正交单形及其中心
Orthocentric simplices and their centers
摘要：
如果一个单形的高度相交于一个公共点，称为它的正交心，则称它是正交心的。本文证明了：如果正交单形（在任意维上）的任意两个传统中心重合，则该单形是正则的。在此基础上，刻画了所有小平面具有相同外接圆半径的正交单形，并精确描述了正交中心可能的重心坐标。特别地，这些重心坐标用于参数化正交单形的形状。大量的，但广泛的，关于正交单形的文献进行了简要的调查，以便将新的结果在其适当的背景下，和一些以前已知的结果给出了新的证明，从目前的角度来看。
英文摘要：
A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given new proofs from the present perspective.
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