How to calculate control points for

It’d be useful to know by the documentation of JUCE, how to calculate the control points for the Path::quadraticTo and Path::cubicTo functions.

Or perhaps I can use the parametric versions?

Well, you posted exactly the link to the graphic to solve that question:

Point<float> p0, p1, p2, p3;
Path p;
p.startNewSubPath (p0);
p.cubicTo (p1, p2, p3);

Does that help?