Geometric Tests - 3D Math Primer for Graphics and Game Development - page 733

Figure A.13
A sphere moving towards a plane

The problem is greatly simplified by realizing that no matter where on
the surface of the plane the intersection occurs, the point of contact on the
surface of the sphere is always the same. That point of contact p 0 is given
by c − r n , as shown in Figure A.14.

Figure A.14
Point of contact between a sphere and a plane

Now that we know which point on the sphere first contacts the plane,
we can use a simple ray-plane intersection test from Section A.9. We start
with our solution to the ray-plane intersection test from (A.4) and then
substitute c − r n for p 0 :

t = d − p 0

n

Dynamic intersection of
a sphere and plane

,

d n

= d − ( c − r n ) n
d n

,

= d − c n + r
d n

.