Sphere-Sphere Intersection

Let two spheres of radii  and  be located along the x-axis centered at  and , respectively. Not surprisingly, the analysis is very similar to the case of the circle-circle intersection. The equations of the two spheres are

			(1)
			(2)

Combining (1) and (2) gives

(3)

Multiplying through and rearranging give

(4)

Solving for  gives

(5)

The intersection of the spheres is therefore a curve lying in a plane parallel to the -plane at a single -coordinate. Plugging this back into (◇) gives

			(6)
			(7)

which is a circle with radius

			(8)
			(9)

The volume of the three-dimensional lens common to the two spheres can be found by adding the two spherical caps. The distances from the spheres'centers to the bases of the caps are

			(10)
			(11)

so the heights of the caps are

			(12)
			(13)

The volume of a spherical cap of height  for a sphere of radius  is

(14)

Letting  and  and summing the two caps gives

			(15)
			(16)

This expression gives  for  as it must. In the special case , the volume simplifies to

(17)

In order for the overlap of two equal spheres to equal half the volume of each individual sphere, the spheres must be separated by a distance

			(18)
			(19)

Sphere to Plane Contact Force (3D)

his block implements a contact force between a sphere and a plane. The force is active above and below the plane. This is part of the Simscape Multibody Contact Forces Library

Frame connected to PlaB port:

• Located at midpoint of plane (x, y, and z).
• Z-axis is normal to the surfaces where force is active.

Frame connected to the SphF port:

• Located at center of sphere.
• Orientation does not matter.

Output signal is a bus with intermediate calculations and total force.