# intersection of quadratic surface and plane

The intersection of a sphere with a plane (http://planetmath.org/IntersectionOfSphereAndPlane) is a circle, similarly the intersection of any surface of revolution^{} formed by the revolution of an ellipse^{} or a hyperbola^{} about its axis with a plane perpendicular^{} 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 curves^{} (conics). If for example the ellipsoid^{}

$\frac{{x}^{2}}{{a}^{2}}}+{\displaystyle \frac{{y}^{2}}{{b}^{2}}}+{\displaystyle \frac{{z}^{2}}{{c}^{2}}}=\mathrm{\hspace{0.33em}1$ | (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

$$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}1},$$ |

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

$\frac{{x}^{2}}{{a}^{2}}}-{\displaystyle \frac{{y}^{2}}{{b}^{2}}}=\mathrm{\hspace{0.33em}2}z.$ | (2) |

Cutting it e.g. with the plane $y=b$, which is parallel^{} to the $zx$-plane, the substitution yields the equation

$$2z=\frac{{x}^{2}}{{a}^{2}}-1$$ |

meaning that the intersection curve in the plane $y=b$ has the projection^{} (http://planetmath.org/ProjectionOfPoint) parabola^{} in the $zx$-plane with such an equation, and accordingly is such a parabola.

If we cut the surface (2) with the plane $z=\frac{1}{2}$, the result is the hyperbola having the projection

$$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}1}$$ |

in the $xy$-plane. But cutting with $z=0$ gives $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=\mathrm{\hspace{0.33em}0}$, i.e. the pair of
lines $y=\pm \frac{b}{a}x$ which is a degenerate conic.

Let us then consider the general equation

$A{x}^{2}+B{y}^{2}+C{z}^{2}+2{A}^{\prime}yz+2{B}^{\prime}zx+2{C}^{\prime}xy+2{A}^{\prime \prime}x+2{B}^{\prime \prime}y+2{C}^{\prime \prime}z+D=\mathrm{\hspace{0.33em}0}$ | (3) |

of quadratic surface and an arbitrary plane

$ax+by+cz+d=\mathrm{\hspace{0.33em}0}$ | (4) |

where at least one of the coefficients $a$, $b$, $c$ is distinct from zero. Their intersection equation is obtained, supposing that e.g. $c\ne 0$, by substituting the solved form

$$z=-\frac{ax+by+d}{c}$$ |

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

$$\alpha {x}^{2}+\beta {y}^{2}+2\gamma xy+2\delta x+2\epsilon y+\zeta =\mathrm{\hspace{0.33em}0},$$ |

which a quadratic curve (http://planetmath.org/QuadraticCurves) 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 |