几何图形比较
Comparison of geometric figures
摘要：
虽然人们对图形的几何相等已经有了深入的研究，但对不等图形的比较研究却很少。我们习惯于只比较相似形的数字，但比较不相似的数字有意义吗？在本文中，我们试图建立一个上下文，它是可能的比较，甚至不相似形。采用Klein对欧几里得几何的观点，我们定义了一个关系“<=”：S<=T，只要存在刚体运动f，使得f（S）是T的子集。这个关系不是一个顺序，因为有图形（平面的子集），使S<=T，T<=S和S，T不几何相等。我们的目标是避免这个悖论，并追踪非平凡类的数字，其中关系“<=”至少成为偏序。这样的一类将被称为好的图形类。一个合理的问题是，形成一个好类的图形是否具有某些性质，以及这些图形的代数是否也是一个好类。因此，我们把数字分为引起上述悖论的数字和不引起上述悖论的数字，最后一类数字称为好数字。虽然很简单，但好的图形的定义很难处理，因此我们引入了一个更技术性的，但内在的和方便的定义，即强好的图形。有了这些工具，我们构建了一个新的背景下，我们扩大了我们的视角，不仅在欧几里德，而且在双曲和椭圆几何的几何比较。最后，我们在论文的最后部分提出了一些开放的和具有挑战性的问题。
英文摘要：
Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation "<=" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation "<=" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper.
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。
