While researching how to render 2D bezier curves given the control points, I found a simple formula and the resource where I found this marked this iterative process as a ternary interpolation and ternary approximation .
Using this 2 keywords I can't find basically any hit while searching with my favourite search engine and I'm wondering if this is a jargon or a term that is just made up by the author .
The "philosophy" of this iterative process looks really similar to what happens in the 3D world with a lattice that is subject to the Catmull-Clark subdivision, it also looks similar to what is shown in this youtube video .
Altough all this 3 examples are different, they appear to be connected and I'm wondering if there is a better and more generalized name for that ternary interpolation that can help me with this researches .
For example the formula for 2D cubic bezier curves, considering my original source, is
$$\mathbf{B}(t)=(1-t)^3\mathbf{P}_0 + 3(t-2t^2+t^3)\mathbf{P}_1 + 3(t^2-t^3)\mathbf{P}_2 + t^3\mathbf{P}_3$$
And the thing I noted is that the coefficient in this case are $1,3,3,1$, and it looks like a tuple used in the video I mentioned .
What's the general case and the wording to use here ?
Bézier curves are connected with Catmull-Clark subdivision surfaces because both are fundamentally "corner cutting" algorithms. If you stay away from so-called "extraordinary points, the C-C algorithm will produce a b-spline surface, which is closely related to Bézier curves and surfaces.
The term " ternary" is non-standard, and probably refers to the degree (3) of the Bézier curve. Bézier curves can have any degree, but degree 3 is by far the most common.