B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 115

plane z = 1 and a plane through the origin projects onto line on the plane
z =1.
If ( a, b, c ) is any point in Cartesian 3-dimensional space, then this point
determines a line through the origin whose equations are

x = at
y = bt
z = ct,

where t is a parameter. Any other point ( at, bt, ct ) determines the same line.
So, two points, ( a 1 ,b 1 ,c 1 ) and ( a 2 ,b 2 ,c 2 ), are on the same line through the
origin if

a 2 = a 1 t
b 2 = b 1 t
c 2 = c 1 t.
We say ( a 1 ,b 1 ,c 1 )( a 2 ,b 2 ,c 2 ). The equivalence classes of all triples equiva-
lent to ( a, b, c ), written as [( a, b, c )], are the points of the projective plane.
Any representative ( a 1 ,b 1 ,c 1 ) equivalent to ( a, b, c ) is called the homogeneous
coordinates of the point [ a, b, c ] in the projective plane.
The points of the form ( a, b, 0) are called ideal points of the projective
plane. This comes from the fact that lines in the plane z = 0 project to
infinity. In a similar way, any plane through the origin has an equation n 1 x +
n 2 y + n 3 z = 0. We can observe that kn x + kn 2 y + kn 3 z = 0, where k is a
multiple, also defines the same plane.
Any triple of numbers ( n 1 ,n 2 ,n 3 ) defines a plane through the origin. Now,
( n 1 ,n 2 ,n 3 )( d 1 ,d 2 ,d 3 ) if there is a number k such that d 1 = kn 1 , d 2 = kn 2 ,
and d 3 = kn 3 . The equivalence classes of all triples [ n 1 ,n 2 ,n 3 ] are the lines of
the projective plane. Any representative ( d 1 ,d 2 ,d 3 ) of the equivalence classes
[ n 1 ,n 2 ,n 3 ] is called the homogeneous line coordinate in the projective plane.
If ( x 1 ,y 1 ,z 1 ), z 1

= 0 are the homogeneous coordinates of a point of the
projective plane, the equations x =

y 1

x 1
Z 1 Z 1 define a correspondence

between points P 1 ( x 1 ,y 1 ,z 1 ) of the projective plane and points P ( x, y )ofthe
Cartesian plane. There is no Cartesian point corresponding to the ideal point
( x 1 ,y 1 , 0). But it is convenient to consider it as defining an infinitely distant
point.
Also, it is clear that any Cartesian point P ( x, y ) corresponds to a projective
point P ( x 1 ,y 1 ,z 1 ) whose homogeneous coordinates are x 1 = x , y 1 = y ,and
z 1 = 1. This correspondence between Cartesian coordinates and homogeneous
coordinates is exploited in graphics transformations. Note that even though
there is a correspondence between the points of the projective plane and those
of the Cartesian plane, these planes have different topological properties and
these properties should be taken into account while working with homogeneous
coordinates. Finally, if P 1 ( x 1 ,y 1 ,z 1 ,w 1 ) are the homogeneous coordinates of a
point in the three dimensional projective plane, then the corresponding three
dimensional Cartesian point P ( x, y, z )for w

and y =

= 0 is as follows: