## Ellipse Envelope

Consider the family of Ellipses

 (1)

for . The Partial Derivative with respect to is
 (2)

 (3)

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

 (5)

where the Discriminant is
 (6)

so (5) becomes
 (7)

Eliminating then gives
 (8)

which is the equation of the Astroid. If the curve is instead represented parametrically, then
 (9) (10)

Solving

 (11)
for gives

 (12)

so substituting this back into (9) and (10) gives
 (13) (14)

the parametric equations of the Astroid.