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Ellipse Envelope

AstroidEllipses

Consider the family of ellipses

(x^2)/(c^2)+(y^2)/((1-c)^2)-1=0
(1)

for c in [0,1]. The partial derivative with respect to c is

-(2x^2)/(c^3)+(2y^2)/((1-c)^3)=0
(2)
(x^2)/(c^3)-(y^2)/((1-c)^3)=0.
(3)

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

[1/(c^2) 1/((1-c)^2); 1/(c^3) -1/((1-c)^3)][x^2; y^2]=[1; 0]
(4)
[x^2; y^2]=1/Delta[-1/((1-c)^3) -1/((1-c)^2); -1/(c^3) 1/(c^2)][1; 0]
(5)
=1/Delta[-1/((1-c)^3); -1/(c^3)],
(6)

where the quadratic curve discriminant is

Delta=-1/(c^2(1-c)^3)-1/(c^3(1-c)^2)=-1/(c^3(1-c)^3),
(7)

so (6) becomes

[x^2; y^2]=[c^3; (1-c)^3].
(8)

Eliminating c then gives

x^(2/3)+y^(2/3)=1,
(9)

which is the equation of the astroid. If the curve is instead represented parametrically, then

x=ccost
(10)
y=(1-c)sint.
(11)

Solving

(partialx)/(partialt)(partialy)/(partialc)-(partialx)/(partialc)(partialy)/(partialt)=(-csint)(-sint)-(cost)[(1-c)cost] =c(sin^2t+cos^2t)-cos^2t=c-cos^2t=0
(12)

for c gives

c=cos^2t,
(13)

so substituting this back into (10) and (11) gives

x=(cos^2t)cost
(14)
=cos^3t
(15)
y=(1-cos^2t)sint
(16)
=sin^3t,
(17)

the parametric equations of the astroid.

See also

Astroid, Ellipse, Envelope

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Cite this as:

Weisstein, Eric W. "Ellipse Envelope." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipseEnvelope.html

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