Step By Step Calculus » 3.1 - General Conics

Synopsis
Double Napped Cone and Conic Sections The conic sections are produced when a plane intersects a double napped cone, two cones attached at a vertex. The vertical line through the vertex is called the central axis, considered to have an angle 0^{\circ}0^{\circ}, and the diagonal sides are called generators, considered to have an angle \theta\theta measured clockwise from the central axis. The following table summarizes the conic sections based on the angle \phi\phi between the intersecting plane and the central axis.
\begin{tabular}{l|l|c}\hline Type & Conic & Angle, $\phi$\\\hline & Parabola & $\theta$\\ Non-degenerate conic sections & Circle & $90^{\circ}$\\ (plane does not pass through vertex) & Ellipse & $\theta<\phi<90^{\circ}$\\ & Hyperbola & $0\le\phi<\theta$\\ \hline & Single line & $\theta$\\ Degenerate conic sections &A point & $\theta<\phi\le90^{\circ}$\\ (plane passes through vertex) &Two intersecting lines & $0\le\phi<\theta$\\ & & \\\hline \end{tabular} General and Standard Forms of Conic Sections The general form that represents all conics is given by