0

I recently worked out a very fun theory of what happens when you compactify the complex plane by adding a projective line at infinity, instead of just a single point as in the case of the Riemann Sphere. In this way it's possible to treat the entire complex plane as simultaneously also being a projective plane in the sense of projective geometry.

My question is: has this approach been tried before? If so, then what is a good reference to learn more about this; if not, then is there some kind of mistake I might have made which will make this not such a useful perspective?

Projective transformations can change circles into hyperbola's, as described in this question: Understanding Projective Geometry; images of circles becoming ellipses, parabolas, or hyperbolas

Intuitively, what I am trying to do can be understood in the following way: we place a camera at the origin $(0,0,0)$ in $\mathbb{R^3}$ and are looking at a holomorphic function $f(z)$ from a specified orientation corresponding to the initial values given by $f(z)$. The complex plane is the "screen" of our camera, and we can use projective transformations to change our perspective.

This is the same mathematics used by all modern computer graphics frameworks, for example, openGL. So what we are doing is moving the camera around in three space and using projective geometry to find different ways of "looking at" $f(z)$ without using conformal mappings.

DEFINITION (projective complex plane): The projective complex plane is the complex plane where we add a projective line at infinity, in this way, every point in the complex plane $x+iy \in \mathbb{C}$ is associated with a line passing through the origin in $\mathbb{R^3}$ with homogenous coordinates $(x,y,1)$. In other words, associate every projective line with homogeneous coordinates $(x,y,1) \in \mathbb{P^2(R)}$ with the complex number $x+iy \in \mathbb{C}$

DEFINITION ($\eta$-function of a complex valued function): Given a holomorphic function $f(z)$, the $\eta$-function of $f$, $\eta_f$, is a mapping $$\eta_f : \mathbb{R^3} \rightarrow \mathbb{C}$$ which takes constant values on all projective lines, and is defined by the following equations $$\eta(x,y,1)=f(x + iy)$$ $$\eta(x,y,0)=\lim_{|z| \to \infty}f(z)$$

DEFINITION ($\alpha$-matrix): An $\alpha$-matrix is a projective transformation of $\mathbb{P^2(R)}$ which sends the projective line with homogeneous cooridinates $(x, y, 1)$ to the projective line $(x^{\prime}, y^{\prime}, z^{\prime})$. It can be represented as a matrix by taking homogeneous coordinates in $\mathbb{R^3}$ where the point $(x,y,z)$ is associate with the projective line in $\mathbb{R^4}$ with homogeneous coordinates $(x,y,z,1)$

For example, we can map the projective line with homogeneous cooridinates $(\frac{1}{2}, y, 1)$ to the projective line at infinity $(x, y, 0)$. It can be represented as a matrix by taking homogeneous coordinates in $\mathbb{R^3}$ where the point $(x,y,z)$ is associated with the projective line in $\mathbb{R^4}$ with homogeneous coordinates $(x,y,z,1)$

$$ \alpha = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & -\frac{1}{2} \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

It's easy to check this transoformation sends the line $(\frac{1}{2}, y, 1)$ to the line at infinity $(x, y, 0)$ in the following way

$$ \begin{bmatrix}z \\ y \\ 0 \\ 1 \end{bmatrix} =\begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & -\frac{1}{2} \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}\frac{1}{2} \\ y \\ z \\ 1 \end{bmatrix} $$

Note that since this is a projective transformation which exchanges the axis $x \leftrightarrow z$, it is not a conformal mapping, and will not preserve angles.

The basic idea is to carry out some projective transformation on the projective complex plane to gain more information about holomorphic functions which is not easy or possible to see with other methods, and also to take the inverse mapping of that transformation to return to a conformal mapping settings. Since I have been unable to take it much farther in a meaningful way, I wanted to pause and ask this as a question.

Is this a good idea which I should keep working on, or do you think this is not going to go anywhere because I am missing something important?

Thanks!

Matt Calhoun
  • 4,404
  • 2
    This is not a new idea, although your application might be novel. There are plenty of other projective planes besides $\mathbb{RP}^2$, including the minimal finite Fano plane. Complex numbers crop up on occasion even when studying the real projective plane: degenerate conics, for example, can be understood as pairs of possibly imaginary lines; all circles intersect the line at infinity at a fixed pair of points with complex coordinates, and so on. Indeed, those “circular points” and their images are used in some camera calibration methods. – amd Apr 15 '20 at 18:25
  • 3
    by the uniformization theorem, any 1 (complex) dimensional compact manifold is the Riemann sphere so $\mathbb CP^1$ is the Riemann sphere with different coordinates; if you use $\mathbb RP^2$ you lose the complex structure hence all the important theorems associated with it; if you use $\mathbb CP^2$ on the other hand you get into the complex variable in several dimensions world which has highly non-trivial differences from the one dimensional complex analysis world – Conrad Apr 15 '20 at 18:34
  • @Conrad that is very interesting! I don't recall ever hearing about the uniformization theorem before. I have a question, when you stated "any 1 (complex) dimensional compact manifold is the Riemann sphere", my understanding is that the manifold upon which the so called $\eta$-function of $f$ acts is the projective plane $\mathbb{P^2(\mathbb{R})}$, which is not a 1 (complex) dimensional manifold. Since I am reading about the uniformization theorem now, I wonder if I am perhaps misunderstanding your comment. If complex structure is lost, perhaps other new information is gained is my hope. – Matt Calhoun Apr 15 '20 at 18:38
  • 1
    I am not sure about your definitions because in one you use the projective complex plane, in another the projective real plane, etc; also what makes you think that holomorphic functions have limits at infinity; generally they don't; not on rays, lines etc - there are lots of examples where the image under a ray of a holomorphic function is dense in an open unbounded set – Conrad Apr 15 '20 at 18:43
  • @Conrad good point about limits, I meant for it to be undefined in that case, but your point is well taken. What if I do this: 1.) take any holomorphic function $f(z)$ and do everything in a real projective plane 2.) assign a complex number to every projective line in $\mathbb{R^3}$ via $f(z)$ 3.) Use $\mathbb{P^3(\mathbb{R})}$ as described in my question to do projective transformations 4.) use map $(x,y,z,1)\rightarrow s=x+iy$ 5.) this defines a new complex valued function, idea is to try and discover more information about $f(z)$ and take an inverse transformation. Does that make sense? – Matt Calhoun Apr 15 '20 at 18:52
  • 1
    i have no idea but I am skeptical because complex analyticity is a very rigid concept in many ways, but it is your time and effort - the only thing i would add is to try and make siure you write everything since sometimes things become much clearer when you try and make them make sense formally on paper (cannot count how many "good ideas" turned out to be rubbish when i tried to put them in writing – Conrad Apr 15 '20 at 19:02
  • @Conrad thank you for your encouraging advice. I find the pressure of writing questions and making strong on topic contributions to this community extremely helpful while thinking through abstract ideas outside my comfort zone like this question – Matt Calhoun Apr 15 '20 at 19:03

0 Answers0