## Section3.5Projective geometry

Early motivation for the development of projective geometry came from artists trying to solve practical problems in perspective drawing and painting. In this section, we present a modern Kleinian version of projective geometry.

Throughout this section, $\F$ is a field, $V$ is a vector space over $\F\text{,}$ $\Proj(V)=(V\setminus \!\{0\})/\F^\ast$ is the projective space, and $PGL(V)=GL(V)/\F^\ast$ is the projective transformation group. See Exercise 2.5.3.6 for definitions and details. We will write $[T]$ for the projective transformation that is the equivalence class of the linear transformation $T$ of $V\text{.}$

### Subsection3.5.1Projective points, lines, and flats

Points in projective space correspond bijectively to 1-dimensional subspaces of $V$ via

\begin{equation*} [v] \leftrightarrow \{\alpha v\colon \alpha\in\F\}. \end{equation*}

The set of 1-dimensional subspaces in $V\text{,}$ denoted $G(1,V)\text{,}$ is an alternative model space for projective geometry. We will usually denote points in projective space using capital letters, such as $P\text{,}$ $Q\text{,}$ etc.

A line in projective space is a set of the form

\begin{equation*} \ell_\Pi=\{[v]\colon v\in \Pi\} \end{equation*}

for some 2-dimensional subspace $\Pi$ in $V\text{.}$ Thus, projective lines correspond bijectively to 2-dimensional subspaces of $V$ via

\begin{equation*} \ell_{\Pi} \leftrightarrow \Pi. \end{equation*}

The set of 2-dimensional subspaces in $V$ is denoted $G(2,V)\text{.}$ We will usually denote projective lines using lower case letters, such as $\ell\text{,}$ $m\text{,}$ etc.

There is an offset by 1 in the use of the word "dimension" in regards to subsets of $\Proj(V)$ and the corresponding subspace in $V\text{.}$ In general, a $k$-dimensional flat in $\Proj(V)$ is a set of the form $\{[v]\colon v\in G(k+1,V)\}\text{,}$ where $G(d,V)$ denotes the set of $d$-dimensional subspaces of $V\text{.}$ 1  Flats are also called subspaces in projective space, even though projective space is not a vector space.

The set $G(d,V)$ is called the Grassmannian of $d$-dimensional subspaces of $V\text{,}$ named in honor of Hermann Grassmann.

Points $P_1=[v_1],P_2=[v_2],\ldots,P_k=[v_k]$ are said to be in general position if the vectors $v_1,v_2,\ldots,v_k$ are independent in $V\text{.}$

### Subsection3.5.2Coordinates

For the remainder of this section, we consider $V=\F^{n+1}\text{.}$ For readability, we will write $P=[v]=[x_0,x_1,x_2,\ldots,x_{n}]$ (rather than the more cumbersome $[(x_0,x_1,x_2,\ldots,x_n)]$) to denote the point in projective space that is the projective equivalence class of the point $v=(x_0,x_1,x_2,\ldots,x_{n})$ in $\F^{n+1}\text{.}$ The entries $x_i$ are called homogeneous coordinates of $P\text{.}$ If $x_0\neq 0\text{,}$ then

\begin{equation*} P=[x_0,x_1,x_2,\ldots,x_n]=\left[1,\frac{x_1}{x_0},\frac{x_2}{x_0},\ldots,\frac{x_n}{x_0}\right]. \end{equation*}

The numbers $x_i/x_0$ for $1\leq i\leq n$ are called inhomogeneous coordinates for $P\text{.}$ The $n$ degrees of freedom that are apparent in inhomogeneous coordinates explain why $\Proj(\F^{n+1})$ is called $n$-dimensional. Many texts write $\F\Proj(n)\text{,}$ $\F\Proj_n\text{,}$ or simply $\Proj_n$ when $\F$ is understood, to denote $\Proj(\F^{n+1})\text{.}$

### Subsection3.5.3Freedom in projective transformations

In an $n$-dimensional vector space, any $n$ independent vectors can be mapped to any other set of $n$ independent vectors by a linear transformation. Therefore it seems a little surprising that in $n$-dimensional projective space $\F\Proj_n=\Proj(\F^{n+1})\text{,}$ it is possible to map any set of $n+2$ points to any other set of $n+2$ points, provided both sets of points meet sufficient "independence" conditions. This subsection gives the details of this result, called the Fundamental Theorem of Projective Geometry.

