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Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves

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Abstract

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝn whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝn are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.

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Authors and Affiliations

  1. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, USA

    Rida T. Farouki & Chang Yong Han

  2. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Joel Hass

Authors
  1. Rida T. Farouki
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  2. Chang Yong Han
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  3. Joel Hass
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Correspondence to Rida T. Farouki.

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Communicated by C. Brezinski

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Farouki, R.T., Han, C.Y. & Hass, J. Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves. Numer Algor 40, 251–283 (2005). https://doi.org/10.1007/s11075-005-4565-9

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  • DOI: https://doi.org/10.1007/s11075-005-4565-9

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