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<p>This article introduces the problem of finding the minimal enclosing cone (MECo) of a finite number of non-reflexive right-circular cones sharing the same vertex and presents a semi-analytical method for solving the same. As an important geometric step in the overall algorithm, another untouched problem is studied and solved, namely the computation of Apollonius circles on a sphere. The efficacy of the algorithm is demonstrated by constructing minimum enclosing cones of 1,000 sets of randomly generated input cones, each consisting of up to one million elements. It is found that the median value of the computation time is about 57.9 ms for one such set of one million cones. The results are verified numerically. This algorithm finds applications in the analysis and design of robots and mechanisms with spherical joints. It may find use in other domains of science and engineering in the future.</p>

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Determination of the minimum enclosing cone of a finite collection of right-circular cones sharing the same vertex

  • 期刊论文

  • 发表日期：

    2025年5月3日

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  • Bibekananda Patra,
  • Sandipan Bandyopadhyay

摘要

This article introduces the problem of finding the minimal enclosing cone (MECo) of a finite number of non-reflexive right-circular cones sharing the same vertex and presents a semi-analytical method for solving the same. As an important geometric step in the overall algorithm, another untouched problem is studied and solved, namely the computation of Apollonius circles on a sphere. The efficacy of the algorithm is demonstrated by constructing minimum enclosing cones of 1,000 sets of randomly generated input cones, each consisting of up to one million elements. It is found that the median value of the computation time is about 57.9 ms for one such set of one million cones. The results are verified numerically. This algorithm finds applications in the analysis and design of robots and mechanisms with spherical joints. It may find use in other domains of science and engineering in the future.