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:
“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)
“Every man has to learn the points of the compass again as often as he awakes, whether from sleep or any abstraction.”
—Henry David Thoreau (1817–1862)