Construction of Hyperfigure Theory and Its Applications

我们发现，世界的数字可以扩展到这样一种方式，闵可夫斯基操作总是承认其逆，并命名为新的对象“hyperfigures”。该项目的目标是研究超数的数学结构，并建立超数理论的基础。针对这一目标，我们取得了以下成果。(1)对象世界是从斜率单调曲线的世界中扩展出来的，因此任何多边形都包括在内。这是通过斜率单调曲线可以是有限数量凸图形的联合的边界并且折线作为这些曲线的极限而获得的属性来实现的。(2)对超形进行了物理解释。在传统的闵可夫斯基代数中，有时运算的结果会变成一个空集。在我们的新理论中，这样的结果可以用一个非平凡的超形来表示，并且可以解释为，例如，数控机床的能力或水传播车辆的能力。(3)给出了三维立体Minkowski和的一个有效算法。考虑了旋转二维斜率单调闭曲线得到的旋转曲面，设计了一种算法，求出具有相同外法线的点的对应关系，从而计算Minkowski和。该算法也被应用到两个移动的固体的最短距离的问题。

We found that the world of figures can be extended in such a way that the Minkowski operation always admits its inverse, and name the new objects "hyperfigures". The goal of this project was to study the mathematical structures of hyperfigures and establish a basis of hyperfigure theory. Aiming at this goal, we obtained the following results.(1) The object world is extended from the world of slope-monotone curves so that any polygons are included. This was achieved by the property that a slope-monotone curve can be the boundary of the union of a finite number of convex figures and that the polygonal lines are obtained as the limits of those curves.(2) Physical interpretation was given to hyperfigures. In the conventional Minkowski algebra, sometimes the result of the operation becomes just an empty set. In our new theory, such a result can be represent by a nontrivial hyperfigure, and can be interpreted as, for example, the ability of a numerical control machine or the ability of a water-spread vehicle.(3) An efficient algorithm was obtained for the Minkowski sum of three-dimensional solids. We considered surfaces of revolution obtained by rotating two-dimensional slope-monotone closed curves, and designed an algorithm for finding the correspondence of points with the same outward normal, and thus computing the Minkowski sum. This algorithm was also applied to the problem of finding the shortest distance of two moving solids.