In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. (Wikipedia) The fuck?

Okay, basically, it's a function which takes itself as an argument. Yea. This allows us to make a function recursive. Instead of referring to the function to recurse by name, it is passed to the function as a parameter.

This allows us to make inline-recursive functions. This is useful for scenarios where a recursive function simply represents a conditional statement, making a recursive call. This allows the function to be simplified (well, * simplified*) and be squeezed into a single line. It's actually very elegant.

Thanks FP!

The first e is actually accented, but I'll be using the word a lot, and typing it quickly becomes annoying. So I'll omit it for convenience sake. just pretend it's there.

export const U = <T>(f: typeof U, ...args: T[]) => f(f, ...args);
export const inter = (a: Coordinate, b: Coordinate, alpha: number): Coordinate => ({
    x: a.x + (b.x - a.x) * alpha,
    y: a.y + (b.y - a.y) * alpha
});

for (let alpha = 0; alpha <= 1; alpha += 1 / resolution)
    paint(U((f, points: Coordinate[]) => points.length > 1 ? f(f, points.slice(1).map((i, a) => inter(i, points[a], alpha))) : points, points)[0]);

A Bezier curve is curve which uses 1 or more control points to define the trajectory of the curve. These points act as pulling forces, bending a line towards them. This effect is achieved by interpolating between all pairs of adjacent control nodes, using the resultant set of coordinates as control points, until the list is reduced to a single point. This is the value of the curve at point x.

Typically, this is not solved recursively, as recursion can become intensive on resources, however, the simplicity of the layers (interpolation), combined with the most common use-case for bezier curves with very few control points, allows this to become feasible. The snippet above uses the base function inter, which takes two points, and an interpolation value, alpha. The use of the U combinator, allows the snippet to create a list of interpolated points, essentially reducing the list by one. (Consider a list of points, and returning a list of gaps between them). This can be repeated until a single point remains. This is painted at resolution intervals between the root points, to produce the curve.