TY - EJOU
AU - Shinohara, Kazunori
TI - Lemniscate of Leaf Function
T2 - Computer Modeling in Engineering \& Sciences
PY - 2021
VL - 126
IS - 1
SN - 1526-1506
AB - A lemniscate is a curve defined by two foci, *F*_{1} and *F*_{2}. If the distance between the focal points of *F*_{1} − *F*_{2} is 2a
(a: constant), then any point P on the lemniscate curve satisfy the equation *PF*_{1} · *PF*_{2} = *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 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.
KW - Geometry; lemniscate of Bernoulli; leaf functions; lemniscate functions; Pythagorean theorem; triangle similarity
DO - 10.32604/cmes.2021.012383