Elements
The elements of the projective linear group can be understood as "tilting the plane" along one of the axes, and then projecting to the original plane, and also have dimension n.
A more familiar geometric way to understand the projective transforms is via projective rotations (the elements of PSO(n+1)), which corresponds to the stereographic projection of rotations of the unit hypersphere, and has dimension Visually, this corresponds to standing at the origin (or placing a camera at the origin), and turning one's angle of view, then projecting onto a flat plane. Rotations in axes perpendicular to the hyperplane preserve the hyperplane and yield a rotation of the hyperplane (an element of SO(n), which has dimension ), while rotations in axes parallel to the hyperplane are proper projective maps, and accounts for the remaining n dimensions.
Read more about this topic: Projective Linear Group
Famous quotes containing the word elements:
“Kitsch is the daily art of our time, as the vase or the hymn was for earlier generations. For the sensibility it has that arbitrariness and importance which works take on when they are no longer noticeable elements of the environment. In America kitsch is Nature. The Rocky Mountains have resembled fake art for a century.”
—Harold Rosenberg (19061978)
“Barbarisation may be defined as a cultural process whereby an attained condition of high value is gradually overrun and superseded by elements of lower quality.”
—Johan Huizinga (18721945)
“The popularity of that baby-faced boy, who possessed not even the elements of a good actor, was a hallucination in the public mind, and a disgrace to our theatrical history.”
—Thomas Campbell (17771844)