# Monotone Operator Visualization for exboldg

## Description of the Monotone Operator exboldg

The operator $T : R2\to R2$ where T is defined by

## How to Interpret the Visualization

Rendered above is the visualization of the monotone operator from $R2\to R2$, along the Red, Green, and Blue axes and in Colourspace according to the following format:

 T : (x, y) → (x*,y*) T : (x, y) → (x*,y*) T : (x, y) → r*( cosθ*, sinθ*) T : (x, y) → r*( cosθ*, sinθ*)

... where variables in this colour are represented by colour, from low to high, except in the case of θ, which is represented by hue (where, by colour, θ=0, θ=2π/3, and θ=4π/3).

Along the x-axis, values range from -3 to 3 , and along the y-axis, values range from -3 to 3 .

To paraphrase, in the upper left quadrant, x* in the range is mapped to z , and y* in the range is mapped to colour , and in the upper right quadrant vice-versa for x* and y*. In the lower left quadrant, the norm (r*) of the image is mapped to z , and the polar angle (θ*) is mapped to colour , again vice-versa for the lower right quadrant for the norm (r*) and polar angle (θ*).

## Notes on visualization

The monotone operator itself is a 4-dimensional object, which is why for the best intuition four ways of visualizing it are given concurrently. You can click on the rendering and while holding the mouse button move the mouse. This will rotate all perspective equally. Compare the table above with the coloured axes.

Regions that should be perfectly circular may not appear such, and colours may appear to bleed, because values were calculated on a small mesh of 64x64.

For a different perspective still, click on exboldg_diffId in the monotone operator selection links above on click on the link below to see a visualization of the difference between this operator and the identity.