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:
“It is a life-and-death conflict between all those grand, universal, man-respecting principles which we call by the comprehensive term democracy, and all those partial, person-respecting, class-favoring elements which we group together under that silver-slippered word aristocracy. If this war does not mean that, it means nothing.”
—Antoinette Brown Blackwell (1825–1921)
“The two elements the traveler first captures in the big city are extrahuman architecture and furious rhythm. Geometry and anguish. At first glance, the rhythm may be confused with gaiety, but when you look more closely at the mechanism of social life and the painful slavery of both men and machines, you see that it is nothing but a kind of typical, empty anguish that makes even crime and gangs forgivable means of escape.”
—Federico García Lorca (1898–1936)