This 3-part series attempts to explain a classical result in algebraic geometry to a lay-audience, using only high-school level mathematics.
From ancient Greece to the modern era: Cones, curves, and coordinates
In the 2nd century BC, Apollonius of Perga studied and classified the so-called conic sections: geometric curves obtained by intersecting a flat 2-dimensional plane with a hollow cone. If the plane is intersected with the cone at a shallow angle, the resulting curve is an ellipse, the circle being a special case.
On the other hand, intersecting with a steep inclination gives us a hyperbola.
Lying directly between these two, when the plane is parallel to the cone, we have the parabola.
Apollonius considered these figures as living the plane as described by Euclid of Alexandria: a flat 2d world where overlapping circles and straight lines give rise to the notions of distance and angle, the angles of a triangle sum to 180 degrees, and all lines meet once except for parallel lines. It’s the plane we all remember from school.
These three families of curves are still studied by all modern students of geometry, and indeed physics, as Newton’s theory of gravitation predicts that the orbits of planets and comets all follow the path of a conic section. The three curves share many similar characteristics. In this series, I’ll explore a more modern, abstract view of conic sections, leading to a beautiful interpretation of the relationship between them.
When Descartes introduced coordinate systems to geometry, it became possible to describe these curves with algebraic equations — that is, equations that use only numbers, and some chosen letters like and , combined using only addition, subtraction, and multiplication. When an equation involves two or more letters, there are generally infinitely many ways to solve it. Thus, instead of trying to “solve for ” as we did in high school, it makes more sense to understand the geometry of the solutions.
The geometry of equations
If the Euclidean plane is given the usual coordinate system by choosing some central point to act as together with a pair of perpendicular coordinate axes, then every conic section can be described by giving an equation involving and . If a point satisfies the equation — that is, the equation is true when and — then that point lies on the conic section, and otherwise it does not. For example, the points of an ellipse satisfy an algebraic equation of the form:
where and are any two numbers that determine the size and shape of the ellipse1
Let’s be more concrete. If the ellipse in question is the circle , then the point lies on the circle, because
On the other hand, the point does not lie on the circle, because
The family of hyperbolas are almost the same, except that they satisfy an equation of the form
Finally, the points on a parabola satisfy an equation that looks something like
Again in these two latter cases, the choice of and determine the overall shape of the curve.
The ability to express geometric ideas using algebraic equations — and vice versa, to express the relationships described by algebraic equations as geometrical figures — is the inspiration for algebraic geometry. The goal this of series is to explain a modernized2 version of Apollonius’ classification of conic sections: the classification of conics in the projective plane. Along the way, we’ll have to understand what is meant by the terms conic and projective plane.
The algebro-geometric definition of conic is easy: a conic is any plane curve described by equations where the coordinates are multiplied together in pairs, like , , or , but never any higher products (for example, not or ). All of Apollonius’ conic sections are conics. The equation for the parabola has appearing by itself without being multiplied by or . That’s okay too, but the equation must have at least one “double” term to be considered a conic equation. To use the correct lingo, we say that a conic is a curve of degree 2.
Breaking Euclid’s plane
The projective plane is a weird alternative to the Euclidean plane; understanding precisely what it is will take the remainder of this post and the next. We’ll begin by re-considering the Euclidean plane. One of the most fundamental properties of any geometric space is what it means for two figures in the space to be essentially equivalent. Consider the following two rectangles.
Are these rectangles equivalent? From the Euclidean perspective, the answer is is yes, as one is just a rotated version of the other. Two shapes in the Euclidean plane are equivalent if one can be transformed into the other in such a way that all distances—and hence also all angles—are preserved. Another perspective is that one can be turned into the other by tilting your head. If the plane is given Descartes’ coordinate system, the mathematical equivalent of tilting your head is changing that coordinate system by moving and rotating the coordinate axes together. In mathematics, changing a figure while leaving it equivalent and changing the coordinate system used to describe it are two sides of the same coin, just as in the rectangle example above. Consequently, what it means for two figures to be equivalent can often be specified by describing the valid coordinate changes in that space. The first step toward the projective plane will be to relax the Euclidean criteria for equivalence of figures.
The problem is that, when studying algebraic curves, lengths and angles are not really an aspect of the algebraic structure of the equations, so saying that figures can only be considered equivalent if their lengths and angles are the same is too restrictive for algebraic geometry. Thinking back to the example of the ellipse above, we recall that ellipses of all shapes and sizes are described by an equation that looks like
for some and . So two ellipses maybe have different sizes and shapes when we draw them on paper, but because the “shape” of their algebraic equations are the same (just with different numbers and )3, the algebraic geometer has no need to distinguish between them. Hence, we will consider two figures to be equivalent if one can be turned into the other by means of rotating, reflecting, or moving—these are the Euclidean transformations—but also by shearing, or uniformly stretching the entire plane. In Euclid’s world, two triangles are equivalent if each corresponding angle and side is the same—in this new space we are describing, all triangles are equivalent. This can be summarized by saying that our transformations of the plane must take all evenly space parallel lines to evenly spaced parallel lines.
Thankfully in these articles, we’ll never have to move our curves around the plane, so it’s even simpler for us. I can now explain what equivalence means by describing the allowable coordinate transformations. In this space, two figures described in coordinates are considered equivalent if we can turn one into the other with a transformation like so:
This is enough to rotate, reflect, shear, and stretch any figure—moving the figure would be controlled by also adding an extra constant in each equation. We’ll also add an extra condition, that , a condition that I’ll call “non-degeneracy”. This guarantees that the entire plane does not get squashed into a single line or point. Such a transformation would still leave all parallel lines parallel — they’d just all be the same line, which is no good. Hence, by change of “change of coordinates” or “transformation”, I mean anything that can be described by a substitution in form of , subject to the non-degeneracy rule.
As an example of this in action, consider the rather grisly algebraic equation
Since the equation is of degree , it’s a conic. But which of Apollonius’ conic sections is it? Well,
and so if we write
as per the prescription, we obtain
so the figure was a hyperbola all along4.
We have now developed a new geometric space, more suitable for algebraic geometry, called the affine plane. It is similar to the Euclidean plane, in that it has a similar notion of parallel lines, but “angle” and “length” are meaningless concepts. Shapes that differ in angle and size are considered to be the same shape, just like in the Euclidean plane, a pair of shapes related by movement and rotation are considered to be the same. Let’s give these two spaces symbols. I’ll call the Euclidean plane , and the affine plane , where the meanings of and are obvious, and the represents that these are 2d spaces.
Geometers and philosophers from an earlier time may have objected to the mathematical validity of such a space because it is unintuitive. Intuitively, lengths and angles are what makes two shapes equivalent. It’s easy to see this by drawing them. But this is the nature of modern mathematics. It is abstract, often defying our everyday intuition and accurate pictorial representation. We will still use pictures to aid our intuition, but must remember there are mathematical characteristics of the affine space that cannot be fully captured by the picture.
We obtained the affine plane from the Euclidean plane by removing structure, namely, the concepts of length and angle, but retaining the concept of parallel lines. But for reasons that will be explained soon, this is still not enough. In the next post, we must explore the even stranger projective plane, in which even parallel lines “eventually meet”. If the affine plane was constructed by disfiguring the Euclidean space, we will construct the projective plane by disfiguring affine space. But until then, please add your questions and comments below.
- If , then the ellipse the equation describes is a circle.
- Not modern by today’s standards, but about 2000 years ahead of Apollonius.
- For the maths students: what I’m really saying here is that their coordinate rings are isomorphic
- See for yourself here.