Re: How to visualize Riemann surfaces
From: Prasanna (prasan8181_at_dacafe.com)
Date: 12/23/04
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Date: 23 Dec 2004 15:56:57 -0800
Atlast I think I get the general picture. But still there are a couple
of things not clear here and there. There seems to be two ways of
visualising the compactified Riemann surface. One way is to take the
real{f(z)} or imag{f(z)} and apply the stereographic projection on it.
If I try that using mathematica, I did that for a sorta elliptic curve
y^2 = x^3 + x + a, what I get in no way looks like a torus. It is like
a sphere with some more surfaces inside the sphere. I could not even
spot any holes in that. But that is how the mathematica article I
mentioned in the previous post does it. What could be wrong? It is even
sufficient for me if I could see some hole in my resulting surface. I
dont need an explicit homeomorphism that maps it into a torus. I just
want to know if this way of projecting the im{f(z)} is correct (for
getting the torus).
Another way is your previous post I read at
http://www.math.niu.edu/~rusin/known-math/99/RS
That explanation looks quite good. But still I dont understand what you
mean by
"If you could "graph" this function in C x C, you'd get something
which is
a (real) 2-dimensional surface, with one point attached to every point
of the complex plane except for the points in your barrier. Viewed from
above, then, this graph is just (a wavy form of) the slit complex plane
itself."
This seems to be very confusing. All I can do to plot and visualize a
surface is use a real x coordinate, a real y coordinate and a real z
coordinate. So, what does looking from above mean to a surface plotted
in C x C?
When I use a mathematical software to plot f(z) = sqrt(z^4-z^2) as a
complex plane, I get contours which are continous everywhere except a
slit passing through zero. And if I plot the
|f(z)| then I get something like a cone which is rough near origin. But
that does not enlighten much either. Can you try to explain it a bit
more?
If this does not help too, I guess it should be better for me to
hibernate for a while and then come back to this again. :)
> PS -- no one calls me Dr. Rusin except my mom, who gets a kick out of
it.
What might your students be calling you as then...
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