# 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

 $\displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\;=\;1$ (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

 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\;=\;1,$

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

 $\displaystyle\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\;=\;2z.$ (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

 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\;=\;1$

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

Let us then consider the general equation

 $\displaystyle Ax^{2}+By^{2}+Cz^{2}+2A^{\prime}yz+2B^{\prime}zx+2C^{\prime}xy+2% A^{\prime\prime}x+2B^{\prime\prime}y+2C^{\prime\prime}z+D\;=\;0$ (3)

of quadratic surface and an arbitrary plane

 $\displaystyle ax\!+\!by\!+\!cz\!+\!d\;=\;0$ (4)

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

 $\alpha x^{2}+\beta y^{2}+2\gamma xy+2\delta x+2\varepsilon y+\zeta\;=\;0,$

which a quadratic curve (http://planetmath.org/QuadraticCurves) or some of the degenerated cases of them.

Title intersection of quadratic surface and plane IntersectionOfQuadraticSurfaceAndPlane 2013-03-22 18:31:38 2013-03-22 18:31:38 pahio (2872) pahio (2872) 7 pahio (2872) Result msc 51N20 QuadraticSurfaces QuadraticCurves Conic EquationOfPlane ConicSection