Open Access
ARTICLE
Lemniscate of Leaf Function
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 sleaf
2 (l) and cleaf
2 (l) is derived using the geometrical properties of the lemniscate, similarity of
triangles, and the Pythagorean theorem. In the literature, the relation equation for sleaf
2 (l) and cleaf
2 (l) (or the
lemniscate functions, sl (l) and cl (l)) has been derived analytically; however, it is not derived geometrically.
Keywords
Cite This Article
Shinohara, K. (2021). Lemniscate of Leaf Function.
CMES-Computer Modeling in Engineering & Sciences, 126(1), 275–292.