Pythagorean-Hodograph curve's control points

Question

In the Farouki's book "Pythagorean-hodograph curves: algebra and geometry inseparable" it is said that control points of the cubic Bezier curves with Pythagorean-hodograph qualities are set as follows:
$p_1 = p_0 + 1/3(u_0^2 - v_0^2, 2u_0 v_0)$
$p_2 = p_1 + 1/3(u_0 u_1 - v_0 v_1, v_0 v_1 + u_1 v_0)$
$p_3 = p_2 + 1/3(u_1^2 - v_1^2, 2u_1 v_1)$
These control points based on linear polynomials:
$u(t)=u_0(1-t)+u_1t$; $v(t)=v_0(1-t)+v_1t$
So the questions are: how these $(u_0, u_1)$ and $(v_0, v_1)$ should be calculated? And what is the meaning of these pairs on the plot?

Answer 1

They are just numbers, and they can be anything you like (except for a few corner cases). Whatever numbers you choose, you'll get a PH cubic.

These numbers don't have any geometric significance (as far as I know) -- they come from algebraic reasoning about the curve.

If you want a more geometric discussion, look at theorem 18.1 in Farouki's book.