First, L divides the plane into two pieces, or "components", which are interchanged by R$_L$(z), Second, each point on the boundary between the components remains fixed; Third, St/, (z) is involutory or self-inverse, meaning that R$_L$(z) o R$_L$(z) is the identity mapping, leaving every point fixed. To put this last property differently, consider a point z and its reflection z_= R$_L$(z) in L. Such a pair are said to be "mirror images", or to be "symmetric with respect to L".
cross-ratio: w = [z, q, r, s]
The introduction of homogeneous coordinates thereby accomplishes for algebra what the Riemann sphere accomplishes for geometry—it does away with the exceptional role of infinity.
dot product ----(generalization)----> inner product