Geometrical Applications

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


Taking an ellipse to have parametric equations
$$x = a\,\sin \theta ,\quad \in y = b\,\cos \theta ,$$
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
$$d{s^2} = \sqrt {(d{x^2} + d{y^2})} = \sqrt {({a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta )} d\theta = a\sqrt {(1 - {e^2}{{\sin }^2}\theta )} d\theta ,$$
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
$$s = a\int_0^\phi {\sqrt {(1 - {e^2}{{\sin }^2}\theta )} } d\theta = aE(u,e),$$
where φ= am(u, e) or sn(u, e) = sin φ(vide equation (3.4.27)). Note that the modulus equals the eccentricity.


Parametric Equation Semimajor Axis Addition Theorem Spherical Triangle Geometrical Application 
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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Derek F. Lawden
    • 1
  1. 1.University of Aston in BirminghamBirminghamUK

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