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Is there any place to go on line for good graphics of how a complex elliptic curve sits as an affine curve in $\mathbb{C}^2$?

The mathematics is well discussed in Drawing elliptic curve and Is the real locus of an elliptic curve the intersection of a torus with a plane?. But I'd like to find graphics.

Mercio asks for pointers. Of course we are all accustomed to drawing 3 real dimensions on a 2-d screen or paper. And the big help here is that the curve has only 2 real dimensions.

So the most direct approach would be to give a 2-d drawing of how the curve would look projected into 3-d. The value of this would depend on finding an good pair of angles to show what is going on. See any number of drawings of tessaracts and other regular 4-d solids online done this way -- some interactive to allow rotation. A graphically simpler approach would be to draw a few intersections of the curve with flat 3-dim sections of $\mathbb{C}^2$, using gradation of colors to indicate successive sections.

For more discussion and great graphics of shapes in 4-d space see https://en.wikipedia.org/wiki/Four-dimensional_space and https://en.wikipedia.org/wiki/3-sphere

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Colin McLarty
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    well if you give me some $4$-dimensional paper I could try to draw one for you – mercio Aug 16 '14 at 08:08
  • @mercio If you are serious then I can give you pointers on representing higher dimensional shapes. Or you could Google "four dimensional shapes." – Colin McLarty Aug 16 '14 at 08:11
  • if so could you include that in your question so that we have an idea of what kind of answer you are expecting ? (although $\Bbb P^2(\Bbb C)$ would be preferrable, not $\Bbb C^2$) – mercio Aug 16 '14 at 08:12
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    FWIW, here is (half of) a smooth cubic, and here is a nodal cubic, with the real points highlighted on each. – Andrew D. Hwang Aug 16 '14 at 18:27
  • @user86418 Yes, this is just the sort of thing I'd like. And perhaps these natural projections are the clearest to visualize as well. Given $y^2=f(x)$ graph (in 3-d) the real part of $y$ as a function of the real and imaginary parts of $x$ -- or alternatively graph the imaginary part of $y$. – Colin McLarty Aug 16 '14 at 22:52

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http://www.math.purdue.edu/~dvb/ graphs the Weierstrass function of an elliptic curve, and its derivative, using three spatial dimensions plus one color dimension. He gives similar graphics for nodal and cuspidal cubics.

He even gives Maple code for it. Color graphics as four dimensional paper!

Colin McLarty
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