Sphere and plane

The relative position of a sphere and a plane is established by comparing the radius $R$ with the distance $d$ of the centre from the plane.

Property — Relative positions of sphere–plane

Let $d=d(C,\pi)$:

• $d>R$: plane external to the sphere;
• $d=R$: plane tangent; the point of contact is the foot of the perpendicular from $C$ to $\pi$;
• $d<R$: plane secant; the section is a circle of radius $r=\sqrt{R^2-d^2}$.

Section of a sphere with a secant plane: $R^2=d^2+r^2$.

Remark — Centre of the section

The centre of the circular section is the foot of the perpendicular from $C$ to the plane: one writes the line through $C$ with direction $\vec n$, puts it in a system with the plane and solves.

Links

Topics: Analytic geometry in space
Concepts: Point-plane distance · Cartesian plane · Circular section · Sphere
Skills: Analytic geometry · Reasoning by cases · Using formulae