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Ellipse Envelope


Consider the family of Ellipses

{x^2\over c^2}+{y^2\over (1-c)^2}-1=0
\end{displaymath} (1)

for $c\in [0,1]$. The Partial Derivative with respect to $c$ is
-{2x^2\over c^3}+{2y^2\over (1-c)^3}=0
\end{displaymath} (2)

{x^2\over c^3}-{y^2\over(1-c)^3}=0.
\end{displaymath} (3)

Combining (1) and (3) gives the set of equations
\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} (4)

$\displaystyle \left[\begin{array}{c}x^2\\  y^2\end{array}\right]$ $\textstyle =$ $\displaystyle {1\over\Delta}\left[\begin{array}{cc} -{1\over(1-c)^3} & -{1\over...
... & {1\over c^2}\end{array}\right]\left[\begin{array}{c}1\\  0\end{array}\right]$  
  $\textstyle =$ $\displaystyle {1\over\Delta}\left[\begin{array}{c}-{1\over(1-c)^3}\\  -{1\over c^3}\end{array}\right],$ (5)

where the Discriminant is
\Delta =-{1\over c^2(1-c)^3}-{1\over c^3(1-c)^2} = -{1\over c^3(1-c)^3},
\end{displaymath} (6)

so (5) becomes
\left[{\matrix{x^2\cr y^2\cr}}\right]=\left[{\matrix{c^3\cr (1-c)^3\cr}}\right].
\end{displaymath} (7)

Eliminating $c$ then gives
\end{displaymath} (8)

which is the equation of the Astroid. If the curve is instead represented parametrically, then
$\displaystyle x$ $\textstyle =$ $\displaystyle c\cos t$ (9)
$\displaystyle y$ $\textstyle =$ $\displaystyle (1-c)\sin t.$ (10)


${\partial x\over\partial t}{\partial y\over\partial c}-{\partial x\over\partial c}{\partial y\over\partial t} = (-c\sin t)(-\sin t)-(\cos t)[(1-c)\cos t]$
$ = c(\sin^2 t+\cos^2 t)-\cos^2 t=c-\cos^2 t=0\quad$ (11)
for $c$ gives

c=\cos^2 t,
\end{displaymath} (12)

so substituting this back into (9) and (10) gives
$\displaystyle x$ $\textstyle =$ $\displaystyle (\cos^2 t)\cos t =\cos^3 t$ (13)
$\displaystyle y$ $\textstyle =$ $\displaystyle (1-\cos^2 t)\sin t=\sin^3 t,$ (14)

the parametric equations of the Astroid.

See also Astroid, Ellipse, Envelope

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© 1996-9 Eric W. Weisstein