In the area of automatic parallelization of programs, analyzing and transforming loop nests with parametric affine loop bounds requires fundamental mathematical results. The most common geometrical model of iteration spaces, called the polytope model, is based on mathematics dealing with convex and discrete geometry, linear programming, combinatorics and geometry of numbers. In this paper, we present automatic methods for computing the parametric vertices and the Ehrhart polynomial, i.e., a parametric expression of the number of integer points, of a polytope defined by a set of parametric linear constraints. These methods have many applications in analysis and transformations of nested loop programs. The paper is illustrated with exact symbolic array dataflow analysis, estimation of execution time, and with the computation of the maximum available parallelism of given loop nests.