Discretization of a Continuous Curve

We consider the problem of approximating a finite-length continuous curve by a piecewise linear one whose segments are assumed to be connected by 2 DOF joints. We solve the problem under the assumption that the endpoints of the line segments lie on the continuous curve. Analytical expressions for the relative orientations of each pair of line segments as a function of a single rotational DOF are found. This angle can be chosen arbitrarily or used to optimize a secondary task. The motivating application for this paper is the control of a snake-like robot using gaits designed from shape primitives.