Change of Coordinate Systems
Just as the selection of axes in the Cartesian coordinate is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore it is useful to know how the different systems are related to each other.
Let (x, y, z) be the homogeneous coordinates of a point in the projective plane and for a fixed matrix
with det(A) ≠ 0, define a new set of coordinates (X, Y, Z) by the equation
Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for points in the projective plane. If z is fixed at 1 then
are proportional to the signed distances from the point to the lines
(The signed distance is the distance multiplied a sign 1 or −1 depending on which side of the line the point lies.) Note that for a = b = 0 the value of X is simply a constant, and similarly for Y and Z.
The three lines,
in homogeneous coordinates, or
in the (X, Y, Z) system, form a triangle called the triangle of reference for the system.
Read more about this topic: Homogeneous Coordinates
Famous quotes containing the words change and/or systems:
“Time heals griefs and quarrels, for we change and are no longer the same persons. Neither the offender nor the offended are any more themselves.”
—Blaise Pascal (1623–1662)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)