Discrete approximations to continuous curves

We consider the problem of approximating a continuous curve by a piecewise linear one whose segments are assumed to be connected by universal joints. Rather than taking a least-squares approach, we require that the endpoints of the line segments lie on the continuous curve. We show that under these assumptions a single rotational degree of freedom remains. An algorithm is derived to determine the set of angles characterizing the relative orientation of each consecutive pair of line segments as a function of this rotational degree of freedom. Two examples are given to illustrate the procedure. The motivating application for this work is the control of a snake-like robot using a set of gaits designed from shape primitives