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CLP-3 Multivariable Calculus

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*}
being a polynomial of degree two 2 . By rotating and translating our coordinate system the equation of the conic section can be brought into one of the forms 3 
  • \(\al x^2 + \be y^2 =\ga\) 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.
name
elliptic cylinder
parabolic cylinder
hyperbolic cylinder
sphere
equation in standard form
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
\(y=ax^2\)
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
\(x^2\!+\!y^2\!+\!z^2=r^2\)
\(x=\)constant cross-section
two lines
one line
two lines
circle
\(y=\)constant cross-section
two lines
two lines
two lines
circle
\(z=\)constant cross-section
ellipse
parabola
hyperbola
circle
sketch
Figure A.8.1. Table of conic sections
name
ellipsoid
elliptic paraboloid
elliptic cone
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
ellipse
parabola
two lines if \(x=0\text{,}\) hyperbola if \(x\ne 0\)
\(y=\) constant cross-section
ellipse
parabola
two lines if \(y=0\text{,}\)hyperbola if \(y\ne 0\)
\(z=\) constant cross-section
ellipse
ellipse
ellipse
sketch
Figure A.8.2. Table of quadric surfaces-1
name
hyperboloid of one sheet
hyperboloid of two sheets
hyperbolic paraboloid
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
hyperbola
hyperbola
parabola
\(y=\) constant cross-section
hyperbola
hyperbola
parabola
\(z=\) constant cross-section
ellipse
ellipse
two lines if \(z=0\text{,}\) hyperbola if \(z\ne 0\)
sketch
Figure A.8.3. Table of quadric surfaces-2
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.