Consider the family of Ellipses
![\begin{displaymath}
{x^2\over c^2}+{y^2\over (1-c)^2}-1=0
\end{displaymath}](e_610.gif) |
(1) |
for
. The Partial Derivative with respect to
is
![\begin{displaymath}
-{2x^2\over c^3}+{2y^2\over (1-c)^3}=0
\end{displaymath}](e_612.gif) |
(2) |
![\begin{displaymath}
{x^2\over c^3}-{y^2\over(1-c)^3}=0.
\end{displaymath}](e_613.gif) |
(3) |
Combining (1) and (3) gives the set of equations
![\begin{displaymath}
\left[{\matrix{{1\over c^2} & {1\over(1-c)^2}\cr
{1\over c^3...
...rix{x^2\cr y^2\cr}}\right] = \left[{\matrix{1\cr 0\cr}}\right]
\end{displaymath}](e_614.gif) |
(4) |
where the Discriminant is
![\begin{displaymath}
\Delta =-{1\over c^2(1-c)^3}-{1\over c^3(1-c)^2} = -{1\over c^3(1-c)^3},
\end{displaymath}](e_618.gif) |
(6) |
so (5) becomes
![\begin{displaymath}
\left[{\matrix{x^2\cr y^2\cr}}\right]=\left[{\matrix{c^3\cr (1-c)^3\cr}}\right].
\end{displaymath}](e_619.gif) |
(7) |
Eliminating
then gives
![\begin{displaymath}
x^{2/3}+y^{2/3}=1,
\end{displaymath}](e_620.gif) |
(8) |
which is the equation of the Astroid.
If the curve is instead represented parametrically, then
Solving
|
|
|
(11) |
for
gives
![\begin{displaymath}
c=\cos^2 t,
\end{displaymath}](e_625.gif) |
(12) |
so substituting this back into (9) and (10) gives
the parametric equations of the Astroid.
See also Astroid, Ellipse, Envelope
© 1996-9 Eric W. Weisstein
1999-05-25