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 R_d be the displacement along R.

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

When D=R this places O_c 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.