An Introduction to Taxicab Geometry

https://spin.atomicobject.com/2015/08/31/taxicab-geometry/#.VeRT_Yz-SFs.hackernews

by philk10 • 11 years ago

49 13 11 years ago

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StavrosK2• 11 years ago

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.

exupero2• 11 years ago

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.

JadeNB4• 11 years ago

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.)

StavrosK2• 11 years ago

In that case, why does the ellipse look like that? Why are the angles where they are? Why wouldn't it be curved?

chriswarbo• 11 years ago

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.

DArcMattr• 11 years ago

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

JadeNB4• 11 years ago

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• 11 years ago

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).

Someone• 11 years ago

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).

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.

An Introduction to Taxicab Geometry

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