# Seeking graphics of elliptic curves as surfaces

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

• well if you give me some $4$-dimensional paper I could try to draw one for you – mercio Aug 16 '14 at 8:08
• @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 8:11
• 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 8:12
• 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