1.9: Quadric Surfaces

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Another named class of relatively simple, but commonly occurring, surfaces is the quadric surfaces.

Definition 1.9.1. Quadrics

A quadric surface is surface that consists of all points that obey $Q(x,y,z)=0\text{,}$ with $Q$ being a polynomial of degree two¹.

Technically, we should also require that the polynomial can't be factored into the product of two polynomials of degree one.

For $Q(x,y,z)$ to be a polynomial of degree two, it must be of the form

\[\begin{gather*} Q(x,y,z) = Ax^2 +By^2 +Cz^2 +Dxy +Eyz +Fxz +Gx +Hy +Iz +J \end{gather*}\]

for some constants $A\text{,}$ $B\text{,}$ $\cdots\text{,}$ $J\text{.}$ Each constant $z$ cross section of a quadric surface has an equation of the form

\[\begin{gather*} Ax^2 + Dxy +By^2 +gx +hy +j =0,\quad z=z_0 \end{gather*}\]

If $A=B=D=0$ but $g$ and $h$ are not both zero, this is a straight line. If $A\text{,}$ $B\text{,}$ and $D$ are not all zero, then by rotating and translating our coordinate system the equation of the cross section can be brought into one of the forms²

$\alpha x^2 + \beta y^2 =\gamma $ with $\alpha ,\beta \gt 0\text{,}$ which, if $\gamma \gt 0\text{,}$ is an ellipse (or a circle),
$\alpha x^2 - \beta y^2 =\gamma $ with $\alpha ,\beta \gt 0\text{,}$ which, if $\gamma \ne0\text{,}$ is a hyperbola, and if $\gamma =0$ is two lines,
$x^2 = \delta y\text{,}$ which, if $\delta\ne 0$ is a parabola, and if $\delta=0$ is a straight line.

There are similar statements for the constant $x$ cross sections and the constant $y$ cross sections. Hence quadratic surfaces are built by stacking these three types of curves.

We have already seen a number of quadric surfaces in the last couple of sections.

We saw the quadric surface $4x^2+y^2-z^2=1$ in Example 1.7.1.
$hyperboloid1.svg$

Its constant $z$ cross sections are ellipses and its $x=0$ and $y=0$ cross sections are hyperbolae. It is called a hyperboloid of one sheet.
We saw the quadric surface $x^2+y^2=1$ in Example 1.8.2.
$cylinder.svg$

Its constant $z$ cross sections are circles and its $x=0$ and $y=0$ cross sections are straight lines. It is called a right circular cylinder.

Appendix A.8 contains other quadric surfaces.

Technically, we should also require 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 algebra course.