Similarities and Differences between Cylinder Equations

Question

Describe the intersection of x^2 + y^2 ≤ 1 and y^2 + z^2 ≤ 1 and x^2 + z^2 ≤ 1.

How is this intersection the same as x^2 + y^2 + z^2 ≤ 1? How is it different?

When I graphed them in GeoGebra, they all laid right on top of one another as if they had all points in common. How are they the same and different?

Answer 1

The first 3 equations are cylinders, it is the intersection of 3 perpendicular cylinders of radius 1 with their axis on the x,y and z axis. $x^2+y^2+z^2 \leq 1$ describes a sphere of radius 1.

Answer 2

Note that $x^2+y^2 + z^2 \le 1 \implies x^2+y^2 \le 1$. Similarly for the other inequalities.

However, consider points like $(\sqrt{0.5}, \sqrt{0.5}, \sqrt{0.5})$, it is inside the intersection of the cylinders but it is not inside the unit sphere.