Lemniscate of Leaf Function

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 = a². 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 were 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 θ 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.