@Article{cmes.2021.012383,
AUTHOR = {Kazunori Shinohara},
TITLE = {Lemniscate of Leaf Function},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {126},
YEAR = {2021},
NUMBER = {1},
PAGES = {275--292},
URL = {http://www.techscience.com/CMES/v126n1/40872},
ISSN = {1526-1506},
ABSTRACT = {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.},
DOI = {10.32604/cmes.2021.012383}
}