# 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.
Or,

• 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.  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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