The equation for a sphere produces the following graphs and I have questions on them. I consecutively squared the radius of a sphere and graphed at an offset of the radius. As shown below, the spheres touch at a point. Is there a theorem that describes how the spheres of squared radius touch at a point between them?
I considered graphing in 2D to compare circles of $x,y$ at coordinate $\theta$, however couldn't figure out how to find the how to graph the second circle at the same angle where the first touches.
Equation for a Sphere in Cartesian Coordinates:
$x = x_0 + r sin(\theta)cos(\phi)$, $y = y_0 + rsin(\theta)sin(\phi)$, $z = z_0 + r cos(\phi)$
Equations:
Graphs:
