intersection of sphere and plane

TheoremMathworldPlanetmath.  The intersection curve of a sphere and a plane is a circle.

Proof.  We prove the theorem without the equation of the sphere.  Let c be the intersection curve, r the radius of the sphere and OQ be the distanceMathworldPlanetmath of the centre O of the sphere and the plane.  If P is an arbitrary point of c, then OPQ is a right triangleMathworldPlanetmath.  By the Pythagorean theoremMathworldPlanetmathPlanetmath,

PQ=ϱ=r2-OQ2= constant.

Thus any point of the curve c is in the plane at a distance ϱ from the point Q, whence c is a circle.

Remark.  There are two special cases of the intersectionMathworldPlanetmath of a sphere and a plane:  the empty setMathworldPlanetmath of points (OQ>r) and a single point (OQ=r); these of course are not curves.  In the former case one usually says that the sphere does not intersect the plane, in the latter one sometimes calls the common point a zero circle (it can be thought a circle with radius 0).

Title intersection of sphere and plane
Canonical name IntersectionOfSphereAndPlane
Date of creation 2013-03-22 18:18:39
Last modified on 2013-03-22 18:18:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 22
Author pahio (2872)
Entry type Theorem
Classification msc 51M05
Related topic ConformalityOfStereographicProjection
Defines zero circle