Lemniscate of Leaf Function

TL;DR

This paper explores the geometric properties of leaf functions related to the lemniscate curve, providing a geometric derivation of the relationship between leaf functions sleaf2 and cleaf2, extending previous analytical results.

Contribution

It presents a geometric analysis of leaf functions at n=2 and derives the relationship between sleaf2 and cleaf2 using lemniscate properties and geometric methods.

Findings

Derived the geometric relationship between sleaf2 and cleaf2.

Presented the geometric properties of leaf functions at n=2.

Connected the angle theta with lemniscate arc length l geometrically.

Abstract

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.