Approximating the helix with rational cubic Bézier curves (Q1902420)

Helices, being given by transcendental functions, cannot be used directly in CAD programs based on rational surfaces. The author shows explicitly how to construct a cubic rational approximation to an arc of helix such that either given tangent directions are observed at the endpoints or, more importantly, that rotated copies of such arcs can be composed to yield a \(G^2\) (geometrically second order smooth) curve. He carefully discusses the error and produces charts for the selection of the optimal angle (giving minimal error) for a given parameter of the helix.