曲面上的Steiner问题
The Steiner Problem on a Surface
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摘要: 本文将平面上的Steiner问题的一些结果推广到一般的正则曲面上,主要结果是 定理1 设A,B,C是正则曲面Σ上的三个点,若Σ上另一点P使Σ上的光滑弧长之和达到极小,则此三弧中每两弧在P之交角皆为120°。Abstract: In this paper we generalize the Steiner problem on planes to general regular surfaces. The main result is: Theorem 1. If A,B,C are three points on a regular surface Σ and if another point P on Σ such that the sum of the lengths of the smooth arcs reaches the minimum, then the angles formed by every two arcs at P are all 120°.
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[1] Courant,R.and H.Robbins,What Is Mathematics?Chapter 7,§5,Oxford University Press,New York(1964). [2] 克莱因,M.,《古今数学思想》,第三卷,科技出版社(1981), 246-250. [3] Melzak.Z.A.,On the problem of Steiner,Ganad Math.Bull.,4(1961),143-148. [4] Pollak,H.O.,Some remarks on the Steiner problem,J. Combinational Thy.,A,24(1978),278-295. [5] 王凯宁,凸n边形内Fermat点问题的初等证明,中国科学技术大学学报,11, 4 (1981), 139-141. [6] 黄光明,最短网络,运筹学杂志,2, 2 (1983), 18-25. -
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