Κωνικά, i.e. the ugly shadow of algebraic geometry

When one imagines the paths along which various space objects travel what ordinarily comes into mind are circles - all the popular pictures of our solar system picture planets (the fact that Pluto has been reclassified strengthens my point because it was the only object (except for a random comet which an artist threw in) which was difficult) as encircling our sun for example. It’s less circles and more of stuff like this

and this

Indeed, the solutions to Newton’s equations of motion for one gravitating body are much richer than ordinary, boring circles. Circular orbits are but a special case of shapes that encompass ellipses, hyperbolas, parabolas etc.

To learn about those, lets go back to the Hellenistic world, the time when Ptolemaic Aegypt and Kingdom of Pergamon were the most important scientific centres of the world. One can remember both of those for impressive libraries, but they were housing also various different institutions of higher learning like Musaeum and University of Alexandria. In similar way as today mathematicians, scientists and philosophers were working and doing research there, using the financing provided by the governments of Hellenistic countries (not only that, but they also often moved between those centres, as scientists of today who are always trying to find a place where they can be paid a living wage in the sad reality of under-financed science).

Apollonius of Perga was one of those mathematicians. We know him from his multiple tomes treatise *Κωνικά* (gr. *Conics*), which topic is all about ellipses, parabolas and hyperbolas. Essentially, those are curves that one can obtain by intersecting a cone and a plane (hence the name “conics”). In modern parlance, if you are algebraic geometrically inclined (and if you are indeed, I’ll be judging you), those are examples of algebraic curves, which are described by polynomial equations.

However, Apollonius was not only interested in pure mathematics - among his interests was also astronomy, in particular the movement of the Moon and the theory of epicycles. It is not a stretch to think that the abstract treatment of geometry and the possible applications to astronomy were intertwined, although it is somewhat difficult to tell because much of this knowledge were irreparably lost with the fall of the ancient Greek civilisation. With the hindsight it is easy for us to see that those two things are really closely related - all of us learn in the school about Johannes Kepler (and Isaac Newton) and his three laws of planetary motion, and the first of those is that planets travel along ellipses. So, in order to figure out how planets and satellites and space stuff move it is essential to look into ellipses (the topic of hyperbolas and parabolas we will tackle another time) more in-depth.

As previously in the case of a circle, ellipse is described by the quadratic equation

$$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 = 1,$$

where $a$ and $b$ are real numbers which describe the exact shape (how “egg-like” the ellipse is). Obviously, setting $a=b=R$ returns us to the case of a circle, so we see that it is indeed a special case. We can plot it to have something pretty to look at, by e.g. using the following Mathematica code

`ellipse = x^2/a^2 + y^2/b^2 == 1;`

ContourPlot[(ellipse /. a -> 5 /. b -> 1) // Evaluate, {x, -5,

5}, {y, -1.5, 1.5}, Axes -> True, AxesStyle -> Black,

Frame -> False]

which gives us the graph like this

This equation is implicit - however, we can parametrise it, using an angle $\omega t$ or time $t$ as

$$ x = a \sin(\omega t + \phi),$$ $$ y = b \cos(\omega t + \phi), $$

where $\phi$ is the angle from which we start, and $\omega$ is angular velocity. Now, to verify that this parametrisation is indeed of a parametrisation of an ellipse, it is sufficient to plug this to the implicit equation and use the the Pythagorean identity.

Now, looking back at the “space” pictures, one can notice that the planet is not in the centre of the coordinate system. In fact, it is located in somewhat special point called a focus. But that will be a subject of another post…