Summary of Quadric Surfaces

A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
The intersection of a three-dimensional surface and a plane is called a trace. To find the trace in the $xy$ -, $yz$ -, or $xz$ -planes, set $z=0$ , $x=0$ , or $y=0$ , respectively.
Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form $Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0$ .
To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
Important quadric surfaces are summarized in Figure 8 and Figure 9.

Glossary

cylinder: a set of lines parallel to a given line passing through a given curve

ellipsoid: a three-dimensional surface described by an equation of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ all traces of this surface are ellipses

elliptic cone: a three-dimensional surface described by an equation of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$ traces of this surface include ellipses and intersecting lines

elliptic paraboloid: a three-dimensional surface described by an equation of the form $z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ traces of this surface include ellipses and parabolas

hyperboloid of one sheet: a three-dimensional surface described by an equation of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ traces of this surface include ellipses and parabolas

hyperboloid of two sheets: a three-dimensional surface described by an equation of the form $\frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ traces of this surface include ellipses and parabolas

quadric surfaces: surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)

trace: the intersection of a three-dimensional surface with a coordinate plane