Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Question

Conic sections seem to get special attention in early math classes.

My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensional) object?

I have a couple of guesses:

• Studying a particular "simple" example can provide insight into the general idea (i.e. cross sections of higher dimensional objects). And conic sections are deemed simple.
• The applications of ellipses, parabolas, and hyperbolas are just so vast that their graphs and properties deserve special studying (e.g. elliptical orbits).

I'd really appreciate some outside thoughts on this, even if it is just speculation. I've been giving cross sections some special study attention recently and have done a handful of google searches to try and understand why conic sections keep coming up (as can be seen in a lot of math curriculum).

Thank you!

Answer 1

One of the things that makes a cone simpler than a cube is that it is an “algebraic object” that can be defined by a simple polynomial identity ($x^2 + y^2 - z^2 = 0$). Taking cross-sections preserves this algebraic nature (since an infinite plane is also algebraic) so we end up with a quadratic curve in two variables, which is a fairly nice object. In some sense these are the “simplest” possible shapes beyond straight lines.

At the same time, you are probably aware that the shape of a quadratic curve varies drastically depending on where the minus signs live. The equation of a cone has enough minus signs that it is able to represent pretty much the entire spectrum of such curves, in contrast to, say, a sphere.

Lastly, there is some extent to which we study these because they were studied classically by the ancient Greek geometers. This kind of speaks to the utility angle in that there would be more applications of things that are well studied. But the preceding two points show that there are objective (non-historical) reasons to consider conic sections interesting. They are simple enough to be studied very thoroughly, and this simplicity also increases the chances that they would emerge naturally in many situations.

Answer 2

Elementary algebra begins with the study of linear equations. Quadratic equations naturally come next. Their graphs are ellipses, parabolas and hyperbolas. The Greek geometers knew them as sections of a cone. That geometry is often left out of beginning math courses today.

The quadratic functions are particularly useful in physics. I think they are not nearly as useful in other applications, or in understanding numbers in the news. When I teach mathematics to students who are not planning to go on in science we work on exponential functions instead.

Answer 3

My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensional) object?

Because there's nothing else. Look at the common 3D solids. The cross sections of a cuboid (and in fact any other polytope) are just a bunch of straight lines connected together, so this is just a piecewise linear graph, and piecewise linear graphs don't need to be motivated as cross-sections of three-dimensional objects. The cross sections of a spheroids (ellipsoids) are ellipses, which come up in cross sections of cones anyway. The cross sections of a cylinder are either trapeziums or ellipses. These are much less interesting and rich than conic sections!

Answer 4

There are several types of objects that can be analyzed as a curve defined by the zeroes of a second-degree polynomial in two variables: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. This includes circles, ellipses, parabolas, hyperbolas; all with the same basic equation form, but differing in their relationship between A, B, and C.

It's handy to have one term that covers all of these, and since all* of these shapes happen to be planar cross-sections of a cone, “conic sections" works nicely.

(* Well, except for the degenerate case of two parallel lines, as in $(y - x)^2 = 1$, which is actually not a conic section but a cylindrical section. Just say that a cylinder is a cone that doesn't slant.)

“Cubic sections", “pyramidal sections", etc., can just be called “polygons", so there's less need for a special word for them.

Answer 5

TL;DR: ultimately you do study cross-sections of other cone-like shapes, you just don't know about it unless you've studied some algebraic geometry.

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The cross-sections of a cone lie in perspective. Imagine you are sat at the origin in $\mathbb{R}^{3}$ and you look out along a ray (=half-line) which lies inside the cone. Fix a plane $\Pi$ which does not go through the origin in $\mathbb{R}^{3}$ and look at the conic section $C$ given by $\Pi.$ From your point of view, the conic section $C$ looks like some vaguely oval shape. Now rotate the plane $\Pi$ about a fixed point on $C$; amazingly enough, your point of view doesn't change! Your vaguely oval shape stays completely still throughout the entire process. Even though, from an extrinsic perspective, the conic section $C$ is changing dramatically during the process - maybe at first it was a circle, and then as $\Pi$ rotates, it becomes an ellipse, then very briefly a parabola, and then a hyperbola - the whole time, what you see from the origin looks the same.

Roughly speaking, the study of this phenomenon used to be known as projective geometry. Modern projective geometry has its roots in this, but has grown far outside these bounds.

