Every functional-analysis teacher who introduces the \ell^1 norm mentions that the resulting geometry is called taxi-cab geometry, and may even draw a picture or two, or tell a fanciful story, to explain it; but it had never occurred to me to try drawing the analogues of conic sections before. This is a very nice way of both illustrating the (geo)metric (as opposed to algebro-geometric, i.e., via equations) definitions of those sections, and to show that they can give unfamiliar answers when the underlying metric changes.
(I could have done without constantly being told the many ways that I was arguing with the author, but, then again, I'm a mathematician, so maybe not the intended audience. I also found it strange that, despite using the name 'taxicab geometry', the author never mentions the connection to taxicabs!)
EDIT: Oh, sorry, the taxi connection does merit a brief sentence:
> All distances are measured not as the shortest distance between two points, but as a taxi driver might count the distance between Point A and Point B: so many blocks one way plus so many blocks the other way.
I've seen this distance measurement mode referred to as Manhattan distance (https://en.wiktionary.org/wiki/Manhattan_distance), but it's interesting to see the full theory around it (and of course taxi-cab geometry is the more properly accepted name).
For example, one could draw circles with the same centre and radius for norms with p=1, 1.1, 1.2, ..., 1.9, 2.0, and then up to large p in a single plot. Bonus points for making the plots interactive, allowing us to move the loci around.
That looks wrong to me. Why does the ellipse have diagonal lines and the circle/square doesn't? It seems to me like nothing should have diagonal lines in this geometry.
The shapes are not constrained by taxicab geometry. Not only can you have diagonal lines, but you can have curves as well. The only difference from regular geometry is how the length of lines and curves is measured.
Consider creating taxicab shapes on graph paper. If you color in cells that satisfy the definition of a circle or ellipse, you're right that you won't see diagonal lines, only colored squares. If, however, you shrink the size of the grid further and further until it's infinitesimally small, those points will take on the appearance of diagonal lines.
> Consider creating taxicab shapes on graph paper. If you color in cells that satisfy the definition of a circle or ellipse, you're right that you won't see diagonal lines, only colored squares. If, however, you shrink the size of the grid further and further until it's infinitesimally small, those points will take on the appearance of diagonal lines.
This is an interesting idea, but I'm not sure how it helps understanding. As you point out in your first paragraph, taxicab geometry differs from ordinary geometry only in the way that it measures distances (I shy away from saying 'lengths', particularly of curves, because it's not clear to me that the Euclidean theory of rectifiable curves has a nice analogue in taxicab geometry); and distance is a point-point property, not a property of cells. How could one decide whether or not a cell satisfies the definition, except by picking a point in it? (That's not a rhetorical question.)
Instead of shading in cells, consider a field of discrete, equally spaced points. For instance, mark the intersections on the graph paper instead of the open spaces.
That can dramatically affect the shapes involved. For example, if you try the same thing on a rectangular Euclidean grid, then you'll find that the 'circle' through two adjacent points has only four points!
(Of course, as you say, the error involved can be made 'small', in some sense, by making the grid suitably fine; but at that point, with such a simple metric as the taxicab one, it's not clear to me what you're gaining over just looking at the entire plane all at once.)
Just as the drawing of a physical point on a piece of paper has area while the actual Euclidean "point" that it is trying to represent does not, a specific grid point on a piece of graph paper has area whereas the actual taxicab "point" does not. It is not that some sort of error is made to be small by making the grid more fine, rather the coarseness of the grid is simply to aid visualization. In reality, taxicab geometry is continuous, rather than discrete, just as Euclidean geometry is.
Edit: For clarification, the "points" in both the Euclidean and taxicab case can be represented in Cartesian coordinates (e.g. (x, y)). What is different is how you define the distance between those points. In Euclidean geometry, the distance is r = Sqrt((x - x')^2 + (y - y')^2), whereas in taxicab geometry the distance is r = |x - x'| + |y - y'|.
> It is not that some sort of error is made to be small by making the grid more fine, rather the coarseness of the grid is simply to aid visualization.
Indeed, if one were trying to understand, say, the locus of e^(x + y) = x^2 + y^2, which is an unfamiliar shape then I would say by all means to discretise it; that's what visualisation software would do, after all.
However, conic-section analogues defined via linear constraints on distances will, in the taxicab metric, always consist of unions of line segments, and it seems to me that discretisation is likely to hurt, not help, visualisability of such shapes.
Notice that the foci of that example ellipse are not on the same horizonal line: since the foci are on a diagonal, the ellipse becomes partly diagonal.
(I could have done without constantly being told the many ways that I was arguing with the author, but, then again, I'm a mathematician, so maybe not the intended audience. I also found it strange that, despite using the name 'taxicab geometry', the author never mentions the connection to taxicabs!)
EDIT: Oh, sorry, the taxi connection does merit a brief sentence:
> All distances are measured not as the shortest distance between two points, but as a taxi driver might count the distance between Point A and Point B: so many blocks one way plus so many blocks the other way.