Frenet Apparatus of Curves in n-Dimensional Generalized Dual Space and Its Application to Rational Space Curves

This paper is devoted to describing and investigating the geometry of a generalized dual curve in the generalized dual space ${\mathbb{D}}_p^n$. Analogous to the classical theory of curves, we construct a dual frame for generalized dual curves, which we call the generalized dual Frenet apparatus. Moreover, using the framework developed in the paper, we propose an algorithm that symbolically computes the similarities between two rational algebraic curves in the generalized 3-dimensional dual space ${\mathbf{D}}_p^3$. The algorithm relies on the relation between the dual curvatures and torsions of the curves. We provide an implementation of the algorithm in the computer algebra system Maple, and perform extensive experimentation of the proposed method.