Let $e_1,e_2,\ldots, e_n,e_{n+1}$ denote the standard basis vectors for $\F^{n+1}$ and let $e_0=\sum_{i=1}^{n+1}e_i\text{.}$ Let $v_1,v_2,\ldots,v_n$ be another basis for $\F^{n+1}$ and let $c_1,c_2,\ldots,c_{n+1}$ be nonzero scalars. Let $T$ be the linear transformation $T$ of $\F^{n+1}$ given by $e_i\to c_iv_i$ for $1\leq i\leq n+1\text{.}$ Projectively, $[T]$ sends $[e_i]\to [v_i]$ and $[e_0]\to [\sum_i c_iv_i]\text{.}$

Now suppose there is another map $[S]$ that agrees with $[T]$ on the $n+2$ points $[e_0],[e_1],[e_2],\ldots,[e_{n+1}]\text{.}$ Then $[U]:=[S]^{-1}\circ [T]$ fixes all the points $[e_0],[e_1],[e_2],\ldots,[e_{n+1}]\text{.}$ This means that $Ue_i=k_ie_i$ for some nonzero scalars $k_1,k_2,\ldots,k_{n+1}$ and that $Ue_0=k'e_0$ for some $k'\neq 0\text{.}$ This implies

\begin{equation*} (k_1,k_2,\ldots,k_{n+1})=(k',k',\ldots,k') \end{equation*}

so we have $k'=k_1=k+2=\cdots k_{n+1}\text{.}$ Therefore $[U]$ is the identity transformation, so $[S]=[T]\text{.}$ We have just proved the following existence and uniqueness lemma.

### Subsection3.5.4The real projective plane

The remainder of this section is devoted to the planar geometry $\Proj(\R^3)=\R\Proj_2$ called the real projective plane. It is of historical interest because of its early practical use by artists. Lines through the origin in $\R^3$ model sight lines in the real world as seen from an eye placed at the origin. A plane that does not pass through the origin models the "picture plane" of the artist's canvas. Figure 3.5.3 shows a woodcut by Albrecht Dürer that illustrates a "perspective machine" gadget used by 16th century artists to put the projective model into practice for image making.

A two dimensional subspace $\Pi$ in $\R^3$ is specified by a normal vector $n=(n_1,n_2,n_3)$ via the equation $n\cdot v=0\text{,}$ that is, a point $v=(x,y,z)$ lies on $\Pi$ with normal vector $n$ if and only if $n\cdot v=n_1x+n_2y+n_3z=0\text{.}$ Any nonzero multiple of $n$ is also a normal vector for $\Pi\text{,}$ so the set $G(2,\R)$ of 2-dimensional subspaces in $\R^3$ is in one-to-one correspondence $\R^3/\R^\ast\text{.}$ We will write $\ell=[n]=[n_1,n_2,n_3]$ to denote the projective line $\ell$ whose corresponding 2-dimensional subspace in $\R^3$ has normal vectors proportional to $(n_1,n_2,n_3)\text{.}$ Beware the overloaded notation! Whether the equivalence class $[v]$ of a vector $v$ in $\R^3$ denotes a projective point or a projective line has to be specified.

The equation $n\cdot v=0$ makes sense projectively. This means that if $n\cdot v=0$ for vectors $n,v\text{,}$ then

\begin{equation} (\alpha n)\cdot (\beta v)=0 \;\;\text{for all}\;\; \alpha,\beta\in \F^\ast\text{,}\label{projdotprodzero}\tag{3.5.1} \end{equation}

even though the value of the dot product is not well-defined for projective equivalence classes! Thus we will write $\ell\cdot P=[n_1,n_2,n_3]\cdot [x,y,z]=0$ for a projective line $\ell=[n_1,n_2,n_3]$ and a projective point $P=[x,y,z]\text{,}$ to mean (3.5.1), and we make the following interpretation of the dot product as an incidence relation in $\R\Proj_2\text{.}$

\begin{equation} \ell \cdot P = 0 \;\; \Leftrightarrow \;\;P\;\;\text{lies on}\;\;\ell\;\;\Leftrightarrow \;\;\ell\;\;\text{contains}\;\;P. \tag{3.5.2} \end{equation}

