Cross-ratio definitions.

cross-ratio projective-geometry richter-gerbert

I am reading Richter-Gebert’s book (1, Richter-Gebert, 2011) and I keep on going back to the cross ratio formula. I’ll sketch some notes here for my own reference. Book reference: see pages 73 and 77.

The cross ratio is projectively invariant.

The cross ratio is a magnitude that is computed from an ordered quadruple of points on a line.

  • Input: ordered quadruple of points.
  • Output: scalar magnitude.

Now, some notational items. Richter-Gebert’s book uses a bracket notation for determinants. Another notational item: \(\mathbb{K}\mathbb{P}^1\) for field \(\mathbb{K}\) is the \(\mathbb{K}\) Projective 1-dimensional space (page 52 from the book). Members of \(\mathbb{K}\mathbb{P}^1\) are two-dimensional, non-zero vectors of \(\mathbb{K}\). The standard embedding in the real projective one-dimensional space \(\mathbb{R}\mathbb{P}^1\) is \(\begin{bmatrix} a\\ 1 \end{bmatrix}\).

Defining the bracket notation for \(\mathbb{K}\mathbb{P}^1\):

\[[a, b] = \det\begin{pmatrix} a_{x} & b_{x}\\ a_{y} & b_{y} \end{pmatrix} \label{eq:detkp1}\]

And for \(\mathbb{K}\mathbb{P}^2\),

\[[a, b, c] = \det\begin{pmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{pmatrix} \label{eq:detkp2}\]

The cross ratio of four collinear, non-zero points in \(\mathbb{K}\mathbb{P}^1\) is then

\[(a, b; c, d) = \frac{[a, c][b, d]}{[a, d][b, c]}. \label{eq:crkp1}\]

[Lemma 4.6 from the book, two-dimensional projective space] : The cross ratio of four collinear, non-zero points \(a, b, c, d\) in \(\mathbb{K}\mathbb{P}^2\) with point \(o\) not on the line. Then the cross ratio is

\[(a, b; c, d) = \frac{[o, a, c][o, b, d]}{[o, a, d][o, b, c]}. \label{eq:crkp2}\]

Cited references

  1. [1] Jürgen Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, 1st ed. Springer Publishing Company, Incorporated, 2011.

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