Suppose you give me a polynomial $f(x,y)$ in two variables; then the equation $f(x,y)=0$ defines a set of points in $\mathbb{R}^{2},$ and this set of points forms a curve. The curve might be a "conic section", like in the case $f(x,y)=x^{2}+y^{2}-1,$ or it might not be, like in the case $f(x,y)=y^{2}-x^{3}-x^{2}+1.$

Here's a neat trick: make a new polynomial $F(x,y,z),$ this time in three variables, as follows: $F(x,y,z)=f(x/z,y/z)\cdot z^{\deg{f}}.$ Let's look at the two examples I mentioned above, to get a feel for this. If $f(x,y)=x^{2}+y^{2}-1,$ then $$F(x,y,z)=f(x/z,y/z)\cdot z^{2}=z^{2}\left(\frac{x^{2}}{z^{2}}+\frac{y^{2}}{z^{2}}-1\right)=x^{2}+y^{2}-z^{2}.$$ In the same way, if $f(x,y)=y^{2}-x^{3}-x^{2}+1,$ then $$F(x,y,z)=z^{3}\left(\frac{y^{2}}{z^{2}}-\frac{x^{3}}{z^{3}}-\frac{x^{2}}{z^{2}}+1\right)=y^{2}z-x^{3}-x^{2}z+z^{3}.$$ By construction, what we get at the end is always a homogeneous polynomial, that is, every monomial term in the polynomial is of the same fixed degree (in the above examples, $2$ and $3$, respectively).

Why is this cool? Consider the set of points in $\mathbb{R}^{3}$ satisfying $F(x,y,z)=0.$ What does this look like? In the case of the circle $f(x,y)=x^{2}+y^{2}-1,$ we got $F(x,y,z)=x^{2}+y^{2}-z^{2},$ and $\{(x,y,z):x^{2}+y^{2}=z^{2}\}$ is exactly the standard cone in $\mathbb{R}^{3}$! The "conic section" $x^{2}+y^{2}=1$ is just what you get when you intersect this surface with the plane $\Pi=\{z=1\}.$ And in fact, more generally, if you give me any homogeneous polynomial $F(x,y,z),$ then the set $\{(x,y,z):F(x,y,z)=0\}$ will be a cone over the curve $f(x,y)=F(x,y,1)=0.$ Make no mistake: this "cone" doesn't look like the kind of cone you may be used to; it will be weirdly shaped, so that the intersection with the plane $\Pi=\{z=1\}$ will give back the curve $F(x,y,1)=0.$

What does this mean? It means that whenever you consider an algebraic curve $f(x,y)=0$ in in the plane (and I wager that you do this a lot, whether or not you realize it), you are in fact already considering a "conic" section; it's just the cone is not quite what you are expecting! But it does have the same crucial "perspective" property that I point out before.

In algebraic geometry, you would study the homogenization $F(x,y,z)=0,$ but the modern viewpoint is not to think of this as a surface in space, but instead as a curve in the projective plane $\mathbb{P}^{2}$: each line through the origin in $\mathbb{R}^{3}$ is represented by a point in $\mathbb{P}^{2}.$ I'll leave it to you to imagine a line through the origin rotating around in space, and sometimes it is contained, as a subset, in the surface $F(x,y,z)=0$ (think of the special case of the ordinary cone $x^{2}+y^{2}=z^{2}$ with which you're already familiar), but usually it only intersects this surface at the origin; those times when it is contained in the surface are the "points" on the "curve" $F(x,y,z)=0$ in $\mathbb{P}^{2}.$

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So why the focus on classical conic sections? This has a simple answer: because they are the easiest case! They are one of the few truly accessible cases in algebraic geometry, in the sense that it is very easy to classify them: you have circles, hyperbolas, and ellipses (plus some degenerate cases). With more general "cones", this classification is not always so easy! And moreover, there is a fair amount to be said, many ways to say those things, and the maths relates to lots of other things (besides geometry, the study of conic sections is by definition also the study of quadratic forms).

Answer 6

I would suggest they get attention not "because they are conics" but because in two dimensions, pictures of something which is very important in analysis and applied math just happen to be conics, and drawing pictures is a good way to understand something that could be taught later using only algebra and calculus.

The first non-zero term of the Taylor expansion of a smooth function $f(x)$ about a maximum or minimum value $x = x_0$ is quadratic: $$f(x) \approx f(x_0) + \frac 1 2 f''(x_0) (x-x_0)^2.$$ Generalizing $x$ to an $n$-dimensional vector $\mathrm{x}$ gives the result $$f(\mathrm{x}_0) \approx f(\mathrm{x}_0) + \frac 1 2 (\mathrm{x} - \mathrm{x}_0)^T \mathrm{A}\, (\mathrm{x} - \mathrm{x}_0)$$ where $\mathrm{A}$ is the Hessian matrix of partial derivatives of $\mathrm{x}$.

In two dimensions, this is a quadratic form whose graph is a conic section.

The graphical approach illustrates the general idea that there is a "special" coordinate system in which the "cross terms" in the product disappear, or in the general case, the matrix $\mathrm{A}$ is diagonal. That notion will reappear in the context of linear algebra and eigenvalue problems.

So the long-term motivation is certainly not clear to the students at this stage, but "studying conics" is a way to introduce them to some important ideas for the future.

The fact that curves defined by the above equations are also cross sections of a circular cone is just an "interesting factoid" at this stage of the students' math education, though it will turn up again for those who eventually study differential geometry! A modern math course is unlikely to spend much time on purely geometrical proofs of the properties of conics, though comparing classical (3D) geometry with analytic geometry in that situation could be interesting.

Looking at it from this perspective, the fact that "conics are cross sections of a solid" is not the important issue here, so studying "cross sections of other solids" is beside the point.