intersection of quadratic surface and plane

The intersection of a sphere with a plane ( is a circle, similarly the intersection of any surface of revolutionMathworldPlanetmath formed by the revolution of an ellipseMathworldPlanetmathPlanetmath or a hyperbolaMathworldPlanetmath about its axis with a plane perpendicularPlanetmathPlanetmathPlanetmath to the axis of revolution is a circle of latitude.

We can get as intersection curves of other quadratic surfaces and a plane also other quadratic curvesMathworldPlanetmath (conics).  If for example the ellipsoidMathworldPlanetmath

x2a2+y2b2+z2c2= 1 (1)

is cut with the plane  z=0 (i.e. the xy-plane), we substitute  z=0  to the equation of the ellipsoid, and thus the intersection curve satisfies the equation

x2a2+y2b2= 1,

which an ellipse.  Actually, all plane intersections of the ellipsoid are ellipses, which may be in special cases circles.

As another exaple of quadratic surface we take the hyperbolic paraboloid

x2a2-y2b2= 2z. (2)

Cutting it e.g. with the plane  y=b,  which is parallelMathworldPlanetmathPlanetmath to the zx-plane, the substitution yields the equation


meaning that the intersection curve in the plane  y=b  has the projectionMathworldPlanetmath ( parabolaMathworldPlanetmath in the zx-plane with such an equation, and accordingly is such a parabola.

If we cut the surface (2) with the plane  z=12, the result is the hyperbola having the projection

x2a2-y2b2= 1

in the xy-plane.  But cutting with  z=0  gives  x2a2-y2b2= 0, i.e. the pair of lines  y=±bax  which is a degenerate conic.

Let us then consider the general equation

Ax2+By2+Cz2+2Ayz+2Bzx+2Cxy+2A′′x+2B′′y+2C′′z+D= 0 (3)

of quadratic surface and an arbitrary plane

ax+by+cz+d= 0 (4)

where at least one of the coefficients a, b, c is distinct from zero.  Their intersection equation is obtained, supposing that e.g.  c0, by substituting the solved form


of (4) to the equation (3).  We then apparently have the equation of the form

αx2+βy2+2γxy+2δx+2εy+ζ= 0,

which a quadratic curve ( or some of the degenerated cases of them.

Title intersection of quadratic surface and plane
Canonical name IntersectionOfQuadraticSurfaceAndPlane
Date of creation 2013-03-22 18:31:38
Last modified on 2013-03-22 18:31:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Result
Classification msc 51N20
Related topic QuadraticSurfaces
Related topic QuadraticCurves
Related topic Conic
Related topic EquationOfPlane
Related topic ConicSection