples, and used to analyze the root structure of quadratic quaternion polynomials that generate quintic curves with rational rotation–minimizing frames (RRMF curves). The degenerate (i.e., linear or planar) quintic RRMF curves correspond to the case of a double root. For polynomials with distinct roots, generating non–planar RRMF curves, the cubic always factors into linear and quadratic terms, and a closed–form expression for the quaternion roots in terms of a real variable, a unit vector, a uniform scale factor, and a real parameter τ ∈ [−1,+1 ] is derived. ∗Corresponding author. Phone number: 530–752-1779