---
documentclass: scrartcl
classoption: DIV=15
title: Projective Geometry
subtitle: After [N. J. Wildberger's YouTube lecture series
et al
author: Martin Pitt ()
header-includes: \usepackage[all]{xy}
...
Projective geometry is the geometry involving only a straightedge. It is
the fundament for affine, Euclidean, spherical, hyperbolic, and other
geometries.
Classification of geometries
============================
\xymatrix{
& & & *+[F]\txt{\textbf{Euclidean}}\\
*+[F]\txt{\textbf{Projective}\\
only tool: straightedge\\
objects: points, lines\\
ops: join of points,\\
meet of lines}
\ar[r]
& *+[F]\txt{\textbf{Affine}\\
introduce parallel lines\\
translation\\
scaling along a line\\
relative area\\
vectors\\
linear transformations}
\ar[r]
& *+[F]\txt{\textbf{Metrical structure}\\
introduce null conic\\
defines a dot product\\
quadrance (distance)\\
spread (angle)\\
perpendicularity\\
circles}
\ar[ur]
\ar[r]
\ar[dr]
& *+[F]\txt{\textbf{Hyperbolic}}\\
& & & *+[F]\txt{\textbf{Spherical}/\\
\textbf{Elliptical}}\\
}
Notation
========
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 $b$
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
\pagebreak
Important Theorems
==================
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.
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.
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).
Perspectivity
=============
In arts, photography, etc., one often has a horizon and perspective. In
these projections, these rules hold:
- Lines are projected to lines.
- 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.
- 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.
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 **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 $[x:y]$ (**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 **two-dimensional geometry**, the projective plane $\mathbb{P}^2$ is
described with a three-dimensional vector space $\mathbb{V}^3$,
**projective points** $a=[x:y:z]$ (lines through the origin) and
**projective lines** $A=(l:m:n) \Leftrightarrow lx+my+nz = 0$ (planes
through the origin).
In particular, we introduce a **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.
Calculations with homogenous coordinates
========================================
Join of points:
: If $a = [x_1:y_1:z_1]$ and $b = [x_2:y_2:z_2]$, then
$ab = (y_1z_2 - y_2z_1 : z_1x_2 - z_2x_1 : x_1y_2 - x_2y_1)$
Meet of lines:
: If $L = (l_1:m_1:n_1)$ and $M = (l_2:m_2:n_2)$, then
$ab = [m_1n_2 - m_2n_1 : n_1l_2 - n_2l_1 : l_1m_2 - l_2m_1]$
Incidence:
: The point $[x:y:z]$ and the line $(l:m:n)$ are incident
$\Leftrightarrow lx + my + nz = 0$
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
$[x:y:z] \to [x':y':z'] = [x\: y\: z]\left[\begin{array}{ccc}a & b & 0 \\c & d & 0\\ 0 & 0 & 1\end{array}\right]$
Affine transformation:
: A translation (non-linear) $(x, y) \to (x + a, y + b)$ is
represented homogeneously with
$[x:y:z] \to [x':y':z'] = [x \: y \: z]\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0\\ a & b & 1\end{array}\right]$
$= [x + az: y + bz : z]$
Metrical structure
==================
Add a “null conic” ${\mathcal C}$ 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 (**dot/inner product**)
on $\mathbb{V}^3$: $v\cdot w := vAw^T$
The usual **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 **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 **perpendicularity**:
$p \perp q \Leftrightarrow p\cdot q = pAq^T = 0$. This is well-defined
projectively and in $\mathbb{P}^2$.
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 **polarity**.
Pole:
: The pole $a$ of a line $L$ is the inversion point of $L$’s closest
point to ${\mathcal C}$. $\bf a = L^\perp$
Polar:
: The line $L$ is the polar of the pole point $a$. $\bf L = a^\perp$
\pagebreak
**Construction of polar of $a$:**
- Find “null points” $\alpha, \beta, \gamma, \delta$ on
${\mathcal C}$ so that
$(\alpha\beta)(\gamma\delta) = a$ (i. e. select two secants of
${\mathcal C}$ which pass through $a$)
- Define $b := (\alpha\delta)(\beta\gamma)$ and $c :=
(\alpha\gamma)(\beta\delta)$, i. e. the two intersections of
the other diagonals of the quadrangle $\alpha\beta\gamma\delta$
- 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)
- **Polar Independence Theorem:** The polar $A = a^\perp$ does not
depend on the choice of lines through $a$, i. e. the choice of
$\alpha, \beta, \gamma, \delta$.
**Construction of pole of $A$:** Choose any two points $b, c\in A$. Then
$a = (b^\perp c^\perp)$.
**Construction of polar of null point $\gamma$:**
- Chose any two secants $A$, $B$ that pass through $\gamma$
- Construct their poles $A^\perp$, $B^\perp$
- Polar $\Gamma = \gamma^\perp$ is the tangent $(A^\perp B^\perp)$
**Polar Duality Theorem:** For any two points $a$ and $b$:
$a\in b^\perp \Leftrightarrow b\in a^\perp$ (as $A$ is symmetric)