tangent plane of quadratic surface

The common equation of all quadratic surfaces in the rectangular (x,y,z)-coordinate systemMathworldPlanetmath is

Ax2+By2+Cz2+2Ayz+2Bzx+2Cxy+2A′′x+2B′′y+2C′′z+D=0 (1)

where A,B,C,A,B,C,A′′,B′′,C′′,D are constants and at least one of the six first is distinct from zero.  The equation of the tangent plane of the surface, with (x0,y0,z0) as the point of tangency, is


This is said to be obtained from (1) by polarizing it.

Example.  The tangent plane of the elliptic paraboloid4x2+9y2=2z  set in the point  (x0,y0,z0)  of the surface is  4x0x+9y0y=z+z0,  and especially in the point  (12,13, 1)  it is  2x+3y-z-1=0.

Title tangent plane of quadratic surface
Canonical name TangentPlaneOfQuadraticSurface
Date of creation 2013-03-22 14:58:48
Last modified on 2013-03-22 14:58:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Result
Classification msc 51N20
Related topic TangentOfConicSection
Related topic QuadraticSurfaces