Projective Geometry

1. Homogeneous coordinate
In Cartesian coordinate system, the coordinates of a point measures distance relatively, but homogeneous coordinate
system serves for different purpose. The main propose of this system is calculation of infinitimum cum extension of
affine coordinate.
If we consider a plane and take a point at infinity, then coordinate (x, y) (in general both) are infinite. The
calculations of such infinite quantities are interesting with homogeneous coordinate system.
The point (x, y) of plane in homogeneous system is denoted by triple
class of triples with
which preserves proportionality
If k is any real number except zero, the homogeneous coordinates
and
represents exactly the same point in the same way that we normally reduce our rational numbers to
lowest terms. Therefore a simple representation for homogeneous coordinates is always preferred
that we can.
If
is not equal to zero, we can multiply every coordinate of
by 1/
to obtain equivalent point
which is same as
i.e. the Euclidean point (x,y) can be extended to homogeneous coordinate simply by adding „1” as third coordinate.
For example, the points (2,1,1), (4,2,2) (200,100,100)… all corresponds to same Euclidean point (2,1).
(200,100)
(10,5)
(4,2)
Figure. 1.21
(2,1)
Note
Euclidean coordinate (x,y) corresponds to homogeneous coordinate (x,y,1)
If we plot the points (2,1), (4,2) (6,3), (20,10), (200,100)… where the second coordinate is always double the first.
Thus all the points along line x=2y can be written as (2y, ), where is a real number.
But in homogeneous system
homogeneous coordinates [
i.e. if
, we can let
be zero, and we obtain a “point at infinity” with
,0]
, then
is point at infinity but ratio of x and y is finite.
Summary
(1) Let
be a point in
,
then homogeneous coordinate of
is denoted by
with
,
[The pair
(2) Let
Since
The coordinate
can be a point in R if and both are reals]
be a line in R, then homogeneous coordinates of
is equivalent to
is same as
is denoted by

2. The triple
can be homogeneous coordinates of a line
in R if
not zero.
[
can be a line in
if and both are not zero]
(3) The homogeneous coordinates of a point
lies on the coordinate of line
lies on the line
i.e.
or
The point
[Since
lies on line
and
both are
if and only if
]
if and only if
Note
The incidence condition of a point
on
on the line
implies same condition of the coordinate
(4) If
be coordinates of two lines
point of intersection is given as
If the lines given above are parallel, then
intersection becomes as
in
with the possibility of
R then their
, then the point of
Since the triple
can be a homogeneous coordinates of
in R if
,
can‟t be
homogenous coordinate of any point. Thus we adopt the point of the form
as an ideal point (point
at infinity) in
where two parallel lines meet.
(5) Since the coordinate
satisfy the incidence condition for all ideal points of the form
we
adopt
as an ideal line (line at infinity) in .
Things to Remember
1. Homogeneous coordinate of a line in Euclidean plane is
2. Homogeneous coordinate of a point in Euclidean plane is
where x3 is not zero
3. If x3 is zero then
represent a point at infinity
4. Homogeneous coordinate Preserves proportionality class
5. Homogeneous coordinate Is invented by Poncelete
6. Homogeneous coordinate
represents line at infinity
7. Homogeneous coordinate
represents point at infinity
8. By homogeneous coordinate calculation of infinitesimal is possible
9. Felix Klein provided an algebraic foundation for projective geometry in terms of
"homogeneous coordinates," which had been discovered independently by K. W. Feuerbach
and A. F. Mobius in 1827.
10. Euclidean coordinate (x,y) can be extended to homogeneous coordinate (x,y,1)
11. Homogeneous coordinate Preserves proportionality class
Homogeneous coordinate
Is invented by Poncelete