Given two independent vectors $v,w$ in $\R^3\text{,}$ their cross product $v\times w$ is a normal vector for the 2-dimensional space spanned by $v,w\text{.}$ Given two 2-dimensional subspaces $\Pi,\Sigma$ in $\R^3$ with normal vectors $n,m\text{,}$ the cross product $n\times m$ is a vector that lies along the 1-dimensional subspace $\Pi\cap\Sigma\text{.}$ The bilinearity of cross product implies that cross product is well-defined on projective classes, i.e., we can write $[u]\times [v]:=[u\times v]\text{.}$ Thus we have the following.

### Exercises3.5.5Exercises

###### 1.

Use Lemma 3.5.1 to prove the Fundamental Theorem of Projective Geometry.

###### 2.Coordinate charts and inhomogeneous coordinates.

To facilitate thinking about the interplay between the projective geometry $\Proj(\F^{n+1})=\F\Proj_n$ and the geometry of $\F^{n}$ (rather than $\F^{n+1}\text{!}$) it is useful to have a careful definition for "taking inhomogeneous coordinates in position $i$". Here it is: Let $U_i$ be the subset of $\F\Proj_n$ of points whose homogeneous coordinate $x_i$ is nonzero. Let $\pi_i\colon U_i\to \F^n$ be given by

\begin{equation*} [x_0,x_2,\ldots x_{i-1},x_i,x_{i+1},\ldots,x_n] \to \left(\frac{x_0}{x_i},\frac{x_2}{x_i},\ldots \frac{x_{i-1}}{x_i},\frac{x_{i+1}}{x_i},\ldots,\frac{x_n}{x_i}\right). \end{equation*}

The one-sided inverse $\F^n\to \F\Proj_n$ given by

\begin{equation*} (x_0,x_1,\ldots x_{i-1},\widehat{x_i},x_{i+1},\ldots,x_n) \to [x_0,x_1,\ldots x_{i-1},1,x_{i+1},\ldots,x_n] \end{equation*}

(where the circumflex hat indicates a deleted item from a sequence) is called the $i$-th coordinate chart for $\F\Proj_n\text{.}$ What is the map that results from applying the $0$-th coordinate chart $\C\to \C\Proj_2$ followed by taking homogeneous coordinates in position 1?

###### 3.Möbius geometry is projective geometry.

Show that Möbius geometry $(\extC,\M)$ and the projective geometry $(\Proj(\C^2),PGL(2))$ are equivalent via the map $\mu\colon \Proj(\C^2) \to \extC$ given by

\begin{equation} \mu([\alpha,\beta]) = \left\{ \begin{array}{cc} \alpha/\beta \amp \beta\neq 0\\ \infty \amp \beta=0 \end{array} \right..\label{mobiusmodelequivmap}\tag{3.5.3} \end{equation}

Comment: Observe that $\mu$ is an extension of $\pi_1\colon U_0\to \C$ given by $\pi_1([x_0,x_1])=\frac{x_0}{x_1}$ (defined in Exercise 3.5.5.2).

###### 4.Cross ratio.

The projective space $\Proj_1=\Proj(\F^2)$ is called the projective line). The map $\mu\colon \Proj_1\to \extF\text{,}$ given by $\mu([x_0,x_1])=\frac{x_0}{x_1}$ (defined in Exercise 3.5.5.3, but where $\F$ may be any field, with $\extF=\F\cup \{\infty\}$) takes the points

\begin{equation*} [e_0]=[1,1],[e_2]=[0,1],[e_1]=[1,0] \end{equation*}

in $\Proj_1$ to the points $1,0,\infty$ in $\extF\text{,}$ respectively. Let $(\cdot,P_1,P_2,P_3)$ denote the unique projective transformation $[T]$ that takes $P_1,P_2,P_3$ to $[e_0],[e_2],[e_1]\text{.}$ The cross ratio $(P_0,P_1,P_2,P_3)$ is defined to be $\mu([T](P_0))\text{.}$

