Projective GeometryProjective Geometry

Hu Zhan Yi

Entities At InfinityEntities At Infinity

The ordinary space in which we lie is Euclidean space. The parallel lines usually do not intersect. If let the parallel lines extend infinitely, we have vision of their intersecting at a point, which is a point at infinity.

There is a unique point at infinity on any a line.

All points at infinity in a plane make up of a line, which is the line at infinity of the plane.

All points at infinity in space make up of a plane, which is the plane at infinity.

Projective SpaceProjective Space

With no differentiation between finite points and infinite points, n-dimensional Euclidean space and the entities at infinity make up of a n-dimensional projective space.

Homogeneous CoordinatesHomogeneous Coordinates

In order to study the entities at infinity, homogeneous coordinate is introduced.

After setting up a Euclidean coordinate system, every finite point in n-dimensional space can be represented by its coordinate

. Let be any scalars that satisfying:

Then is called the homogeneous coordinate of that point.

),...,( 1 nmm

),,...,( 01 xxx n

.,...,,00

10

10 n

n mx

xm

x

xx

01 ,,..., xxx n

Relative to homogeneous coordinate,

is called non-homogeneous coordinate of that point.

Vectors

are defined to be the homogeneous coordinates of points at infinity.

),...,( 1 nmm

)0,,...,( 1 nxx

Projective ParameterProjective Parameter

For a line in any dimensional projective space, any points on it can be linearly generated by two fixed points on it:

where are the homogeneous coordinates of respectively, are two scalars that are not both zero.

P

21, PP

2211 XcXcX

21,, XXX

21,, PPP 21,cc

The ratio is called the projective parameter of with respect to on the line through them.

By allowing , the projective parameter is .

2

1

c

c

P 21, PP

02 c

Cross RatioCross Ratio

For four collinear points , the ratio

is called the cross ratio of with

respect to , denoted by . Where are the projective parameters of

, .

4321 ,,, PPPP

))((

))((

4132

4231

),( 43 PP

),( 21 PP ),;,( 4321 PPPP

i

iP 4..1i

Projective TransformationProjective Transformation

Let be two n-dimensional projective spaces , be a 1-1 map from

to . If preserves:(i) the incidence relations of points and lines;

i.e. relations: a point on a line, a line through a point, et.al.

(ii) the cross ratio of any four collinear points, then is called a n-dimensional

projective transformation.

', nn SS

TnS '

nS T

T

The two projective spaces may be identical.

A n-dimensional projective transformation can be represented by a (n+1)-(n+1) matrix:

', nn SS

0

1

)1)(1(1)1(

)1(111

'0

'

'1

,,

,,

x

x

x

tt

tt

x

x

x

nnnn

n

n

For example: the following map from on the line to on the line is a

1-dimensional projective transformation:

iP

L 'iP 'L

Projective GeometryProjective Geometry

Projective Geometry is the geometry to study the properties in projective space that is invariant under projective transformation.

Harmonic RelationHarmonic Relation

We say that the pairs of points and

are harmonic if

),( 21 PP

),( 43 PP

1),;,( 4321 PPPP

The pairs of and are harmonic if and only if

where are the projective parameters of

, .

),( 21 PP ),( 43 PP

)(2))(( 43214321

i

iP 4..1i

ConicConic

A conic is the totality of points in a projective plane whose homogeneous

coordinates satisfy the following equation:

where at least one of is nonzero.

)(03

1,jiij

jijiij aaxxa

),,( 321 xxx

ija

The above equation in the definition of a conic has the equivalent form:

.

The matrix is symmetric, and its rank

is not changed under a projective transformation.

0

3

2

1

333231

232221

131211

321

x

x

x

aaa

aaa

aaa

xxx

)( ija

If the determinant of is zero, then the conic is two lines or one line, called degenerate conic.

Circles, ellipses, hyperbolas and parabolas are all non-degenerate conics.

)( ija