Exploring the Intersection of Ellipsoids and Spherical Shells

[Ark236]

Homework Statement

I would like to know how the boundary of the inequality change when the origin of the coordinate system changes.

Homework Equations

The original inequality is[/B]
$$ r_0 \le x^2+y^2+z^2 \le R^2$$

I would like to know the boundary of the following term, considering the previous inequality
$$ (2x-1)^2+(2y-1)^2+z^2 $$

The Attempt at a Solution

I write

$$(2x-1)^2+(2y-1)^2+z^2=4[ (x-0.5)^2+(y-0.5)^2+z^2/4] $$[/B]

but I do not know how to proceed with the problem

 

[haruspex]

Start with a simpler problem. If x²<a², what bounds can you put on (x-1)²?
It may help to play around with some examples.

 

[Ark236]

in this case $$ (x-1)^2 \le (a+1)^2 $$ and $$ (x-1/2)^2 \le (a+1/2)^2 $$

 

[fresh_42]

Both sets of numbers also describe two different shapes. One is a spherical shell, the other ellipsoids. So the first question is:
'Do you consider them as a coordinate transformation, and you want to know how the shell is transformed?' or 'Do you want to know which part of the ellipsoids intersects with the shell, i.e. both hold within the same coordinate system?'

 

[haruspex]

Both sets of numbers also describe two different shapes. One is a spherical shell, the other ellipsoids. So the first question is:
'Do you consider them as a coordinate transformation, and you want to know how the shell is transformed?' or 'Do you want to know which part of the ellipsoids intersects with the shell, i.e. both hold within the same coordinate system?'

I think what is required is the range of ellipsoids (i.e. the values of c in $c=(2x-1)^2+(2y-1)^2+z^2$) which fit between the two spherical shells.