1. Show that this definition of cross ratio in projective geometry corresponds to the cross ratio of Möbius geometry for the case $\F=\C\text{,}$ via the map $\mu\text{,}$ that is, show that the following holds.
\begin{equation*} \mu(P_0,P_1,P_2,P_3)=(\mu(P_0),\mu(P_1),\mu(P_2),\mu(P_3)) \end{equation*}
2. Show that
\begin{equation*} (P_0,P_1,P_2,P_3)=\frac{\det(P_0P_2)\det(P_1P_3)}{\det(P_1P_2)\det(P_0P_3)} \end{equation*}
where $\det(P_iP_j)$ is the determinant of the matrix $\twotwo{a_i}{a_j}{b_i}{b_j}\text{,}$ where $P_i=[a_i,b_i]\text{.}$ Suggestion: First find $[T]^{-1}\text{.}$ Solve
\begin{equation*} (a_1,b_1) = x(a_3,b_3) + y(a_2,b_2) \end{equation*}
to get $x=\det(P_1P_2)\text{,}$ $y=\det(P_3P_1)\text{.}$ From this, show the following.
\begin{align*} [T]^{-1}\amp=\twotwo{xa_3}{ya_2}{xb_3}{yb_2}\\ [T] \amp= \twotwo{yb_2}{-ya_2}{-xb_3}{xa_3} \end{align*}
Now just calculate $\mu(T(a_0,b_0))\text{.}$
###### 5.Condition for collinearity in $\R\Proj_2$.

Let $u=(u_1,u_2,u_3),v=(v_1,v_2,v_3),w=(w_1,w_2,w_3)$ be vectors in $\R^3\text{,}$ and let $M$ be the matrix $M=\left[\begin{array}{ccc} u_1 \amp v_1 \amp w_1\\ u_2 \amp v_2 \amp w_2\\ u_3 \amp v_3 \amp w_3\\\end{array}\right]$ Show that $[u],[v],[w]$ are collinear in $\R\Proj_2$ if and only if $\det M=0\text{.}$

###### 6.

The following is a famous theorem of classical geometry.

###### Pappus' Theorem.

Let $A,B,C$ be three distinct collinear points in $\R\Proj_2\text{.}$ Let $A',B',C'$ be another three distinct collinear points on a different line. Let $P,Q,R$ be the intersection points $P=BC'\cap B'C\text{,}$ $Q=AC'\cap A'C\text{,}$ $R=AB'\cap A'B\text{.}$ Then points $P,Q,R$ are collinear. See Figure 3.5.5.

Follow the outline below to prove Pappus' Theorem under the additional assumption that no three of $A,A',P,R$ are collinear. Applying the Fundamental Theorem of Projective Geometry, we may assume $A=[e_1]\text{,}$ $A'=[e_2]\text{,}$ $P=[e_3]\text{,}$ and $R=[e_0]\text{.}$

• Check that $AR=[0,-1,1]$ and $A'R=[1,0,-1]\text{.}$
• Explain why it follows that $B'=[r,1,1]$ and $B=[1,s,1]$ for some $r,s\text{.}$
• Explain why $C=[rs,s,1]$ and $C'=[r,rs,1]\text{.}$
• Explain why $Q=[rs,rs,1]\text{.}$
• Observe that $P,Q,R$ all lie on $[1,-1,0]\text{.}$
A quadric in $\Proj(\F^{n+1})$ is a set of points whose homogeneous coordinates satisfy an equation of the form
A quadric in $\R\Proj_2$ is called a conic.
1. Explain why (3.5.4) is a legitimate definition of a set of points in $\Proj(\F^{n+1})\text{.}$
2. Consider the conic $C$ given by
What are the figures in $\R^2$ that result from taking inhomogeneous coordinates (see Exercise 3.5.5.2) on $C$ in positions $0,1,2\text{?}$