Scalar fields for gradients and other graphic patterns

The Mathaesthetics flagship app will support a number of modes of mathematical image creation. I spent most of yesterday working on and improving the scalar field renderer.

In a scalar field, a function f(x, y) produces a value for each point. This scalar value is then mapped to a color gradient. I'm exploring both different modes of color assignment and different function definitions and the results they yield.

Here's a simple example, of the function f(x, y) = x • y. The color scale is white-to-black, and here I'm only allowing 16 different distinct shades:

Here's the same function with the color mapping changed to orange-to-blue:

And again, but with the discrete shading limit removed so we get a smooth gradient:

Producing gradients that aren't simply linear or radial is one potential graphic application of scalar fields with algebraic functions. For example, a logarithmic gradient...