Geometry Notes part 1: Line at Infinity
Author: Samuel Peterson
Date Published
2023 - 07 - 09 ISO 8601
78 - 07 - 09 PB
One of the nice things about mathematics is that it has a long shelf-life. By that I mean that a textbook on geometry written in 1970 is unlikely to be full of outdated material. Now most of these older textbooks fall out of publication for good, some are enshrined as US mathematics educational canon (e.g. Munkres' Topology); and some don't quite make the cut for such institution fixture, but are good enough to be picked up by Dover publishing.
I love browsing the offerings of Dover. One of my posts (the one on tensor analysis) was indeed about a book published by them. The books from dover are usually really cheap, sometimes less than $10. They all seem to be of good quality and, since the demand curve slopes downward, I have acquired a number of them. Recently, I purchased a book on geometry which I have started working through. It's called "Geometry: A Comprehensive Course" by Dan Pedoe.
I cannot give my full thoughts on the work since I have only begun to work through it. At any rate, I do not intend to give a full review of the book after I'm done. Rather, I intend to remark on small pieces of the book which I find interesting enough as I go along.
What have I found already that's interesting enough to write a post about? Why it's none other than ...
The line at infinity
This is a thing that's sitting in chapter 0, the stuff the book thinks you should know already. Indeed, much of it is familiar to me except the line at infinity and the section on affine geometry. The line at infinity is an extension of the 2-dimensional Euclidean plane. It is not the only such extension, indeed I do remember from my studies at university the notion of the point at infinity. This is a one point (sequential) compactification of either \(\mathbb{R}^2\) or \(\mathbb{C}\), which is probably best visualized via stereographic projection.
In my opinion, the easiest way to define it is to start with the set of ideal points. The ideal points of \(\mathbb{R}^2\) is a set of equivalence classes of lines on the real plane. Here, two lines are equivalent if they are parallel. The concept being that for any line on the plane, we invent some point outside the real plane which that line (and all other lines parallel to it) intersect. Some fun things to note about the set of ideal points, \(\mathbb{R}^2\), and lines:
- A point in \(\mathbb{R}^2\) together with an ideal point define a line.
- For all lines in \(\mathbb{R}^2\), there is a unique ideal point which it intersects.
The above properties justify us calling the set of ideals the line at infinity. I like to think of the line at infinity as a point which represents direction. Compare, for instance the case of defining a line using two points in \(\mathbb{R}^2\) (as it is done in Euclid's elements), and the case of defining a line using a point in \(\mathbb{R}^2\) and an ideal point. In the former case we are picking two points and saying, right, take a straight edge and draw a line through these two points. Clearly this is fine way of drawing a line. In the latter case, we are picking a point and just picking a direction and draw. This is also a fine way to draw a line.
That's it. The articles I write regarding my studies in this text are not necessarily going to have a point deeper than "Hey, that's neat". So it is with this one.