A Short Lesson on Quadrics

In 1D, the familiar quadratic equation $ ax^2 +bx +c =0 $ was studied to death in high school. But we’re not quite done with it. It has to serve as the humble bottom rung in a dimensional ladder, sometimes called the dimensional dialectic in honor of Plato’s method of reaching a glimpse of the Ideas. It is a process of seeing into higher dimensions by carefully developing analogies from lower dimensions. In this case, we wish to have a peek into the 10D parameter space for the quadrics.

Unfortunately, in 1D the entire solution set (locus) of the quadratic equation is described by the familiar

which consists of a single point when $ b^2 = 4ac $, and has two or no (real) solutions, depending on which side of the discriminant surface $ b^2 = 4ac $ the point $ (a,b,c) $ is on.

This surface is called the discriminant of the quadratic equation, and for $ (a,b,c) $ on the discriminant surface, there is only one solution. On one or the other sides of the surface there are none, or two solutions.

The DPGraph visualizes $ y^2=4xz $ very well, but because the axis of this double cone is not aligned with the xyz-axes, it is easily mistaken for some other quadric. If we make the substitution

we get the more recognizable equation $ y^2=x^2-z^2 $ or $ x^2 +y^2 = z^2 $ whose locus has xy-sections the circles of radius $ |z|$.

For $ x=0 $ we factor $ 0= z^2-y^2=(z-y)(z+y) $ for the equation of two crossing lines.

Rotating these lines about the z-axis shows that the discriminant surface for the quadratic equation is, in fact, a double cone.

Now, lets (try) to do the same thing in 2D. Note that the 1D-quadratic equation is just a special case of the equation for a quadric surface, namely $ b=c=d=e=f=h=k=0$ . The loci of the 2D quadratic equation

are called the conic sections, because these curves are all obtained by cutting a (double) cone in 3D by a plane (you learned that in high school, right?)

One approach to solving quadratic equations is very geometrical (as will be apparent much later.) It consists of turning the quadratic equation into a homogeneous equation. We do this by first setting a new variable $ w=1 $ and then mutliplying through by 1 a number of times.

Let’s first see how this works in 1D. We write

Note that this is a special case for the 2D quadratic equation (4) with $ d=e=f=0 $, provided we release $ w=1 $ and let it be 2nd variable in the 2D xw-plane. Our letters may vary from paragraph to paragraph, but not their roles. Letters at the end of the alphabet are spatial coordinates, and the parameters are at the beginning of the alphabet.

We now solve the 1D homogeneous quadratic equation as follows.

If $ a=0 \and b \ne 0 $ (if it were, we’d have nothing to look at, right?) then the LHS factors into

which is true if $ w=0$ or if $ (bx+cw)=0 $ , both of which are equations of straight lines in xw-space. To get back to 1D, we set $ w=1 $ (which is still another line) and look at the intersection the locus (loci) makes with the special line $ w=1 $. Draw a figure illustrating this.

If $ a\ne0 $ then we can divide $ a $ through equation (5). Note that the RHS $ 0/a=0 $ doesn’t change. So the locus can’t change either when we solve

The last line is the equation of two lines crossing at the origin. When we set $ w=1 $ again, then these two lines cross this horizontal at the two solutions to the quadratic equations.

The foregoing was an example of how to solve an inhomogenous equation in some dimension, by solving a homogeneous equation in one dimension higher. Working in homogeneous coordinates is an essential aspect of computer graphics as it has been in algebraic geometry for hundreds of years.

Note that by solving the 1D quadratic equation in the foregoing way, also solved the special case of the 2D quadratic equations $ x^2 + bxy + cy^2 = 0 $ , only with different letters.

We next consider the "other half" of the entire quadratic equation, namely the case of linear equations in the plane, $ dx +ey +f =0$, except that we rewrite this equation again, as $ax+by+c=0$. This time we want to know how to characterize each line such an equation defines. In high school you knew lines by their point-slopes equation, or by two points the line passes through, or some other geometrical property of the line. Now we want to understand the lines in the plane as the points in some space, which we shall call the the moduli space of the lines in the plane. More familiar names for this might be configuration space. In the present case, it is not the same as the parameter space, which is the entire 3D abc-space. Why? Because the point $ (a,b,c) $ is not unique to the line. The equation $ tax + tby +tc =0 $ describes the exact same line, provided that $ t \ne 0 $, of course. (Think: what is the locus when $ t=0 $ .)

The Greeks would have said that it isn’t the triple $ (a,b,c) $ that determines the line, but their proportion $ a:b:c $. So, now we want to know how to imagine the space of all ratios $ a:b:c $ , except $ a=b=c=0 $. In modern mathematics every object should be a set, so we can define a proportion as

and the space of all proportions to be

To solve this visualization problem we shall need some topology. First, we chose a special value, $ t = \frac{1}{\sqrt{a^2+b^2+c^2}}$, so that $ (a/r,b/r,c/r) $ is where the line pierces the unit sphere. We write $ S^n $ for the n-dimensional unit sphere. $ S^1 $ is a circle.

Unfortunately, we still have two points specifying the same line, because $ (-a/r, -b/r, -c/r) $ is a point on the unit sphere in 3D abc-space that represents the same line in the $xy$-plane. These two points are antipodes on the sphere, because they are opposite each other through the center of the sphere.

In some circles of topology people would content themselves by saying that the moduli space we are seeking to visualize is the sphere with antipodes indentified to single points, which is written officially so:

But this too hard to visualize. So we proceed by throwing away one of each pair of antipodes, leaving only one equatorial hemisphere. Each of the points strictly below the equator correspond to a unique line in the xy-plane. But on the equator we still need to identify antipodes. We could throw away half the equator, and we would have a picture of the moduli space. But it have a very ``raw edge" on the equator. Sow we proceed to try and suture this raw edge. While there is a nice visualization of this procedure, it’s not the easiest to appreciate without computer graphics

\footnote{ See the illiSnail real-time interactive compuer animation.}.

So we back up a little and start with the sphere. But this time we decompose it into three pieces, a belt along the equator that extends the same distance above as below the equator. What remains are two disjoint polar caps. Every point in one cap has its antipode in the other. So we discard one of the caps and keep the other. Just remember we have to sew it back to the equatorial belt eventually.

The belt also contains antipodes. This time we cut it in half across the belt. This leaves an ``east strip" and and ``west strip". Again each point in one strip has its antipode in the other (and vice versa), so we discard one of them.

We’re not done. The vertical (short) cuts on the west strip are antipodes. But now we can identify these, provided we put a half-twist into the strip. We get a Moebius band. This has a single circle for an edge, to which we propose to glue the remaining cap. Of course we can’t do that in 3-space.

If we contort the Moebius strip and the cap just right, and allow the surface to pass through itself, then we can, and so obtain what is known as a Boy surface, because Werner Boy first figured out how to do that.

Now we are done, for now. We have proved that the moduli space for the lines in the plane is a Boy surface. For entirely different reasons, it is also called the projective plane. Visualizing a Boy surface is visualization problem left for another time.

However, we are far from having identified the moduli space of other conic sections, let alone classified all quadrics, the solutions to the 3D quadratic equation.

This lesson is, therefore, a sampler of how mathematical visualization fits into serious mathematics.