Parametric representation of the intersection of spheres

Question

Goal:

I am trying to find the curve of intersection of two spheres.

$\begin{align*}x^2+y^2+z^2 &= 9 \\ (x-3)^2+y^2+(z-1)^2 &= 4 \end{align*}$

What I have done:

One of the ways of achieving this is to do the following.

1. Eliminate one variable, in this case $y$, and obtain an $xz$-relation.
2. Simplify down to a linear expression $6x+2z-15=0$. This is the plane that contains the circle where they intersect.
3. Set $x=t$ for some parameter $t$, and then find $z(t)$ and then $y(t)$.
4. Parameterization complete.

I get $y(t)$ being a plus/minus root since $y$ is defined implicitly, and below is the diagram showing both spheres and half of the circle of where they intersect.

Problem:

There has to be a better way that uses some kind of trigonometric parameterization in the above example. How can I do this without using spherical coordinates?

Answer 1

Observe that:

1. The center of the circle lies on the line connecting the 2 centers of the sphere.
2. Hence, find the radius of the circle: $ r^2 + s^2 = 9, r^ 2 + (\sqrt{10}-s)^2 = 4 $.
3. Hence, find the center of the circle: $\frac{s}{\sqrt{10}}(3,0,1)$.