## Homography between the points of two lines

Having arbitrary line coordinates (see CrossRatio0.html ) on two lines e, e', a homographic relation between the points of these lines is a relation of the form x'=(a*x+b)/(c*x+d). A basic fact is that the geometric locus of intersection points P of the lines XY', X'Y is a third line f (red line).

Since homographic relations preserve the cross ratio, line bundles X(X',Y',Z',V') and X'(X,Y,Z,V) define the same cross ratio on every line that intersects them. The two bundles having in common the line XX', have, by the well known property of line bundles (see CrossRatioLines.html ), the other pairs of corresponding lines intersect allong a line. Fixing Z, V and their images Z', V' by the homographic relation we get the proof of the above proposition.
Notice that this property implies trivially the theorem of Pappus (see PappusLines.html ), since three points X, Y, Z on line e and three other points X', Y', Z' on line e' uniquely determine a homographic relation mapping X to X' etc..
Notice that this property generalizes to conics, almost verbatim, and defines what is called the Homography Axis of an Homography preserving a conic (see HomographyAxis.html ).
To understain the real story with this line see also its connexion with the conic enveloping lines XX', YY' etc., contained in the file LineHomographyAxis2.html .