- #1
jtleafs33
- 28
- 0
Homework Statement
I need to isolate the expressions for ellipsoidal coordinates (see below)...
I'm given:
x2=[itex]\frac{(a^2+\lambda)(a^2+\mu)(a^2+\nu)}{(a^2-b^2)(a^2-c^2)}[/itex]
y2=[itex]\frac{(b^2+\lambda)(b^2+\mu)(b^2+\nu)}{(b^2-a^2)(b^2-c^2)}[/itex]
z2=[itex]\frac{(c^2+\lambda)(c^2+\mu)(c^2+\nu)}{(c^2-b^2)(c^2-a^2)}[/itex]
For [itex]-\lambda<c^2<-\mu<b^2<-\nu<a^2[/itex]
And, I need to transform this to:
[itex]\frac{x^2}{a^2+\lambda}[/itex]+[itex]\frac{y^2}{b^2+\lambda}[/itex]+[itex]\frac{z^2}{c^2+\lambda}[/itex]=1
[itex]\frac{x^2}{a^2+\mu}[/itex]+[itex]\frac{y^2}{b^2+\mu}[/itex]+[itex]\frac{z^2}{c^2+\mu}[/itex]=1
[itex]\frac{x^2}{a^2+\nu}[/itex]+[itex]\frac{y^2}{b^2+\nu}[/itex]+[itex]\frac{z^2}{c^2+\nu}[/itex]=1
The Attempt at a Solution
I've tried to solve the 3 equations in the first part as a system of equations to end up with 3 new equations, one each for [itex]\lambda,\mu,\nu[/itex] in terms of only [itex]x,y,z,a,b,c[/itex], but this just keeps getting more and more complicated. Solving in this 'traditional' way gives me expressions that even Maple or Mathematica refuse to isolate for one variable. I think there maybe is something I'm supposed to note about the geometry or something that helps me develop the second 3 equations?. Any help would be greatly appreciated!