The inverse problem in nonlinear (incompressible) elastica theory, where the end positions and inclinations rather than the forces and moment are specified, is considered. Based on the globally convergent Chow-Yorke algorithm, a new homotopy method which is simple, accurate, stable, and efficient is developed. For comparison, numerical results of some other simple approaches (e.g., Newton's method based on shooting or finite differences, standard embedding) are presented. The new homotopy method does not require a good initial estimate, and is guaranteed to have no singular points. The homotopy method is applied to the problem of a circular elastica ring subjected to N symnetrical point loads, and numerical results are given for N = 2,3,4.