In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of $p$. Thanks to the theory of elimination, we compute the total degree and the partial degrees of this polynomial, and we solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest degree polynomial whose graph is a non-circular curve of constant width. Computations of partial degrees of the defining polynomial of algebraic surfaces of constant width are also provided in the same way.