Before revision:  Given z>0, n vectors p_1,\cdots,p_n on the plane, and a vector b\in\mathbb{R^n, compute a unit vector x and a permutation \pi:
After revision:  Given z>0, n vectorsPoints-to-Lines alignment p_1,\cdots,p_n on the plane, and a vector b\in\mathbb{R^n, compute a unit vector x and a permutation \pi: