Ellipse Envelope

$\begin{figure}\begin{center}\BoxedEPSF{AstroidEllipses.epsf}\end{center}\end{figure}$

Consider the family of Ellipses

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

(1)

for $c\in [0,1]$ . The Partial Derivative with respect to

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

(2)

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

(3)

Combining (1) and (3) gives the set of equations

$\begin{displaymath} \left[{\matrix{{1\over c^2} & {1\over(1-c)^2}\cr {1\over c^3... ...rix{x^2\cr y^2\cr}}\right] = \left[{\matrix{1\cr 0\cr}}\right] \end{displaymath}$

(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

$\begin{displaymath} \Delta =-{1\over c^2(1-c)^3}-{1\over c^3(1-c)^2} = -{1\over c^3(1-c)^3}, \end{displaymath}$

(6)

so (5) becomes

$\begin{displaymath} \left[{\matrix{x^2\cr y^2\cr}}\right]=\left[{\matrix{c^3\cr (1-c)^3\cr}}\right]. \end{displaymath}$

(7)

Eliminating

then gives

$\begin{displaymath} x^{2/3}+y^{2/3}=1, \end{displaymath}$

(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)

Solving

${\partial x\over\partial t}{\partial y\over\partial c}-{\partial x\over\partial c}{\partial y\over\partial t} = (-c\sin t)(-\sin t)-(\cos t)[(1-c)\cos t]$

$= c(\sin^2 t+\cos^2 t)-\cos^2 t=c-\cos^2 t=0\quad$ (11)

for gives

$\begin{displaymath} c=\cos^2 t, \end{displaymath}$

(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

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