---
title: "Mapping a square to a circle"
author: "Stéphane Laurent"
date: '2023-02-25'
tags: R, graphics, maths, special-functions
rbloggers: yes
output:
  md_document:
    variant: markdown
    preserve_yaml: true
  html_document:
    highlight: kate
    keep_md: no
highlighter: pandoc-solarized
---

Denoting by $F(z \,|\, m)$ the incomplete elliptic function of first
kind, the function $$
\varphi(z) = -\sqrt{i} \, F\bigl(i \sinh^{-1}(\sqrt{i} \, z) \,|\, -1 \bigr).
$$ is a conformal mapping from the square $[-1,1] \times [-1,1]$ to a
centered circle. I learnt that
[here](https://math.stackexchange.com/a/3159827/38217).

Then, if one has a square image, we can transform it to a circular image
with the help of this function $\varphi$. I will give an example.

One can use instead the function $\psi(z) = \varphi(z)/\varphi(1)$ to
get a conformal mapping from this square to the unit circle. I will use
this one. And I will call this square the unit square.

First, let's plot a beautiful complex function on a square. I take the
Weierstrass zeta function.

``` r
library(jacobi)     # to get the `wzeta` function
library(RcppColors) # to get the `colorMap1` function
# vectorize a Weierstrass zeta function
f <- Vectorize(function(x, y){
  z <- complex(real = x, imaginary = y)
  wzeta(w, omega = c(1/2, 1i/2))
})
# compute it on a 2-dimensional grid
x_ <- y_ <- seq(-1.5, 1.5, length.out = 512L)
ZETA <- outer(x_, y_, f)
# map this complex matrix to a matrix of colors
img <- colorMap1(ZETA)
# background color for the plot (only appears in R)
bkgcol <- rgb(21, 25, 30, maxColorValue = 255)
# plot
opar <- par(mar = c(0, 0, 0, 0), bg = bkgcol)
plot(c(-100, 100), c(-100, 100), type = "n", asp = 1, 
     xlab = NA, ylab = NA, axes = FALSE, xaxs = "i", yaxs = "i")
rasterImage(img, -100, -100, 100, 100)
par(opar)
```

![](./figures/zetaw_square.png){width="40%"}

I hope you like it. Now we will map this image to a circle.

We firstly compute the values of $\psi$ on a grid of the unit square:

``` r
library(Carlson) # to get the `elliptic_F` function
# define the `psi` function
w <- sqrt(1i)
D <- elliptic_F(1i * asinh(w), -1)
psi <- function(x, y) { # maps the unit square to the unit circle
  z <- complex(real = x, imaginary = y)
  elliptic_F(1i * asinh(w * z), -1, minerror = 1e-10) / D
}
# compute this function on a grid of the unit square
n <- 512L
PSI <- matrix(NA_complex_, nrow = n, ncol = n)
for(i in 1L:n) {
  x <- -1 + 2 * (i-1) / (n-1)
  for(j in 1L:n) {
    y <- -1 + 2 * (j-1) / (n-1)
    PSI[i, j] <- psi(x, y)
  }
}
```

We will need to interpolate the values of the Weierstrass zeta function
stored in the `ZETA` matrix. To do so, we introduce two functions, one
to interpolate the real parts, the other one to interpolate the
imaginary parts:

``` r
library(cooltools) # to get the `approxfun2` function
x_ <- y_ <- seq(-1, 1, length.out = n)
f_re <- approxfun2(x_, y_, Re(ZETA))
f_im <- approxfun2(x_, y_, Im(ZETA))
```

Note that these two functions are defined on the unit square. And the
**cooltools** package is indeed cool.

Now it's easy. With the help of these two functions, we map the values
of $\psi$ that we stored in the `PSI` matrix:

``` r
M_re <- f_re(Re(PSI), Im(PSI))
M_im <- f_im(Re(PSI), Im(PSI))
ZETA_CIRCLE <- complex(real = M_re, imaginary = M_im)
dim(ZETA_CIRCLE) <- c(n, n)
```

Now we transform this complex matrix to a matrix of colors, and we plot
it:

``` r
# map the complex matrix to a matrix of colors
img <- colorMap1(ZETA_CIRCLE)
# plot
opar <- par(mar = c(0, 0, 0, 0), bg = bkgcol)
plot(c(-100, 100), c(-100, 100), type = "n", asp = 1, 
     xlab = NA, ylab = NA, axes = FALSE, xaxs = "i", yaxs = "i")
rasterImage(img, -100, -100, 100, 100)
par(opar)
```

![](./figures/zetaw_circle.png){width="40%"}