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Closed-form characterization of the Minkowski sum and difference of two ellipsoids

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Abstract

This paper makes three original contributions: (1) Explicit closed-form parametric formulas for the boundary of the Minkowski sum and difference of two arbitrarily oriented solid ellipsoids in n-dimensional Euclidean space are presented; (2) Based on this, new closed-form lower and upper bounds for the volume contained in these Minkowski sums and differences are derived in the 2D and 3D cases and these bounds are shown to be better than those in the existing literature; (3) A demonstration of how these ideas can be applied to problems in computational geometry and robotics is provided, and a relationship to the Principal Kinematic Formula from the fields of integral geometry and geometric probability is uncovered.

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Notes

  1. In the case of the Minkowski difference, mild conditions are imposed requiring one body to be containable in the other at all orientations and under all kissing conditions.

  2. Of course the same applies for \(E_2\) in its principal axis frame with \(\mathbf{a}_1 \,\rightarrow \, \mathbf{a}_2\).

  3. Since area takes the place of volume in 2D problems, and we retain the symbol \(V\) when referring to area.

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Acknowledgments

This work was performed while the authors were supported under NSF Grant IIS-1162095. We would like to thank Mr. Joshua Davis, Mr. Qianli Ma, and the anonymous reviewer for their comments.

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  1. Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA

    Yan Yan & Gregory S. Chirikjian

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  1. Yan Yan
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Correspondence to Gregory S. Chirikjian.

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Yan, Y., Chirikjian, G.S. Closed-form characterization of the Minkowski sum and difference of two ellipsoids. Geom Dedicata 177, 103–128 (2015). https://doi.org/10.1007/s10711-014-9981-3

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