Summary: BELTRAMI'S MODELS OF NON-EUCLIDEAN GEOMETRY
Abstract. In two articles published in 1868 and 1869, Eugenio Beltrami pro-
vided three models in Euclidean plane (or space) for non-Euclidean geometry.
Our main aim here is giving an extensive account of the two articles' content.
We will also try to understand how the way Beltrami, especially in the first
article, develops his theory depends on a changing attitude with regards to
the definition of surface. In the end, an example from contemporary mathe-
matics shows how the boundary at infinity of the non-Euclidean plane, which
Beltrami made intuitively and mathematically accessible in his models, made
non-Euclidean geometry a natural tool in the study of functions defined on the
real line (or on the circle).
1. Introduction 1
2. Non-Euclidean geometry before Beltrami 4
3. The models of Beltrami 6
3.1. The "projective" model 7
3.2. The "conformal" models 12
3.3. What was Beltrami's interpretation of his own work? 18
4. From the boundary to the interior: an example from signal processing 21