1. Computing Minkowski Sums for Freeform
Geometry
Research Thesis
Shlomo Geller
Advisor: Iddo Hanniel
Mechanical Engineering - Technion - Israel
Abstract
Minkowski sums have numerous applications in robot path planning, collision detection,
assembly planning and more. While there are algorithms and implementations for comput-
ing Minkowski sums of polygons, implementing such algorithms for freeform geometry
(e.g., B-spline curves) presents new and difficult challenges.
In this work we have implemented algorithms for the computation of Minkowksi sums
of freeform curves. We present our symbolic representation of the boundary of the Minkowski
sum as an implicit bivariate polynomial equation in the parameter space. The solution is a
univariate curve in the parameter space that is mapped to a curve in the Euclidean plane.
This curve is further processed to remove unwanted branches. The computations required
for this processing are performed by defining conditions on the curves, and applying a
constraint solver to compute the result based on these conditions.
We extend our results to the computation of Minkowski sums of freeform surfaces in
R3
. We represent the boundary of the Minkowski sum as a system of implicit multivariate
polynomial equations in the parameter space. The solution is a bivariate manifold in the
parameter space, which is sampled and mapped to the Euclidean space.
Finally, we show how to compute a surface Minkowski sum that is constrained to a
plane. Unlike the full surface Minkowski sum, the solution to this problem is a univariate
curve and therefore easier to process. The result can be applied, for example, in path
planning of a freeform robot moving on a floor while avoiding three-dimensional freeform
obstacles.
Keywords: Minkowksi sums, freeform geometry
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