Let X be a point on the x-axis. A projective transformation can be defined geometrically for this line by picking a pair of points P, Q, and a line m, all within the same x-y plane which contains the x-axis upon which the transformation will be performed.
Points P and Q represent two different observers, or points of view. Point R is the position of some object they are observing. Line m is the objective world which they are observing, and the x-axis is the subjective perception of m.
Draw line l through points P and X. Line l crosses line m at point R. Then draw line t through points Q and R: line t will cross the x-axis at point T. Point T is the transform of point X [Paiva].
The above is a synthetic description of a one-dimensional projective transformation. It is now desired to convert it to an analytical (Cartesian) description.
Let point X have coordinates (x_{0},0). Let point P have coordinates $(P_{x},P_{y})$. Let point Q have coordinates $(Q_{x},Q_{y})$. Let line m have slope m (m is being overloaded in meaning).
The slope of line l is
$P_{y} \over P_{x}-x_{0},$
so an arbitrary point (x,y) on line l is given by the equation
The value of λ must be adjusted so that both sides of vector equation (3) are equal. Equation (3) is actually two equations, one for abscissas and one for ordinates. The one for ordinates is
It is clear from the synthetic definition that the inverse transformation is obtained by exchanging points P and Q. This can also be shown analytically. If P ↔ Q, then α → α′, β → β′, γ → γ′, and δ → δ′, where
$\alpha '=mP_{x}Q_{y}-P_{y}Q_{y}+bP_{y}=\delta ,$
$\beta '=bP_{x}Q_{y}-bP_{y}Q_{x}=-\beta ,$
$\gamma '=m(Q_{y}-P_{y})=-\gamma ,$
$\delta '=mQ_{x}P_{y}+bQ_{y}-Q_{y}P_{y}=\alpha .$
Therefore if the forwards transformation is
$t(x)={\alpha x+\beta \over \gamma x+\delta }$
then the transformation t′ obtained by exchanging P and Q (P ↔ Q) is:
Therefore t′(x) = t^{−1}(x): the inverse projective transformation is obtained by exchanging observers P and Q, or by letting α ↔ δ, β → −β, and γ → −γ. This is, by the way, analogous to the procedure for obtaining the inverse of a two-dimensional matrix:
It remains to show that there is closure in the composition of transformations. One transformation operating on another transformation produces a third transformation. Let the first transformation be t_{1} and the second one be t_{2}:
Since matrices multiply associatively, it follows that composition of projections is also associative.
Projections have: an operation (composition), associativity, an identity, an inverse and closure, so they form a group.
The cross-ratio defined by means of a projectionEdit
Let there be a transformation t_{s} such that t_{s}(A) = $\infty$, t_{s}(B) = 0, t_{s}(C) = 1. Then the value of t_{s}(D) is called the cross-ratio of points A, B, C and D, and is denoted as [A, B, C, D]_{s}:
In equation (10), it is seen that t_{s}(D) does not depend on the coefficients of the projection t_{s}. It only depends on the positions of the points on the "subjective" projective line. This means that the cross-ratio depends only on the relative distances among four collinear points, and not on the projective transformation which was used to obtain (or define) the cross-ratio. The cross ratio is therefore
Transformations on the projective line preserve cross ratio. This will now be proven. Let there be four (collinear) points A, B, C, D. Their cross-ratio is given by equation (11). Let S(x) be a projective transformation: