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) 
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
Solving



(11) 
for gives

(12) 
so substituting this back into (9) and (10) gives
the parametric equations of the Astroid.
See also Astroid, Ellipse, Envelope
© 19969 Eric W. Weisstein
19990525