The cross ratio

Posted on 10 May, 2015 by jesperg

The purpose of this post is to show how the cross ratio can be understood as a generalisation of the corresponding invariants in Euclidean and affine geometry and how it is closely related to the concept of a point frame. For the Euclidean line \mathbf{E}^1, a point frame is a single point x_0 and it allows us to specify a Euclidean invariant of another point x with respect to x_0:

f(x,x_0)=x-x_0.

So if we consider x and x_0 to be coordinates in a Euclidean coordinate system, then f(x,x_0) is simply the coordinate of x in the coordinate system with x_0 as its origin. This is invariant under non-reflection Euclidean coordinate changes.

Moving on to the affine line \mathbf{A}^1, we want to make a corresponding construction that is invariant under affine coordinate changes. Thus we extend the point frame into an affine point frame which consists of two points \{x_0,x_1\} and define an invariant of a point x with respect to this frame to be

\displaystyle f(x,x_0,x_1)=\frac{x-x_0}{x_1-x_0}.

This can be interpreted to be the coordinate of x in the coordinate system with x_0 as its origin and x_1 as its unit point.

For the projective line \mathbf{P}^1 we add a third point x_\infty which makes the frame into a projective frame \{x_0,x_1,x_\infty\}. The point x_\infty is to be understood as the point at infinity and allows us to define an affine coordinate system with x_0 as its origin and x_1 as its unit point. The coordinate of some point x in this coordinate system is

\displaystyle f(x,x_0,x_1,x_\infty)=\left.\frac{x-x_0}{x_1-x_0}\middle/\frac{x-x_\infty}{x_1-x_\infty}\right. .

Thus the cross ratio of four points is the coordinate value of one of the points with respect to the projective frame defined by the other three points. The projective frame gives the additional structure missing in the projective geometry needed to be able to compute an invariant coordinate value and specialises the projective line into an Euclidean line.

A projective frame with three points which we now denote by \{\mathbf{x}_1,\mathbf{x}_2, \mathbf{x}_\text{mid}\} uniquely determines an element of PGL(2). This matrix is constructed by choosing homogenous coordinates of the three points (x_1,y_1), (x_2,y_2), (x_\text{mid},y_\text{mid}) rescaled such that

\displaystyle \begin{pmatrix}x_1\\y_1\end{pmatrix}+\begin{pmatrix}x_2\\y_2\end{pmatrix}=\begin{pmatrix}x_\text{mid}\\y_\text{mid}\end{pmatrix}.

Then the corresponding element in PGL(2) is

\displaystyle \begin{pmatrix}x_1 & x_2\\y_1 & y_2\end{pmatrix}.

Whether we interpret the third point in a projective frame as a point at infinity or as a midpoint of the other two points depends on the circumstance and is a matter of what is most convenient. The two interpretations are equivalent since the midpoint and the infinity point are harmonic conjugates with respect to the two other points of the frame. If we want to emphasis the equivalence of the midpoint and the infinity point we can include both in the frame and thus define a projective frame to be four points with a harmonic cross ratio.

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