Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen--type obstructions) as developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks adde

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed

Chained segment offsetting for ray-based solid representations

International audienceWe present a novel approach to offset solids in the context of fabrication. Our input solids can be given under any representation: boundary meshes, voxels, indicator functions or CSG expressions. The result is a ray-based representation of the offset solid directly used for visualization and fabrication: We never need to recover a boundary mesh in our context. We define the offset solid as a sequence of morphological operations along line segments. This is equivalent to offsetting the surface by a solid defined as a Minkowski sum of segments, also known as a zonotope. A zonotope may be used to approximate the Euclidean ball with precise error bounds. We propose two complementary implementations. The first is dedicated to solids represented by boundary meshes. It performs offsetting by modifying the mesh in sequence. The result is a mesh improper for direct display, but that can be resolved into the correct offset solid through a ray representation. The major advantage of this first approach is that no loss of information – re-sampling – occurs during the offsetting sequence. However, it applies only to boundary meshes and cannot mix sequences of dilations and erosions. Our second implementation is more general as it applies directly to a ray-based representation of any solid and supports any sequence of erosion and dilation along segments. We discuss its fast implementation on modern graphics hardware. Together, the two approaches result in a versatile tool box for the efficient offsetting of solids in the context of fabrication

Minkowski products of unit quaternion sets

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere $S^3$ in $\mathbb{R}^4$, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in $\mathbb{R}^3$ are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.Comment: 29 pages, 1 figur