PIVOTAL INVERSIONS OF A FINITE POINT-SET

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Abstract

For two distinct points $P,$ $Q$ in the plane, let $Q^{P}$ denote the point on the ray $¥overline{PQ}$ such that $PQ¥cdot PQ^{P}=1$ , and let $P^{P}=P$. For a point-set $¥tau$ in the plane and $ P¥in¥tau$ , define $¥tau^{P}=¥{Q^{P}|Q¥in¥tau¥}$ . The transformation $¥tau¥rightarrow¥tau^{P}$ is called the pivotal inversion at $ P¥in¥tau$ . We show that if $n¥geq 4$ then starting from any n-point-set, it is possible, by applying a sequence of pivotal inversions, to produce an n-point-set whose diameter exceeds any prescribed value, but it is impossible to produce more than $n+1$ mutually non-similar $n-point-sets$ . The latter part is proved by showing a group induced by pivotal inversions of ordered $n-point$-sets is isomorphic to the symmetric group of degree $n+1$ .

Journal

  • Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学

    Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学 53(2), 119-126, 2007

    Yokohama City University and Yokohama National University

Codes

  • NII Article ID (NAID)
    120001740837
  • NII NACSIS-CAT ID (NCID)
    AA0089285X
  • Text Lang
    ENG
  • Article Type
    departmental bulletin paper
  • ISSN
    0044-0523
  • Data Source
    IR 
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