论文标题
叶功能
Lemniscate of Leaf Function
论文作者
论文摘要
lemsistate是由两个焦点，F1和F2定义的曲线。如果F1 -F2的焦点之间的距离为2a（a：常数），则s曲线上的任何点p满足方程pf1 pf2 = a^2。雅各布·伯诺利（Jacob Bernoulli）首先描述了1694年的lemen脚。 Euler将Fagnano的公式扩展到了更一般的添加定理（1751）。高斯随后大约在1800年左右提出了尼斯的功能。雅各比作为椭圆函数理论总结了这些见解。叶子功能是扩展的放松功能。以前的论文中已经提出了叶函数的一些公式。其中包括此函数的添加定理及其在非线性方程中的应用。在本文中，叶子函数在n = 2处的几何特性，并使用Lemniscate曲线表示角度theta和Lemniscate弧长L之间的几何关系。叶函数Sleaf2（L）和Cleaf2（L）之间的关系是使用Lemniscate的几何特性，三角形的相似性和毕达哥拉斯定理得出的。在文献中，SLEAF2（L）和CLEAF2（l）的关系方程（或分析得出的Lemaniscate函数，SL（L）和Cl（L））。但是，它不是从几何上得出的。
A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1 - F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1 PF2 = a^2. Jacob Bernoulli first described the lemniscate in 1694. The Fagnano discovered the double angle formula of the lemniscate(1718). The Euler extended the Fagnano's formula to a more general addition theorem(1751). The lemniscate function was subsequently proposed by Gauss around the year 1800. These insights are summarized by Jacobi as the theory of elliptic functions. A leaf function is an extended lemniscate function. Some formulas of leaf functions have been presented in previous papers; these included the addition theorem of this function and its application to nonlinear equations. In this paper, the geometrical properties of leaf functions at n = 2 and the geometric relation between the angle theta and lemniscate arc length l are presented using the lemniscate curve. The relationship between the leaf functions sleaf2(l) and cleaf2(l) is derived using the geometrical properties of the lemniscate, similarity of triangles, and the Pythagorean theorem. In the literature, the relation equation for sleaf2(l) and cleaf2(l) (or the lemniscate functions, sl(l) and cl(l)) has been derived analytically; however, it is not derived geometrically.
