In this section, we consider projective spaces over a commutative field K, although most results may be generalized to projective spaces over a division algebra.
Every frame (p(e_0), \ldots, p(e_n), p(e_0+\cdots+e_n)) allows to define projective coordinates, also known as homogeneous coordinates: every point may be written as p(v); the projective coordinates of p(v) on this frame are the coordinates of v on the base (e_0, \ldots, e_n).