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

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

${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=\mathrm{\hspace{0.33em}2}z.$(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 hyperbolahaving 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 oflines  $y=\pm \frac{b}{a}x$  which is a degenerate conic.

Let us then consider the general equation

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