  This paper proposes a closed-form parametric formula of the Minkowski sum
boundary for broad classes of convex bodies in d-dimensional Euclidean space.
With positive sectional curvatures at every point, the boundary that encloses
each body can be characterized by the surface gradient. The first theorem
directly parameterizes the Minkowski sums using the unit normal vector at each
body surface. 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 further examine the results,
numerical verifications and comparisons of the Minkowski sums between two
superquadric bodies are conducted. The application for the generation of
configuration space obstacles in motion planning problems is introduced and
demonstrated.
