论文标题
理性四边形
Rational quadrilaterals
论文作者
论文摘要
如果四个侧面，两个对角线和该区域都可以通过理性数字表达，则据说四边形是合理的。构建理性四边形的问题可以追溯到七世纪，当时Brahmagupta为问题提供了优雅的解决方案。 1848年，库默（Kummer）提供了一种产生所有理性四边形的方法。在本文中，我们提出了一种产生所有理性四边形的替代方法。对于有理循环四边形，我们获得了一个完整的参数化和非循环有理四边形，我们就二次和四分之一的多项式进行了几个参数化。本文获得的参数化比有理四边形的已知参数化更简单。我们还描述了如何获得理性四边形的进一步参数化。
A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an elegant solution of the problem. In 1848 Kummer gave a method of generating all rational quadrilaterals. In this paper we present an alternative method of generating all rational quadrilaterals. For rational cyclic quadrilaterals, we obtain a complete parametrization and for noncyclic rational quadrilaterals, we give several parametrizations in terms of quadratic and quartic polynomials. The parametrizations obtained in this paper are simpler than the known parametrizations of rational quadrilaterals. We also describe how further parametrizations of rational quadrilaterals may be obtained.
