
An Introduction to Taxicab Geometry - philk10
https://spin.atomicobject.com/2015/08/31/taxicab-geometry/#.VeRT_Yz-SFs.hackernews
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JadeNB
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.

~~~
mmatants
I've seen this distance measurement mode referred to as Manhattan distance
([https://en.wiktionary.org/wiki/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).

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DArcMattr
Shull doesn't mention the book by Krause [1], which provides exercises for the
reader to develop an understanding of Taxicab Geometry.

[1]:
[http://store.doverpublications.com/0486252027.html](http://store.doverpublications.com/0486252027.html)

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Someone
It would be nice to extend this with examples of the same curves for p-norms
([https://en.m.wikipedia.org/wiki/Norm_(mathematics)#p-norm](https://en.m.wikipedia.org/wiki/Norm_\(mathematics\)#p-norm)).

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.

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StavrosK
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.

~~~
exupero
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.

~~~
JadeNB
> 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.)

~~~
exupero
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.

~~~
JadeNB
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.)

~~~
plus
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'|.

~~~
JadeNB
> 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.

