Elements Other Than Points
The equation of a line in the projective plane may be given as sx + ty + uz = 0 where s, t and u are constants. Each triple (s, t, u) determines a line, the line determined is unchanged if it is multiplied by a nonzero scalar, and at least one of s, t and u must be non-zero. So the triple (s, t, u) may be taken to be homogeneous coordinates of a line in the projective plane, that is line coordinates as opposed to point coordinates. If in sx + ty + uz = 0 the letters s, t and u are taken as variables and x, y and z are taken as constants then equation becomes an equation of a set of lines in the space of all lines in the plane. Geometrically it represents the set of lines that pass though the point (x, y, z) and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may be given sets of four homogeneous coordinates, and so on for higher dimensions.
Read more about this topic: Homogeneous Coordinates
Famous quotes containing the words elements and/or points:
“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)
“It’s my feeling that God lends you your children until they’re about eighteen years old. If you haven’t made your points with them by then, it’s too late.”
—Betty Ford (b. 1918)