A [math]\textbf{perspectivity}[/math] from line L to a distinct line M with [math]\textbf{centre}[/math] c not incident to L or M is the function that, for any point x on L, returns the point M[ox].

A [math]\textbf{perspectivity}[/math] from point p to a distinct point q with [math]\textbf{axis}[/math] C not incident to p or q is the function that, for any point X on p, returns the point q[AX].

A [math]\textbf{projectivity}[/math] is a finite composition of perspectivities.

A [math]\textbf{half-perspectivity}[/math] from line L to a non-incident point p is the function that, for any point x on L, returns the line xp. The half-perspectivity from point p to a non-incident line L is the inverse of the function above.

A [math]\textbf{half-projectivity}[/math] is a composition of an odd number of half-perspectivities.

Given two ordered n-tuples of objects from a single incidence set each (X_1, ... , X_n) and (y_1, ... , y_n), we say these tuples are [math]\textbf{projectvely related}[/math] if there is some (half)-projectivity Pr such that Pr(X_k) = y_k for all k such that [math]1\leq k\leq n[/math].

Since we will use (half-) perspectivities and projectivities so often, we need a lot of new notation. We use two notations for (half-) perspectivities and projectivities. The two systems describe the perspectivity from L to M with centre p as follows:

L>p<M

[eqn]L\stackrel{p}{\barwedge}M[/eqn]

The perspectivity from p to q with axis L is noted as follows:

p<L>q

[eqn]p\stackrel{L}{\barwedge}q[/eqn]

Projectivities can be notated by concatenating perspectivities.

p<N>q<L>r

[eqn]A\stackrel{x}{\barwedge}B\stackrel{y}{\barwedge}C\stackrel{z}{\barwedge}D[/eqn]

The LaTeX-based notation does not deal well with half-perspectivities and half-projectivities, but the text-based notation does.

p<L

X>a<Y>b

The next post will contain some more notation, and some examples.