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

Interactive Design and Simulation (Dagstuhl Seminar 19512)

This report documents the program and the outcomes of Dagstuhl Seminar 19512 ""Interactive Design and Simulation". After the executive summary, the collection of abstracts of the presentations forms the core of this report, complemented by an example of working group results that highlights the diversity of backgrounds and approaches

Hierarchichal Representation and Computation of Approximate Solutions in Scientific Simulations

In many applications involving large scale scientific computing a massive amount of data is generated in a typical simulation. Such simulations generally require the numerical solution of systems of differential equations where the results are often generated remotely using special high-performance software and computer systems and then examined and investigated interactively using visualization tools. The visualization packages are usually run on local workstations and make use of colour, lighting, texture, sound and animation to highlight and reveal interesting characteristics or features of the approximate solution. This `interactive' viewing of the data is the `rendering' stage of the modeling process and it can be very selective and local in the sense that only a subset of the variables are rendered and then only in regions where something interesting is happening. The resul

Geometric Modelling, Dagstuhl 2011 (Special Issue of Graphical Model Vol. 76 (6))

International audienc

Geometric Modeling (Dagstuhl Seminar 11211)

This report documents the program and the results of Dagstuhl Seminar 11211 ``Geometric Modeling\u27\u27, taking place May 22-27 2011. The focus of the seminar was to discuss modern and emerging topics in Geometric Modeling by researchers and industrial scientists from all over the world

Geometric Modelling, Interoperability and New Challenges (Dagstuhl Seminar 17221)

This report documents the program and the outcomes of Dagstuhl Seminar 17221 "Geometric Modelling, Interoperability and New Challenges". While previous Dagstuhl seminars on geometric modeling were focused on basic research, this seminar was focused on applications of geometric modeling to four topic areas: big data and cloud computing, multi-material additive manufacturing, isogeometric analysis, and design optimization. For this purpose we brought together participants from industry urgently in need of better solutions, researchers in the above application areas, and researchers in the geometric modeling community

Geometric Modelling, Dagstuhl 2014 (Special Issue of Graphical Model Vol. 82)

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