How can I compute the common points when spheres intersect?

Question

When two spheres intersect they should share one or two points. I want to know how to compute those points?

I was thinking of something like this:

• -check if the spheres intersect
• -calculate radius_1 distance from center_1 in the direction of center_2
• -calculate radius_2 distance from center_2 in the direction of center_1
• -substract the smaller to the larger and have that one as "collision" point

This sounds to me a little too tricky, I wanted to know if there is a simplier way to achieve this?

Answer 1

Let C₁ and C₂ be the sphere centres, and R₁ and R₂ their radiuses. We pick a point A on the intersection circle and we want to find B, the centre of the intersection circle:

Pythagoras gives us the following two equalities:

AB² = AC₁² - BC₁²
AB² = AC₂² - BC₂²

Combining the two and replacing AC₁ with R₁ and AC₂ with R₂:

BC₁² - BC₂² = R₁² - R₂²

We know that C₁B + BC₂ = C₁C₂:

                BC₁² - (C₁C₂ - BC₁)² = R₁² - R₂²
BC₁² - C₁C₂² - BC₁² + 2 * C₁C₂ * BC₁ = R₁² - R₂²
                      2 * C₁C₂ * BC₁ = R₁² - R₂² + C₁C₂²
                                 BC₁ = (R₁² - R₂² + C₁C₂²) / (2 * C₁C₂)

Giving us the final formula for the C₁B vector:

C₁B = (R₁² - R₂² + C₁C₂²) / (2 * C₁C₂²) * C₁C₂

Using pseudocode:

float P = dot(center_2 - center_1, center_2 - center_1);
float Q = (radius_1 * radius_1 - radius_2 * radius_2 + P) / (2.0f * P);
Point B = center_1 + Q * (center_2 - center_1);

Answer 2

Here a simple method to compute the center of the intersection circle between two (hyper)spheres that works for every dimension (but 0D) using some vector notation.

Let C1 and C2 be the center of your (hyper)spheres, r1 and r2 the radii.

D := C2 - C1 is the vector that moves C1 to C2; the leght of D (|D|) is the distance between centers.

Now d := |D| -(r1+r2) is the distance between the (hyper)spheres: positive menans no intersection, 0 menans point intersection, negative means intersection.

In the latter case, -d is an usefull measure of how much the spheres compenetrate.

Now the distance between C1 and the center(C) we are looking for, is r1 - (-d)/2; being -d/2 the half distance and being substracted from r1 because r1 is "responsible" for "half compenetration".

Then C = D /|D| * (r1+d/2)