ABSTRACT: A body, namely, a compact connected subset $K$ of $\R^n$, is said to be of constant width $\alpha$ if its projection on any straight line is a segment of length $\alpha\in\R_+$. We present a complete analytic parametrization of constant width bodies in dimension three, based on the median surface. More precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke's formula. Finally, we present some numerical computations based on the preceding parametrization.