三腳架構形之研究與探討

Abstract

這篇論文在探討三腳架構形，根據Serge Tabachnikov在附錄[1]的第二個定理:給定一個平滑凸閉平面曲線，至少存在兩個三角架構形。我們在這篇論文中想用跟Serge Tabachnikov不太相同的方法去建構三腳架構形，使用另一種比較直覺的幾何去建構出來。我們採取的方法是minimax method，建造一些變形使Y形的距離和漸漸縮短，但不是所有的Y形均會退化，而會收斂到一個沒有退化的臨界點，再說明臨界點即為我們要的三腳架構形。

In this paper, we research the tripod configurations. By Serge Tabachnikov, see Theorem 2 of Appendix [1] says that for any smooth convex closed curve, there exist at least two tripod configurations. In this paper we want to use another way to construct tripod configurations. Use a intuitive way by a geometrical approach to construct it. We use minimax method, and do some deformation such that the distance of the Y-shaped will decrease, but not all of the Y-shaped will degenerate, it will converge to a critical point which will not degenerate, and we explain that this critical point is our tripod configuration.