Homogeneous Coordinates and the Projective Plane

geometric transformations in projective space

scale and reflections

σ

x

0.2

σ

y

0.2

r

x

r

y

shearing

ϕ

0

rotation

θ

0

translation

x

0

y

0

perspective

p

x

0

p

y

0

graph range

r

5

transformation matrix

0.200	0.000	0.000
0.000	0.200	0.000
0.000	0.000	1.000

The points

(a,b,c)

of the projective plane have three homogeneous coordinates, so that

(a,b,c)

and

(at,bt,ct)

,

t∈

are the same point, as long as

t≠0

; these points can be represented as lines in three dimensions passing through the origin (the dotted lines),

x=at

,

y=bt

,

z=ct

. These lines intersect any plane

z=k,k≠0

in points with the two coordinates of the Cartesian plane. This Demonstration shows the action of the transformation matrix on the lines and the points in the plane

z=1

and another such plane.