# Boundary of an inequality

## 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
Homework Helper
Gold Member
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.

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

fresh_42
Mentor
2021 Award
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