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In this paper, we propose a new concept called interval parametrization of algebraic curves, that is, we find an interval Bézier curve which bounds a given algebraic curve such that the bound is as tight as possible. An algorithm is presented to compute the interval parametrization of algebraic curves. The algorithm starts by finding the algebraic conditions that the interval Bézier curve bounds the algebraic curve, and then converts the problem into solving a non-linear programming problem with four variables. The non-linear programming is then approximately simplified to solving two non-linear programming problems with two variables. Some examples are provided to demonstrate the algorithm.

Let PL(F_q) denote the projective line over a Galois field F_q, where q is a prime power. Let G(2, Z) be a free product of two cyclic groups < u > and < υ > of order 3. We have parametrized the conjugacy classes of non-degenerate homomorphisms α with the non-trivial elements of F_q. Also, we have developed a useful machanism by which one can construct a unique coset diagram (attributed to G. Higman) for each conjugacy class, depicting the action of G*(2, Z) on PL(F_q).