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:
“Let us have a fair field! This is all we ask, and we will be content with nothing less. The finger of evolution, which touches everything, is laid tenderly upon women. They have on their side all the elements of progress, and its spirit stirs within them. They are fighting, not for themselves alone, but for the future of humanity. Let them have a fair field!”
—Tennessee Claflin (1846–1923)
“The elements of success in this business do not differ from the elements of success in any other. Competition is keen and bitter. Advertising is as large an element as in any other business, and since the usual avenues of successful exploitation are closed to the profession, the adage that the best advertisement is a pleased customer is doubly true for this business.”
—Madeleine [Blair], U.S. prostitute and “madam.” Madeleine, ch. 5 (1919)
“The three great elements of modern civilization, gunpowder, printing, and the Protestant religion.”
—Thomas Carlyle (1795–1881)