# tangent plane of quadratic surface

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

$A{x}^{2}+B{y}^{2}+C{z}^{2}+2{A}^{\prime}yz+2{B}^{\prime}zx+2{C}^{\prime}xy+2{A}^{\prime \prime}x+2{B}^{\prime \prime}y+2{C}^{\prime \prime}z+D=0$ | (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

$$A{x}_{0}x+B{y}_{0}y+C{z}_{0}z+{A}^{\prime}({z}_{0}y+{y}_{0}z)+{B}^{\prime}({x}_{0}z+{z}_{0}x)+{C}^{\prime}({y}_{0}x+{x}_{0}y)+{A}^{\prime \prime}(x+{x}_{0})+{B}^{\prime \prime}(y+{y}_{0})+{C}^{\prime \prime}(z+{z}_{0})+D=0.$$ |

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

Example. The tangent plane of the elliptic paraboloid $4{x}^{2}+9{y}^{2}=2z$ set in the point $({x}_{0},{y}_{0},{z}_{0})$ of the surface is $4{x}_{0}x+9{y}_{0}y=z+{z}_{0}$, and especially in the point $(\frac{1}{2},\frac{1}{3},\mathrm{\hspace{0.17em}1})$ 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 |