Saturday, November 8, 2008

Loxodromes!

This is a sweet word used to describe a sweet mathematical curve. And by sweet I mean awesome and one of my favorites. Speaking of favorites, I learned about these curves in my favorite mathematical book, Visual Complex Analysis, by Tristan Needham. If this post piques your interest, there is tons more about them there.


Loxodromes are curves of motion you get from certain kinds of Mobius transformations, whose general algebraic definition is the formula (az+b)/(cz+d).  z is a complex number here, that is a point in a plane. Hence, these transformations can be viewed visually by their effects on the points of a plane. However, Mobius transforms have an elegant interpretation when viewed as a corresponding transform obtained by unprojecting the plane onto a sphere (doing the reverse of stereographic projection that I described in a previous post), as the resulting motions on the sphere are much simpler! The sphere is called the Riemann Sphere in honor of Bernhard Riemann.

In this animation, I put a little white fuzzball near the north pole of the sphere to show the light generating the shadows. The shadows are the 3D->2D stereographic projection of the curve on the sphere. So this is two simultaneous views of a loxodrome, both on the sphere and on the plane (fine print: not exactly because the curve I've drawn has a little bit of thickness coming off the sphere surface, but the shadows are still close). If you remember the previous discussion in the soap bubble post about projecting from 4D->3D, hopefully that process is a little more clear by analogy now. We may not be able to look at 4D objects in our world, but we can look at their shadows! Anyway, I hope this shed a little more light (pun intended) on what stereographic projection is.

Loxodromes correspond to one of the more general types of Mobius transform motions, and the animation can help a little in explaining what I mean by that. Watch it for a bit and answer the following: Do you see the curve stretching and shrinking over the sphere like it is moving from one end of the spiral towards the other? Or do you see the curve as unchanging its shape and just rotating as a whole about an axis through the 2 spiral ends?

My perception is biased to the former, but there is no right answer because it could be viewed either way! The first kind of motion is called hyperbolic (unfortunately, the meaning here is not the same as of the last post), and the second is called elliptic. Both are special types of Mobius transformations, and loxodromes are what you get when you do both in combination. Incidentally, it is interesting that I have trouble seeing this as a rotating curve because that is how the POV-Ray script actually generated the sequence of images :)


Recognize the projected shape in this picture? A special case of the projection of a loxodrome onto the plane is a logarithmic, or equiangular spiral! Yes, the spiral of sea shells and galaxies and so much more. Are you feeling the awesomeness yet? By the way, I took advantage of this to simplify making the movie. There are no complex number calculations explicitly going on, just a function that can generate points of a logarithmic spiral on a plane, and a function that can unproject those points from the plane to a sphere. So if you were thinking it was terribly involved to generate the movie and picture, it actually was not so bad (though it did take a long time to render out the frames). The entire definition file is only about 100 lines, with half of it standard required stuff (camera position, etc.). Here it is if you care to check it out.  POV-Ray is great!