PROJECTIVE GEOMETRY COURSE

§ 5: *Projective coordinates*

Given the vector space ℜ^{3} and the real projective plane of the straight lines and planes through the origin.

If the point *p* in *P*^{2} as a straight line through the origin has vector representation λ(p_{1},p_{2},p_{3}), then we call
λ(p_{1},p_{2},p_{3}) *homogeneous* (*projective*) coordinates for *p*.

If the line *l* in *P*^{2} as a plane through the origin has a normal with vector representation λ(l_{1},l_{2},l_{3}), then we call
λ(l_{1},l_{2},l_{3}) projective coordinates for *l*.

The incidence relation is given by: {*p* is incident with *l*} ⇔ {p_{1}l_{1}+p_{2}l_{2}+p_{3}l_{3} = 0}.

(Check that this characterisation is independent of the choice of the representatives (p_{1},p_{2},p_{3}) and (l_{1},l_{2},l_{3}) of *p* and *l*.)

From the symmetry of these characterisations of point, line and incidence relation in terms of projective coordinates we see that the *principle of duality* holds for all
propositions about incidence of points and lines in the real projective plane, among which also the theorems of Pappus andn Desargues (see also the problems 17 through 20).

We get the dual proposition by automatically interchanging the words point and line and the words connect and intersect.

If we use the model of *O5*, we can give a proper point with projective coordinates λ(a,b,c) coordinates (a/c,b/c)
(*inhomogeneous* or *affine*) coordinates). This point has in ℜ^{3} place vector (a/c,b/c,1), so λ(a/c,b/c,1) is an alternative triple of projective coordinates.
See *O21*.

A line *l* whose equation (for the proper points) is, using inhomogeneous coordinates, ax_{1}+bx_{2}+c = 0, has as its equation in homogeneous coordinates
ax_{1}+bx_{2}+cx_{3} = 0 (this is the equation of *l* as a plane in ℜ^{3}). See again *O21*.

A conic *k* whose equation (for the proper points) is, using inhomogeneous coordinates,
a_{11}x_{1}^{2} + a_{12}x_{1}x_{2} + a_{22}x_{2}^{2} + a_{13}x_{1} + a_{23}x_{2} + a_{33} = 0,
has as its equation in homogeneous coordinates
a_{11}x_{1}^{2} + a_{12}x_{1}x_{2} + a_{22}x_{2}^{2} + a_{13}x_{1}x_{3} + a_{23}x_{2}x_{3}
+ a_{33}x_{3}^{2} = 0 (this is the equation of *k* as a conic in ℜ^{3}). See for this §20.

*O17* Find the projective coordinates of the connecting line of two points λ(a_{1},a_{2},a_{3}) and λ(b_{1},b_{2},b_{3}).

Find the projective coordinates of the intersection point of two lines λ(l_{1},l_{2},l_{3}) and λ(m_{1},m_{2},m_{3}).

*O18* Using a determinant, determine the condition for the points λ(a_{1},a_{2},a_{3}), λ(b_{1},b_{2},b_{3}) and
λ(c_{1},c_{2},c_{3}) to be collinear.

Likewise, the the condition for the lines λ(l_{1},l_{2},l_{3}), λ(m_{1},m_{2},m_{3}) and
λ(n_{1},n_{2},n_{3}) to concur.

*O19* Formulate the theorem of Desargues as a theorem from vector analysis: in terms of vectors, inner products, outer products and determinants.

*O20* Formulate the theorem of Pappus as a theorem from vector analysis.

*O21* Given the lines *l:* λ(0,1,2), *m:* λ(1,1,0), *n:* λ(1,0,1), and the points *P:* λ(2,-1,1), *Q:* λ(0,1,-1),
*R:* λ(1,2,0).

a) Make a sketch of these points and lines in the x_{3} = 1 (the plane of your piece of paper), on which a system of axes has been drawn.

b) Let ψ: *l*→*l* be the product of the projections from *l* via *P* onto *m*, from *m* via *Q* onto *n*, and
from *n* via *R* back onto *l*. Calculate the projective coordinates of the invariant points of ψ.