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:
“There surely is a being who presides over the universe; and who, with infinite wisdom and power, has reduced the jarring elements into just order and proportion. Let speculative reasoners dispute, how far this beneficent being extends his care, and whether he prolongs our existence beyond the grave, in order to bestow on virtue its just reward, and render it fully triumphant.”
—David Hume (1711–1776)
“Type of the wise, who soar, but never roam—
True to the kindred points of Heaven and Home!”
—William Wordsworth (1770–1850)