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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 (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*}
being a polynomial of degree two. By rotating and translating our coordinate system the equation of the conic section can be brought into one of the forms
\(\al x^2 + \be y^2 =\ga\) 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.
equation in standard form
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
\(x^2\!+\!y^2\!+\!z^2=r^2\)
\(x=\) constant cross-section
\(y=\) constant cross-section
\(z=\) constant cross-section
Figure A.8.1. Table of conic sections
equation in standard form
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}\)
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}\)
\(x=\) constant cross-section
two lines if \(x=0\text{,}\) hyperbola if \(x\ne 0\)
\(y=\) constant cross-section
two lines if \(y=0\text{,}\) hyperbola if \(y\ne 0\)
\(z=\) constant cross-section
Figure A.8.2. Table of quadric surfaces-1
hyperboloid of two sheets
equation in standard form
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\)
\(\frac{y^2}{b^2}-\frac{x^2}{a^2}=\frac{z}{c}\)
\(x=\) constant cross-section
\(y=\) constant cross-section
\(z=\) constant cross-section
two lines if \(z=0\text{,}\) hyperbola if \(z\ne 0\)
Figure A.8.3. Table of quadric surfaces-2
