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\begin{document}
\title{Projective Geometry}
\subtitle{After N. J. Wildberger's online lecture series\\
          \url{http://www.youtube.com/watch?v=NYK0GBQVngs} et al}
\author{Martin Pitt (\texttt{martin@piware.de})}
\date{}
\maketitle

Projective geometry is the geometry involving only a straightedge. It is the
fundament for affine (Euclidean), spherical, hyperbolic, and other geometries.

\section{Notation}
\begin{description}
    \item [Points:] projective points have lower-case variables ($a$, $p$);
    affine (Euclidean) points have upper-case variables ($A$, $P$);
    the product of two points $ab$ is the line that goes through $a$ and $b$
    \item [Lines:] projective lines have upper-case variables ($L$, $M$);
    affine lines have lower-case variables ($l$, $m$);
        the product of two lines $AB$ is their intersection, i.~e. a point
\end{description}

\section{Important Theorems}
\begin{description}
    \item [Pappus' Theorem:] If $a_1, a_2, a_3$ are collinear and $b_1,
        b_2,b_3$ are collinear:\\
        $c_1 := (a_2b_3)(a_3b_2)$, $c_2:=(a_1b_3)(a_3b_1)$, $c_3 := (a_2b_3)(a_3b_2)$ are collinear.

    \item [Pascal's Theorem:] If $a_1, a_2, a_3, b_1, b_2,b_3$ are points on a conic:\\
        $c_1 := (a_2b_3)(a_3b_2)$, $c_2:=(a_1b_3)(a_3b_1)$, $c_3 := (a_2b_3)(a_3b_2)$ are collinear.

    \item [Desargues Theorem:] If two triangles
        $\overline{a_1a_2a_3}$ and $\overline{b_1b_2b_3}$ are perspective from
        a point $p$, i. e. $a_1b_1$, $a_2b_2$, $a_3b_3$ are concurrent; then they
        are perspective from a line $L$, where $(a_1a_2)(b_1b_2)$,
        $(a_2a_3)(b_2b_3)$, $(a_3a_1)(b_3b_1)$ are collinear).
\end{description}

\section{Perspectivity}

In arts, photography, etc., one often has a horizon and perspective. In these
projections, these rules hold:

\begin{itemize}
   \item Lines are projected to lines.
   \item All parallel lines meet at a point on the horizon (``point at
   infinity''). There is one point on the horizon for each family of parallel lines.
   \item Conics are projected to conics. E. g. a parabola or a hyperbola will
   look like an ellipse. All conics will look elliptic as they originate from
   slicing a cone.
\end{itemize}

\section{Projective homogeneous coordinates}

In affine geometry, lines from a 3D object through the projective plane are
parallel, and projections maintain linear spacing. In projective geometry, they
meet in two points (``observer'' or ''horizon''), and projections distort
linear spacing.

In \textbf{one-dimensional geometry}, the projective line $\mathbb{P}^1$ is the
space of one-dimensional subspaces of the two-dimensional $\mathbb{V}^2$, i. e. a line
through $[0,0]$. This is specified by a proportion $[x:y]$ (\textbf{projective
point}); canonically, either $[x:1]$, or $[x:0]$ for the special case of the
subspace being the x axis (representing ``point at $\infty$'')

In \textbf{two-dimensional geometry}, the projective plane $\mathbb{P}^2$ is
described with a three-dimensional vector space $\mathbb{V}^3$,
\textbf{projective points} $a=[x:y:z]$ (lines through the origin) and
\textbf{projective lines}
$A=(l:m:n) \Leftrightarrow lx+my+nz = 0$ (planes through the origin).

In particular, we introduce a \textbf{viewing plane} $y_{z=1}$. Then any
projective point $[x:y:z], z\neq 0$ meets the viewing plane at
$[\frac{x}{z}:\frac{y}{z}:1]$. Any projective point $[x:y:0]$ corresponds to
the two-dimensional equivalent $[x:y]$, representing the ``point at $\infty$ in
the direction $[x:y]$''. So the projective plane $\mathbb{P}^2$ corresponds to
the affine plane $\mathbb{A}^2$ plus the projective line $\mathbb{P}^1$ for the
points at infinity.

Any projective line $(l:m:n), l,m \neq 0$ meets the viewing plane at the line
$lx+my=n$.

Coordinates in the viewing plane are called $X:=\frac{x}{z}$ and
$Y:=\frac{y}{z}$.

This introduces a duality between lines and points: Any theorem has a dual
counterpart where lines and points, meets and joins etc. are interchanged.

