QUADRIC SURFACES

A Quadric surface is the graph of a Quadratic Relation

$f(x, y, z) = a x^2 + b y^2 + c z^2 + 2h x + 2k y + 2m z +2p xy + 2q yz + 2zx + d = 0.$

This is a degree $2$ equation in $x,\, y,\, z$. We learned earlier that conic sections were the graphs of quadratic relations

$a x^2 + b y^2 +2h x +2k y +2p xy + d = 0$

of two variables $x, \, y$, and as we'll soon be doing all the time, we try to understand the $3$-variable case by reducing everything to the $2$-variable case. But how can that be done? Well, a conic section is a graph in a plane and we know a lot about graphs in the plane - so we take plane slices! In fancy mathematical language, we take the TRACE of the surface on a plane. But which planes shall we take?
    Of course, we could take any plane $Ax + By + C z = D$, but if we are trying to visualize what's happening, it's best to take planes parallel to the coordinate planes. Now the $xy$-plane is the plane $z = 0$, while the $yz$-plane is $x = 0$ and the $xz$-plane is $y = 0$. So

    1. $x = X$ is a vertical plane parallel to the $yz$-plane and shifted by $X$ along the $x$-axis,

    2. $y = Y$ is a vertical plane parallel to the $xz$-plane and shifted by $Y$ along the $y$-axis,

    3. $z = Z$ is a horizontal plane parallel to the $xy$-plane and shifted by $Z$ along the $z$-axis.

Taking the trace of $f(x, y, z) = 0$ on the plane $x = 4$, say, simply amounts to substituting in for $x$ to get $f(4, y, z) = 0$ whose graph will be a conic section - in geometric terms we are taking the intersection of the plane $x = 4$ and the graph of $f(x, y, z) = 0$! That's easy, so long as we know all about graphs in the plane!!

If we are very lucky, then the intersection of the graph of $f(x, y, z) = 0$ with every plane $x = X$ is exactly the same. In other words, $f(X, y, z) = 0$ for all $X$. This can happen only if $f(x, y, z)$ does not depend on $X$. When $f(x, y, z)$ is independent of one of the variables $x, y$ or $z$, we'll say the graph of $f = 0$ is a CYLINDRICAL SURFACE.

Both surfaces to the right are good examples. Can you see that the first one is the graph of $x^2+y^2 = r^2$ for some $r$, while the second one is the graph of $z-y^2 = 0$?

We're also pretty familiar with the case when EVERY plane cuts the surface in a circle except that now the circle may change with the plane. This is more familiar to us if we think of it simply as a surface, not something to slice!

A SPHERE of radius $r$ centered at the origin consists of all points $(x, y, z)$ a distance $r$ from $(0, 0, 0)$. By the distance formula this means that

$x^2 + y^2 + z^2 = r^2.$

If we shift the center to $(h, k, m)$ the equation becomes

$(x-h)^2 + (y-k)^2 + (z-m)^2 = r^2\,,$

which after expansion is

$x^2 + y^2 + z^2 - 2h x - 2k y -2m z + d = 0$

for some constant $d$. Cutting this sphere with the horizontal plane $z = 1$ gives a circle; algebraically, this means setting $z = 1$ which then gives the circle

$x^2 + y^2 = r^2 - 1.$

What does this mean if $r = 1$ or $r < 1$? In fact, every plane cuts a sphere either in a circle or a point, or misses the sphere altogether. The graphic to the right shows the circle case.

Notice that the coefficients of the degree $1$ terms tell us where the sphere is centered. For a general quadric surface we shall say it is in STANDARD POSITION when it is 'somehow' centered at the origin; for a sphere that means $h = k = m = 0$, but there are important cases where these terms are non-zero. There are no such cross terms like $xy$ either - why not? Well, these usually mean that the quadric surface has been rotated, and if we rotate a sphere it still looks exactly the same!! To keep the math simple then from now on we'll only look at quadric surfaces

$f(x, y, z) = a x^2 +b y^2 + c z^2 + 2h x +2k y +2m z + d = 0$

in Standard Position with just a few non-zero cofficients.

Here are two such examples: can you see where to draw the axes? What about the origin? What do the plane slices look like? Can you confirm your answers by using the given quadratic relation?

Is it becoming clearer how the other surfaces go? Here they are along with the corresponding quadratic relations. Try constructing both vertical and horizontal slices.