Geometrical Applications

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.