The basics of parallel paths and an example

Turn off some annoying messages:

A silly parameterized curve:

I want to widen the path. I suppose I could use thicker lines...

... but that' not so great --- look at the bumps where the curvature is highest. The graph consists of a union of fat line segments, so we could simply increase the number of segments. But what if I want to do something more exotic, like have different colored lines running along the edges of this new "road"?

I'll use tangents and normals to find "parallel paths" at a given distance, neglecting the questions of differentiabilty:

I could use a function for making unit vectors...

and a function for making tangent vectors, and normals vectors. (I'll use the "trick" for making normals from tangents.)

After that it's easy to make a parallel path.

Ah, that's what I wanted. (The original curve is the black one in the middle.)