Heron公式向四面体的推广及其零点的射影性质
An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros
AI总结：
本文提出Heron公式到四面体体积的自然推广，将体积四次方表示为六个有理函数的多项式，并分析其零点构成的半代数簇及其射影性质，同时提出推广到高维单纯形的猜想。
摘要：
本文提出了Heron公式（两千年前用于计算三角形面积的公式）向四面体体积的一个自然推广。该推广将体积的四次方表示为六个简单有理函数的多项式，这些有理函数是四面体四个面以及三个中位平行四边形的面积，因此本文将这些中位平行四边形称为“内面”。从几何上看，这些有理函数是四面体的内切球与各面相切时，将外表面分割成的三角形的面积。第一部分概述了这些结果以及面积几何中一些必要但鲜为人知的背景知识。第二部分推导了所承诺的推广，并在结尾提出了关于该公式如何推广到所有n>3维单纯形的猜想。第三部分解释了，对于n=3的情况，该多项式的零点构成一个五维半代数簇，该簇几乎完全由顶点间距离为无穷大的共线四面体组成，但距离比在一般情况下定义良好；该部分进一步证明了这些非常规的欧几里得构型可以与Klein二次曲面在反射群（同构于Z_2^4）作用下的商空间等同，其中仿射平面中的四点构型构成一个特殊的三维子簇。第四部分由五个附录组成，其中展示了仿射平面中零点的代数结构自然定义了相关的四元素、秩为3的chirotope，即仿射定向拟阵。
英文摘要：
A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and of its three medial parallelograms, which are accordingly referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to $n$-dimensional simplices for all $n > 3$. Part III explains how, for $n = 3$, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to $\mathbb Z_2^4$, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank $3$ chirotope, aka affine oriented matroid.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
