What theory describe such images?

Question

I make naive, visual exploration of 2D images, which can be described by simple functions, like:

1. $ y = f(x)$
2. $ y = f(x,y)$
3. $f(z) = \begin{cases} k(x,y) \\ h(x,y) \end{cases}$

I have read about algebraic and differential equations.

These images are used in :

• domain coloring and visualisation of complex functions
• procedural texture
• visualisation of vector fields

Questions:

Is there some math theory behind such images/functions/transformations ?

What is the difference between above functions?