# Geometrical Applications

• Derek F. Lawden
Part of the Applied Mathematical Sciences book series (AMS, volume 80)

## Abstract

Taking an ellipse to have parametric equations
$$x = a\,\sin \theta ,\quad \in y = b\,\cos \theta ,$$
(4.1.1)
where a>b and the eccentric angle θ is measured from the minor axis, if s is the arc length parameter measured clockwise around the curve from the end B of the minor axis, then
$$d{s^2} = \sqrt {(d{x^2} + d{y^2})} = \sqrt {({a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta )} d\theta = a\sqrt {(1 - {e^2}{{\sin }^2}\theta )} d\theta ,$$
(4.1.2)
where $$e = \sqrt {(1 - {b^2}/{a^2})}$$ is the eccentricity. Thus, the length of arc from B to any point P where θ=Ø is given by
$$s = a\int_0^\phi {\sqrt {(1 - {e^2}{{\sin }^2}\theta )} } d\theta = aE(u,e),$$
(4.1.3)
where φ= am(u, e) or sn(u, e) = sin φ(vide equation (3.4.27)). Note that the modulus equals the eccentricity.

## Keywords

Parametric Equation Semimajor Axis Addition Theorem Spherical Triangle Geometrical Application
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.