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:
“psychologist
It is through friendships that teenagers learn to take responsibility, provide support, and give their loyalty to non- family members. It is also in teenage friendships that young people find confidants with whom to share thoughts and feelings that they are not comfortable sharing with their parents. Such sharing becomes one of the elements of true intimacy, which will be established later.”
—David Elkind (20th century)
“An illustrious individual remarks that Mrs. [Elizabeth Cady] Stanton is the salt, Anna Dickinson the pepper, and Miss [Susan B.] Anthony the vinegar of the Female Suffrage movement. The very elements get the “white male” into a nice pickle.”
—Anonymous, U.S. women’s magazine contributor. The Revolution (August 19, 1869)
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—Tennessee Claflin (1846–1923)