@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, <i>F</i><sub>1</sub> and <i>F</i><sub>2</sub>. If the distance between the focal points of <i>F</i><sub>1</sub> − <i>F</i><sub>2</sub> is 2a
(a: constant), then any point P on the lemniscate curve satisfy the equation <i>PF</i><sub>1</sub> · <i>PF</i><sub>2</sub> = <i>a</i><sup>2</sup>. 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 <i>n</i> = 2 and the geometric relation between
the angle <i>θ</i> and lemniscate arc length l are presented using the lemniscate curve. The relationship between the
leaf functions sleaf<sub>2</sub> (l) and cleaf<sub>2</sub> (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<sub>2</sub> (l) and cleaf<sub>2</sub> (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}
}