Section A.8 Conic Sections and Quadric Surfaces
A conic section is the curve of intersection of a cone and a plane that does not pass through the vertex of the cone. This is illustrated in the figures below.
An equivalent 1 (and often used) definition is that a conic section is the set of all points in the \(xy\)-plane that obey \(Q(x,y)=0\) with
\begin{equation*}
Q(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F =0
\end{equation*}
- \(\al x^2 + \be y^2 =\ga\) with \(\al,\be,\ga \gt 0\text{,}\) which is an ellipse (or a circle),
- \(\al x^2 - \be y^2 =\ga\) with \(\al,\be \gt 0\text{,}\) \(\ga\ne0\text{,}\) which is a hyperbola,
- \(x^2 = \delta y\text{,}\) with \(\delta\ne 0\) which is a parabola.
The three dimensional analogs of conic sections, surfaces in three dimensions given by quadratic equations, are called quadrics. An example is the sphere \(x^2+y^2+z^2=1\text{.}\)
Here are some tables giving all of the quadric surfaces.
It is outside our scope to prove this equivalence.
Technically, we should also require that the constants \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) \(D\text{,}\) \(E\text{,}\) \(F\text{,}\) are real numbers, that \(A\text{,}\) \(B\text{,}\) \(C\) are not all zero, that \(Q(x,y)=0\) has more than one real solution, and that the polynomial can't be factored into the product of two polynomials of degree one.
This statement can be justified using a linear algebra eigenvalue/eigenvector analysis. It is beyond what we can cover here, but is not too difficult for a standard linear algeba course.