Open Access

ARTICLE

Lemniscate of Leaf Function

by Kazunori Shinohara

Department of Mechanical Systems Engineering, Daido University, 10-3 Takiharu-cho, Minami-ku, Nagoya, 457-8530, Japan

* Corresponding Author: Kazunori Shinohara. Email: -tokyo.ac.jp

Computer Modeling in Engineering & Sciences 2021, 126(1), 275-292. https://doi.org/10.32604/cmes.2021.012383

Received 28 June 2020; Accepted 14 September 2020; Issue published 22 December 2020

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.

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Keywords

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