  This article derives closed-form parametric formulas for the Minkowski sums
of convex bodies in d-dimensional Euclidean space with boundaries that are
smooth and have all positive sectional curvatures at every point. Under these
conditions, there is a unique relationship between the position of each
boundary point and the surface normal. The main results are presented as two
theorems. The first theorem directly parameterizes the Minkowski sums using the
unit normal vector at each surface point. Although simple to express
mathematically, such a parameterization is not always practical to obtain
computationally. Therefore, the second theorem derives a more useful parametric
closed-form expression using the gradient that is not normalized. In the
special case of two ellipsoids, the proposed expressions are identical to those
derived previously using geometric interpretations. In order to examine the
results, numerical validations and comparisons of the Minkowski sums between
two superquadric bodies are conducted. Applications to generate configuration
space obstacles in motion planning problems and to improve optimization-based
collision detection algorithms are introduced and demonstrated.
