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

 $\displaystyle Ax^{2}+By^{2}+Cz^{2}+2A^{\prime}yz+2B^{\prime}zx+2C^{\prime}xy+2% A^{\prime\prime}x+2B^{\prime\prime}y+2C^{\prime\prime}z+D=0$ (1)

where $A,\,B,\,C,\,A^{\prime},\,B^{\prime},\,C^{\prime},\,A^{\prime\prime},\,B^{% \prime\prime},\,C^{\prime\prime},\,D$ are constants and at least one of the six first is distinct from zero.  The equation of the tangent plane of the surface, with $(x_{0},\,y_{0},\,z_{0})$ as the point of tangency, is

 $Ax_{0}x+By_{0}y+Cz_{0}z+A^{\prime}(z_{0}y+y_{0}z)+B^{\prime}(x_{0}z+z_{0}x)+C^% {\prime}(y_{0}x+x_{0}y)+A^{\prime\prime}(x+x_{0})+B^{\prime\prime}(y+y_{0})+C^% {\prime\prime}(z+z_{0})+D=0.$

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

Example.  The tangent plane of the elliptic paraboloid$4x^{2}+9y^{2}=2z$  set in the point  $(x_{0},\,y_{0},\,z_{0})$  of the surface is  $4x_{0}x+9y_{0}y=z+z_{0}$,  and especially in the point  $(\frac{1}{2},\,\frac{1}{3},\,1)$  it is  $2x+3y-z-1=0$.

Title tangent plane of quadratic surface TangentPlaneOfQuadraticSurface 2013-03-22 14:58:48 2013-03-22 14:58:48 pahio (2872) pahio (2872) 7 pahio (2872) Result msc 51N20 TangentOfConicSection QuadraticSurfaces