Projective Geometry section 2: Perspecivities and Projectivities

To continue, we need tools to examine the projective plane we work with, like circles in ancient Greek geometry. We introduce the most fundamental of those tools here, but before we define it, we need a few definitions.

The set of all points incident to a given line is the [math]\textbf{range}[/math] of that line.

The set of all lines incident to a given point is the [math]\textbf{pencil}[/math] of that line.

In a projective plane, let ranges and pencils be [math]\textbf{incidence sets}[/math].

The incidence sets of the point p and line L can be written as In(,L) and In(p,) respectively.

Before we explicitly define this tool, we demonstrate it with the proof of the following theorem:

Theorem 2.1: Any two distinct ranges from the same plane have the same cardinality.

Proof: We observe that every line has a unique range and vice versa, thus the two resultant lines, which we call A and B, are also distinct. By Axiom 5*, we have a point o not incident to either A or B. Now, consider the following function.

[eqn]\begin{align*}
Pe : In(,\textup{A}) &\to In(,\textup{B})\\
\textup{a} &\mapsto \textup{B[oa]}.
\end{align*}[/eqn]

The construction of the points B[o?] for points on A is shown in the pic.

If d and e are any distinct points on A, then the lines od and oe are distinct lines which meet at o. Since o does not lie on B, then by Axiom 1, the points B[od] and B[oe] are distinct. Since d and e were arbitrary, then Pe is injective.

Now take an arbitrary point x on B. Let z = A[ox]. The point z exists and lies on A, so Pe(z) exists. {z, o, x} is collinear and x lies on B, so Pe(z) = B[oz] = x, thus Pe is surjective.

Since Pe is surjective and injective, it is a bijection between In(,A) and In(,B), thus they have the same cardinality. #

The function used in the proof above is the tool we seek, which we will formally define in the next large post.

Other urls found in this thread:

pastebin.com/50w9MYjD
pastebin.com/fMdRzhTf
dropbox.com/s/ia4m7s0fy59z9jv
dropbox.com/s/6ch0btu3adiz2ob
pastebin.com/Lgt8FB8X
maths.ed.ac.uk/~aar/papers/beutel.pdf.
i.imgur.com/gD02sgz.png.
twitter.com/NSFWRedditVideo

Section 0 text: pastebin.com/50w9MYjD
Section 1 text: pastebin.com/fMdRzhTf

I mis-numbered some theorems in the Section 1 text. They have been properly numbered in the 'cheat sheets' below. I also added and cleaned up some definitions.

Definitions and notations: dropbox.com/s/ia4m7s0fy59z9jv
Axioms, Theorems, and other statements: dropbox.com/s/6ch0btu3adiz2ob

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

I have no idea why the math tags aren't working here.

It doesn't work totally like LaTeX. If you want intermediary text with spacing you need to use "\"text{}" whereas in LaTeX it would just unitalisize it.

The text appears to work properly in the TeX preview, so you can copy the text from pastebin.com/Lgt8FB8X and paste it into the TeX preview to see the post.

Yeah that looks correct I take back , seems like it just fucked up.

If two tuples are (X_1, ... , X_n) and (y_1, ... , y_n) projectively related, we can note this in our two systems:

(X_1, ... , X_n) ~ (y_1, ... , y_n)

[eqn](x_{1},..., x_{n}) \doublebarwedge (Y_{1},..., Y_{n})[/eqn]

If the relating (half-)projectivity is known explicitly, we can combine the notations. Note the tuples are prefixed by the object its elements are all incident to.

p(A,B,C)q

I updated the notation cheat sheet with the new definitions and notations with a few corrections, so you can actually see them.

With those examples done, we can see what the more general (half-)projectivities are like. Since in general, projectivities are compositions of perspectivities, explicitly working them out could be quite messy. Instead, what we will do is show that certain incidence set tuples are projectively related, so demonstrating what certain projectivities can do.

Theorem 2.2: Given three tuples of distinct points (a, b, c) and (a', b', c') with elements taken from I(,L) and I(,M) respectively, then (a, b, c)~(a', b', c').

Proof: If a = a' = LM, then by Axiom 1 b, c, b', and c' are all distinct, and thus [bb'][cc'] = o exists, o does not lie on L or M, and L(a, b, c)>oa'a