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
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“And I looked to be happy, and I was,
As I said, for a while but I don’t know!
Somehow the change wore out like a prescription.”
—Robert Frost (1874–1963)
“The skylines lit up at dead of night, the air- conditioning systems cooling empty hotels in the desert and artificial light in the middle of the day all have something both demented and admirable about them. The mindless luxury of a rich civilization, and yet of a civilization perhaps as scared to see the lights go out as was the hunter in his primitive night.”
—Jean Baudrillard (b. 1929)