高维单形二面角与顶点角的若干关系

为将平面三角学推广到高维空间,最终构建高维三角学理论体系。首先将常见三角形不等式作推广,以正则单形基本性质为依托,引进顶点角概念并建立与顶点角有关的二面角概念,建立n维单形Ωn的n 1个界面的单位外向法向量的度量矩阵,以矩阵理论为基本推理工具,以矩阵特征多项式为契机实现了上述初步构想。在正则单形的基本性质基础上,给出了任何n维单形Ωn的二面角与顶点角之间的不等式关系,其中主要结果包括三个定理与四个推论.通过n维单形Ωn的如上性质的研究可知平面三角学中的相关性质均可拓展到n维空间中。

Some relations between dihedral angle and vertex angle of multidimensional simplex

In order to expand the plane trigonometry theory to the n-dimensions space, and then to set up the theory system of the multidimensional trigonometry, firstly, the writer tries to extend the popular triangle inequalities to the n-dimensions space. Based on the basic nature of the regular simplex and the concepts of dihedral angle and vertex angle, the writer builds a special matrix of the multidimensional simplex s2n, using the matrix theory and the characteristic polynomial of a matrix, gets the work done. The writer gives three theorems and four inferences. Finally the writer points out that in any n-dimensions simplex s2n, there are some relations between the dihedral angle and the vertex angle and then expands the conclusion from the plane trigonometry to the n-dimensions space.