The most important examples of projective lines are the usual lines of the euclidean plane to which we add an additional point, called

Consider the points of line L: y=1, parallel to the x-axis. Each line represented by [x

The set A* of all euclidean lines of the plane passing through the point A generalizes the basic model of projective line. Often A* is called the

Obviously another system (x'

And for the corresponding quotients t'=x'

The points determine three lines represented by corresponding vectors {a,b,c}: A=[a], B=[b], C=[c]. The vectors are selected so that c=a+b. This condition uniquely determines {a,b,c} up to a multiplicative constant. The homogeneous coordinate system results by using the basis {a,b} and writing vor every point D on this line its coordinates with respect to this basis D=[d], where d=d

Obviously in this coordinate system {A,B,C} have correspondingly the coordinates {(1,0), (0,1), (1,1)}.

C is called

This means that selecting any projective bases {A,B,C} on M and {A',B',C'} on N and using the corresponding coordinate systems (x,y) and (x',y'), the transformation Q=F(P) is described in the respective homogeneous coordinates through linear equations:

Since the change of homogeneous coordinates in each projective line is done also through linear maps, a change of homogeneous coordinates replaces the above matrix U with a matrix U'=XUY, where {X,Y} are also invertible matrices related to the coordinate changes in M and N.

The image above illustrates such a map. To verify the defining condition we take R to be the origin of coordinates and the projective bases {A,B,C} on M and {A'=F(A),B'=F(B),C'=F(C)} on N. With respect to this bases map F is described through the identity transformation (x'(Q),y'(Q)) = (x(P),y(P)), which is the simplest invertible linear map.

File RectHypeRelation.html discusses the representation of such a perspectivity referred to more general projective bases on the lines M and N.

The important point is again that this definition, although it uses homogeneous coordinates, it defines a number independent of the particular coordinates used. In fact, changing to another system of homogeneous coordinates, according to (2) changes {p,q,u,v} to {p',q',u',v'} such that:

In other words {p,q,u,v} and {p',q',u',v'} are related through a

Here (u

The ubiquitous

The figure shows how the coordinates of the four points {[a],[b],[c],[x]} depend on the particular vectors {a,b,c,x} representing these points. Vector x is decomposed with respect to {a',b'}, which are determined by the (parallel) projections {a',b'} of c on {a,b} and not by {a,b} themselves. This shows how the

Number d = x

d(X,[a]):d(X,[b]) = (d(x,a)/d(x,b))*(r/s) = (x

In total we have the following possibilities to calculate the cross ratio:

Last expression follows from (5) by dividing both members of the ratio d(X,[b])/d(X,[a]) with distance d(O,X) and both members of ratio d(C,[b])/d(C,[a]) with distance d(O,C). In the last two expressions points {A,B,C,X} are identified with points on the supporting euclidean line.

ProjectiveBase.html

RectHypeRelation.html

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