Homogeneous coordinates

Definition 1. Nonempty set Pn(R) is a projective space over the set of real numbers , iff

1. any point X in the space is attached a unique class [x] of equivalent ordered sets of n+1 real numbers

( x 1 , ..., x n , x 0 )

while two classes (x1,...,xn,x0) and (y1,...,yn,y0) are equivalent, if

x i = h y i , h 0, i = 0, 1,..., n

2. class (0,...,0) consisting of zeros only is attached to no point in the space.

Numbers x0,x1,...,xn are homogeneous coordinates of the point in the projective space.

Arithmetic (algebraic) model of the n-dimensional projective space Pn(R) has the following properties:

  1. There exist no more but at most (n +1) linearly independent points in the space.
  2. All points in the space are linear combinations of any class of (n +1) linearly independent points. Space is therefore indicated also as linear space.
  3. Set of points in the space that are linearly independent from (r +1) linearly independent points form the r-dimensional linear subspace Pr(R) of the space Pn(R).

Extended Euclidean plane E2 is model of the projective plane, and extended Euclidean space E3is model of the projective space.

Homogeneous coordinate system in the projective space E2 is the extension of the Cartesian coordinate system in the Euclidean plane E2(R).

Homogeneous coordinates of the point A=(xA,yA) in the Euclidean subspace E2(R) of the projective space E2 can be defined as any triple of real numbers (a1,a2,a0),a00, while the following holds

x A = a 1 a 0 , y A = a 2 a 0

Any real point in the projective plane E2has the nonzero coordinate a0.

Normal form of the homogeneous coordinates of the real point A is the triple of real numbers A=(xA,yA,1).

obr5
Figure 1: Homogeneous coordinates of ideal point (direction)

Representative of a vector (direction) a=BC is the oriented line segment determined by the real end points B and C. Homogeneous coordinates of the direction a representing the ideal point U can be derived from the homogeneous coordinates of the chosen representative vector endpoints.

a = ( x a , y a , 0 ) = C B = ( x C , y C , 1 ) ( x B , y B , 1 ) =
= ( x C x B , y C y B , 0 ) = ( x U , y U , 0 ) = U

Homogeneous coordinates of an ideal point in the projective plane E2form a triple of real numbers

( x U , y U , 0 ) = h ( x a , y a , 0 ) , h 0

where xa,ya are Cartesian coordinates of the ideal point (direction) chosen representative vector.

Homogeneous coordinates of the point A=(xA,yA,zA) in the Euclidean subspace E3(R) of the projective space E3 can be defined as any quadruple of real numbers (a1,a2,a3,a0),a00, while the following holds

x A = a 1 a 0 , y A = a 2 a 0 , z A = a 3 a 0

Any real point in the projective space E3has the nonzero coordinate a0.

Normal form of the homogeneous coordinates of the real point A is the quadruple of real numbers A=(xA,yA,zA,1).

obr10
Figure 2: Coordinates of vector

Representative of a vector (direction) a=BC is the oriented line segment determined by the real endpoints B and C. Homogeneous coordinates of the direction a representing the ideal point U can be derived from the homogeneous coordinates of the chosen representative vector endpoints.

a = ( x a , y a , z a , 0 ) = C B = ( x C , y C , z C , 1 ) ( x B , y B , z C , 1 ) =
= ( x C x B , y C y B , z C z B , 0 ) = ( x U , y U , z U , 0 ) = U

Homogeneous coordinates of the ideal point in the projective plane E3form a triple of real numbers

( x U , y U , z U , 0 ) = h ( x a , y a , z a , 0 ) , h 0

where xa,ya,za are Cartesian coordinates of the ideal point (direction) chosen representative.

obr11
Figure 3: Homogeneous coordinates of ideal point