# Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $${\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in the Dimensions movie (part 6) or here. Now, is there is a geometric surface (which you could plot with 3dplot) for every complex function which when projected renders the grid transform? (For $$z \rightarrow\ z + k/z$$ this would probably have to cut or overlap itself.) Also what is the further relationship between geometry and complex numbers? And maybe someone knows what software they used for the animations. Is there any sw along these lines able to visualize your own algebras (vs. functions), i.e. not just $(ac - bd), (ad + bc)$ for multiplication.

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Anyway, you can visualize a function $\mathbb R^2\to\mathbb R$ as a surface in $\mathbb R^3$, but a function $\mathbb R^2\to\mathbb R^2$ would have to be a surface in $\mathbb R^4$. People sometimes visualize the real part, the imaginary part, and/or the absolute value of the function as separate surfaces. – Rahul Sep 2 '12 at 21:40
I've gotten this far. The question i.a. is how to represent the grid transform in 3D. The modulus plot doesn't contain all information. – zxh Sep 3 '12 at 7:21

It seems you have software available that produces 3D-plots from parametric represenations of surfaces, as $$(\phi,\theta)\mapsto\bigl(\cos \phi\cos \theta,\ \sin \phi\cos \theta,\ \sin \theta\bigr)\qquad\Bigl(0\leq \phi\leq 2\pi, \ -{\pi\over2}\leq \theta\leq {\pi\over2}\Bigr)$$ (the unit sphere). You can use this software to get "grid plots" of complex analytic functions $f$ as well. When such an $f$ is given as an expression $f(z)$ write it out separating real and imaginary parts: $$f(x+iy)=u(x,y)+i v(x,y)\ ,$$ where now $u$ and $v$ are real functions of the real variables $x$ and $y$. Then let your software produce the 3D-plot corresponding to the parametric representation $$(x,y)\mapsto\bigl(u(x,y),\ v(x,y),\ 0\bigr)\ .$$ In the drawing $\bigl($choose the viewpoint at $(0,0,\infty)\bigr)$ you will see the images of the gridlines $x={\rm const.}$ and $y={\rm const.}$ Similarly you could arrange things such that you see the images of circles $|z|={\rm const.}$ and halflines ${\rm arg}(z)={\rm const.}$
Thanks. Had to take a look back at the plotting functions, Mathematica in my case. There's even an example, referred to as "Yudkowski map" under the ParametricPlot section. Taking $$z \rightarrow z+k/z$$ gives s15.postimage.org/7z4yjfhjf/Yudkowski.jpg. But this leaves the z axis unused (except here used with k). What i was aiming for all along was to use it for a value representing the grid line density. Does this have applications/make sense? How would you compute this? It's not s12.postimage.org/o0eq9fobh/image.jpg. – zxh Sep 6 '12 at 11:33