I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic geometry.
I wanted to add material to the article related to this; however, I would like to add a reference concerning the origin of the cross-ratio in hyperbolic geometry.
Who was the first person to use the cross-ratio to parametrize shapes in hyperbolic geometry?
I've hesitated to attempt an answer to this question because I do not know about shape parameters. However, in the hope that what is really wanted is a history of the cross-ratio, here goes.
The cross-ratio, and its invariance under projection, was discovered by Pappus and rediscovered by Desargues around 1640. It appears in Proposition 129 of Book VII of Pappus' Mathematical Collection from around 300 CE. The Desargues version appears in a 1648 book called Mani`ere universelle de Mr Desargues written by Desargues' disciple Abraham Bosse.
It became standard in projective geometry when the subject flourished in the 19th century. It was used by Cayley in his "Sixth memoir on quantics" of 1859 (which was not his thesis) to introduce a metric into projective geometry. In 1871, Klein realized that this metric was in fact the metric for the hyperbolic plane in his paper "Ueber die sogenannte Nicht-Euclidische Geometrie."
Thurston (even before his lecture series in 1979) was probably the first to use cross-ratios to describe the shapes of hyperbolic tetrahedra. He was certainly the first to use these shapes to compute the hyperbolic volume of knot complements.
Yuri Burman once told me that this was introduced in Cayley's thesis. He conjectured that this must have been the shortest thesis in the history of math as all of the groundwork had been laid.
