Introduction
The conic sections are the curves cut out by second-degree equations in two variables. Asking the same question one dimension higher leads to the quadric surfaces: the surfaces in space described by a second-degree equation in three variables,
\begin{equation*}
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0,
\end{equation*}
where \(A\) through \(J\) are constants.
This general equation carries more freedom than we need. The cross terms \(xy\text{,}\) \(xz\text{,}\) and \(yz\) only tilt the surface away from the coordinate axes, and the linear terms \(Gx\) and \(Hy\) only slide it off the origin; neither affects the shape of the surface. Setting those coefficients to zero therefore costs us nothing essential, and it leaves a surface in standard position. Every example in this chapter comes from the resulting reduced equation
\begin{equation}
Ax^2 + By^2 + Cz^2 + Dz = E.\tag{6.1}
\end{equation}
Each surface below is obtained by one choice of the five remaining coefficients. What tells the surfaces apart is which of them vanish and, above all, the signs of those that do not.
