What does it mean for a line / equation to be parametric?

Great question. Parametric equations introduce a new variable (parameter). Contrast how we learned about lines in high school:

            y = m x + b
            a x + b y + c = 0

Versus a parametric equation:

            x(t) = a1 t + a0
            y(t) = b1 t + b0

The latter nicely generalizes to 3D, doesn't have any special cases (vertical lines), and can easily generate points on the line.

I'm confused on the vertexColors property. What is the difference with vertexColors called from THREE.Geometry() vs THREE.MeshBasicMaterial? Do we only use color interpolation when we want different colors for each vertices of a face? Wouldn't that be for all cases of coloring faces?

Another great question! I did some experiments and it seems that the vertexColors property is only a part of Material, not of Geometry. At least in the current version of THREE.js.

We do have to modify the

Can you go over how to find an equation of a triangle from three points?

For sure. Here's the section on parametric triangles

Key points:

two parameters
One from A to B. P(t)
One from C to P(t): Q(s,t)
the rest is algebra