\section{Calculations with homogenous coordinates}
\begin{description}
    \item [Join of points:] If $a = [x_1:y_1:z_1]$ and $b = [x_2:y_2:z_2]$,
    then 

    $ab = (y_1z_2 - y_2z_1 : z_1x_2 - z_2x_1 : x_1y_2 - x_2y_1)$

    \item [Meet of lines:] If $L = (l_1:m_1:n_1)$ and $M = (l_2:m_2:n_2)$,
    then 

    $ab = [m_1n_2 - m_2n_1 : n_1l_2 - n_2l_1 : l_1m_2 - l_2m_1]$

    \item [Incidence:] The point $[x:y:z]$ and the line $(l:m:n)$ are incident
    $\Leftrightarrow lx + my + nz = 0$

    \item [Linear transformation:] A linear transformation
    $(x, y) \to (x', y') = (x\: y)\left(\begin{array}{cc}a & b\\c & d\end{array}\right)$
    is represented homogeneously with
    $[x:y:z] \to [x':y':z'] = [x\: y\: z]\left[\begin{array}{ccc}a & b & 0 \\c & d & 0\\ 0 & 0 & 1\end{array}\right]$

    \item [Affine transformation:] A translation (non-linear) $(x, y) \to (x + a, y + b)$
    is represented homogeneously with
    $[x:y:z] \to [x':y':z'] = [x \: y \: z]\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0\\ a & b & 1\end{array}\right]$
    $ = [x + az: y + bz : z]$
\end{description}

\section{Metrical structure}

Add a ``null conic'' \nullconic to the geometry by defining a (homogeneous)
symmetric matrix
$A=\left[\begin{array}{ccc}a & d & f\\ d & b & g\\ f & g & c\end{array}\right]$

For any point $p = [x:y:z]$, setting $p A p^T = 0$ expands to the general
equation of a homogeneous conic: $ ax^2+by^2+cz^2 + 2dxy + 2fxz + 2gyz = 0$

When dividing by $z^2$ we get the affine coordinates: $aX^2+bY^2+ 2dXY + 2fX + 2gY + c = 0$

This matrix $A$ defines a symmetric bilinear form (\textbf{dot/inner product}) on
 $\mathbb{V}^3$: $ v\cdot w := vAw^T$

The usual \textbf{Euclidean} one is
   $A_E=\tiny\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]$
but there are other interesting ones too, like the \textbf{relativistic}
(Lorentz, Minkowski, Einstein) structure
  $A_R=\tiny\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\end{array}\right]$.

This defines the notion of \textbf{perpendicularity}:
$p \perp q \Leftrightarrow p\cdot q = pAq^T = 0$.
This is well-defined projectively and in $\mathbb{P}^2$.

\section{Polarity}
$aAb^T = 0$ can be interpreted as $a(Ab^T) = 0$ and since $L := Ab^T$ defines a
line, as $aL = 0$. I. e. the dot product defines a
duality between points and lines, called \textbf{polarity}.

\begin{description}
    \item [Pole:] The pole $a$ of a line $L$ is the inversion point of $L$'s
        closest point to \nullconic. $\bf a = L^\perp$
    \item [Polar:] The line $L$ is the polar of the pole point $a$. $\bf L = a^\perp$
\end{description}

\textbf{Construction of polar of $a$:} 
\begin{itemize}
    \item Find ``null points'' $\alpha, \beta, \gamma, \delta$ on \nullconic so
        that $(\alpha\beta)(\gamma\delta) = a$ (i.~e. select two secants of
        \nullconic which pass through $a$)
    \item Define $b := (\alpha\delta)(\beta\gamma)$ and $c :=
        (\alpha\gamma)(\beta\delta)$, i.~e. the two intersections of the other
        diagonals of the quadrangle $\alpha\beta\gamma\delta$
    \item Then $A = a^\perp = bc$. Also, $b^\perp = ac$ and $c^\perp = ab$ (three-fold symmetry of
        polars and poles produced by the three intersections of diagonals)
    \item \textbf{Polar Independence Theorem:} The polar $A = a^\perp$ does not depend
        on the choice of lines through $a$, i.~e. the choice of $\alpha, \beta,
        \gamma, \delta$.
\end{itemize}

\textbf{Construction of pole of $A$:} Choose any two points $b, c\in A$. Then $a = (b^\perp c^\perp)$.

\textbf{Construction of polar of null point $\gamma$:}
\nopagebreak[2]
\begin{itemize}
    \item Chose any two secants $A$, $B$ that pass through $\gamma$
    \item Construct their poles $A^\perp$, $B^\perp$
    \item Polar $\Gamma = \gamma^\perp$ is the tangent $(A^\perp B^\perp)$
\end{itemize}

\textbf{Polar Duality Theorem:} For any two points $a$ and $b$:
 $a\in b^\perp \Leftrightarrow b\in a^\perp$ (as $A$ is symmetric)


\end{document}

