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 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)
“A few ideas seem to be agreed upon. Help none but those who help themselves. Educate only at schools which provide in some form for industrial education. These two points should be insisted upon. Let the normal instruction be that men must earn their own living, and that by the labor of their hands as far as may be. This is the gospel of salvation for the colored man. Let the labor not be servile, but in manly occupations like that of the carpenter, the farmer, and the blacksmith.”
—Rutherford Birchard Hayes (18221893)