If
and
are two points on an Ellipse
![\begin{displaymath}
{x^2\over a^2}+{y^2\over b^2}=1,
\end{displaymath}](f_239.gif) |
(1) |
with Eccentric Angles
and
such that
![\begin{displaymath}
\tan\phi\tan\phi'={b\over a}
\end{displaymath}](f_242.gif) |
(2) |
and
and
. Then
![\begin{displaymath}
\mathop{\rm arc}\nolimits BP+\mathop{\rm arc}\nolimits BP'={e^2 xx'\over a}.
\end{displaymath}](f_245.gif) |
(3) |
This follows from the identity
![\begin{displaymath}
E(u,k)+E(v,k)-E(k)=k^2\mathop{\rm sn}\nolimits (u,k)\mathop{\rm sn}\nolimits (v,k),
\end{displaymath}](f_246.gif) |
(4) |
where
is an incomplete Elliptic Integral of the Second Kind,
is a complete Elliptic Integral
of the Second Kind, and
is a Jacobi Elliptic Function. If
and
coincide, the point where they coincide is called Fagnano's Point.
© 1996-9 Eric W. Weisstein
1999-05-26