User:Capttwinky

To characterize the circle formed by the intersection when D≤R, let the center of the sphere be the origin of an XY plane P=(x,y,0) and D=0. In this case the sphere and the circle have the same radius and center, making the circle a great circle on the sphere.

Increasing D moves the plane perpendicular to some radius of the sphere R, decreasing the radius of the circle. Let the motion be in the negative Z direction. Let Rd be the displacement along R.

This places the center of the circle, Oc at (0,0,-Rd). The radius of the circle, Rc in terms of Rd is $$R_c = \sqrt {R^2 - R_d ^2 }.$$

When D=R this places Oc at (0,0,-R), which we know is a point since $$\lim_{R_d \to R} \sqrt {R^2  - R_d }  = 0.$$

These results may be generalized to motion along any radius of the sphere via linear transformations.