returns the power product of pw and qw.

Let ${p}^{(w)} = (p,w_p), p\in\mathbb{R}^2, w_p\in\mathbb{R}$ and ${q}^{(w)}=(q,w_q), q\in\mathbb{R}^2, w_q\in\mathbb{R}$ be two weighted points.

The power product, also called power distance between ${p}^{(w)}$ and ${q}^{(w)}$ is defined as

$$\Pi({p}^{(w)},{q}^{(w)}) = {\|{p-q}\|^2-w_p-w_q}$$

where $\|{p-q}\|$ is the Euclidean distance between $p$ and $q$.

The weighted points ${p}^{(w)}$ and ${q}^{(w)}$ are said to be orthogonal iff $\Pi{({p}^{(w)},{q}^{(w)})} = 0$.

Three weighted points have, in 2D, a unique common orthogonal weighted point called the power circle. The power segment will denote the weighted point orthogonal to two weighted points on the line defined by these two points.