Given an ellipse with semi major axis $a$ and semi minor axis $b$. What is the formula to compute the chord length formed by two points, say $P$ and $Q$ on the arc of the ellipse (Euclidean distance between the two points).

  • $\begingroup$ "the arc length between this two points subtend an angle α" - this is a bit unclear. How does one have a subtended angle in an ellipse? $\endgroup$ – J. M. is a poor mathematician Mar 4 '18 at 14:12
  • $\begingroup$ The problem has been rephrased. $\endgroup$ – John Majimboni Mar 4 '18 at 14:14

The parametric equation for the ellipse are $(x,y)=(a \cos \theta, b \sin \theta)$ and the length of the arc between two points $P$ and $Q$ is: $$ l_{PQ}=\int_{\theta _P}^{\theta_Q}\sqrt{dx^2+dy^2}=\int_{\theta _P}^{\theta_Q}\sqrt{\left(\frac{dx}{d \theta}\right)^2+\left(\frac{dy}{d \theta}\right)^2} d \theta = $$ $$ =\int_{\theta _P}^{\theta_Q}\sqrt{a^2\cos^2 \theta +b^2 \sin^2 \theta} d \theta $$

where $\theta_P$ and $\theta _Q$ are the angle from the $x$ axis and the lines that passes thorough the origin and the two points (note that the angle that subtend the arc is not sufficient to fix its length).

This cannot be evaluated with standard functions, it is an elliptic integral of the second kind.

For the length of the chord the result is simpler:

$$ L=\sqrt{(x_P-x_Q)^2+(y_P-y_Q)^2}=\sqrt{a^2(\cos \theta_P-\cos \theta_Q)^2+b^2(\sin \theta_P-\sin \theta_Q)^2} $$ but also in this case we need two angles.

  • $\begingroup$ Thanks for your help. I am however interested in the chord length (the Euclidean distance between the two points). $\endgroup$ – John Majimboni Mar 4 '18 at 14:07
  • $\begingroup$ AH..AH.. OK! I edit and add :) $\endgroup$ – Emilio Novati Mar 4 '18 at 14:13
  • $\begingroup$ If need be, one can use the prosthaphaeresis formulae on the chord length expression. $\endgroup$ – J. M. is a poor mathematician Mar 4 '18 at 14:29

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