Basic definitions

Definition 1.1   The $n$-dimensional real projective space ${\Bbb R}P^n$ is the set of all lines in ${\Bbb R}^{n+1}$ passing through the origin.

Note that if $v$ and $w$ are two nonzero vectors lying on the same line through the origin, then they are multiples of each other. This remark allows us to redefine ${\Bbb R}P^n$ as the quotient of ${\Bbb R}^{n+1} \setminus \{0\}$ by the equivalence relation $v \sim w$ (or $[v] = [w]$) if $v$ and $w$ are multiples of each other.

Exercise 1.1 (00)   Redefine projective space as the quotient of the unit sphere by some equivalence relation. Use this to show that the map $e^{i\theta} \mapsto e^{2i\theta}$ from $S^{1} \subset {\Bbb C}$ to itself can be used to identify ${\Bbb R}P^1$ with a circle.

Exercise 1.2 (*05)   If $V$ is a vector space over a field ${\Bbb F}$ define the associated projective space $P(V)$. If the dimension of $V$ equals $n$ and ${\Bbb F}$ has $p$ elements, how many elements does $P(V)$ have?

Another way to understand ${\Bbb R}P^1$ is to think of it as the real line plus a point at infinity. Indeed, if we identify every line through the origin with its slope, we have that all points in ${\Bbb R}P^1$ except the vertical line can be represented by a number. The vertical line is the point at infinity. It is important to keep in mind that this is not the only way to introduce coordinates on the real projective line: one could also intersect each line through the origin with the line $y = 1$ and measure the $x$-coordinate of the intersection. Now the point at infinity is the horizontal line through the origin.

Exercise 1.3 (05)   If a line through the origin is neither vertical nor horizontal, the constructions above give us two different ways to represent it as a number. If in the first representation this number is $y$, what is the formula for the number in the second representation?

Exercise 1.4 (00)   Show that the action of $GL(n+1,{\Bbb R})$ on ${\Bbb R}P^n$ defined by $(A,[v]) \mapsto [Av]$ is transitive. Show that a matrix $A \in GL(n+1,{\Bbb R})$ fixes all points of ${\Bbb R}P^n$ if and only if it is a multiple of the indentity.

Definition 1.2   The projective group $PGL(n,{\Bbb R}), n \geq 2$, is the quotient of $GL(n,{\Bbb R})$ by the equivalence relation: two matrices are equivalent if they are multiple of each other.

Exercise 1.5 (00)   Use exercise 1.4 to define a transitive action of $PGL(n+1,{\Bbb R})$ on ${\Bbb R}P^n$. Show that if an element of the projective group fixes all points in projective space, then it is the identity.

Exercise 1.6 (00)   Let $y$ be the slope of the line $\ell$ passing through the origin and let

$$A := \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

be an invertible matrix. Verify that the slope of the line $A(\ell)$ equals $(a + by)/(c + dy)$.

This exercise tells us that in suitable coordinates projective transformations have the form $y \mapsto (c + dy)/(a + by)$.

As explained in [2], projective geometry arose from the artists' needs to represent the three-dimensional world on a two-dimensional canvas. An artist in Flatland just has to worry about representing a two-dimensional world on a one-dimensional canvas. The figure below shows a Flatland artist copying a one-dimensional image onto a canvas. Notice how distances are distorted.

This type of correspondence between the points in two lines is called a perspective.

Exercise 1.7 (15)   Could you explain a Flatland artist the relevance of the projective line and projective transformations to his work? Start by explaining why perspectives are projective transformations.

Exercise 1.8 (15)   Characterize perspectives among all projective transformations and show that a composition of perspectives is not necessarily a perspective.