Correspondence between Euclidean and non-Euclidean geometries

Question

I would like to ask correspondence between Euclidean and non-Euclidean geometries.

In the science and hypothesis, Poincare says that non-Euclidean geometry can be translated into Euclidean geometry with the following correspondence.

ref)https://mathshistory.st-andrews.ac.uk/Extras/Poincare_non-Euclidean/

1. Space -> The portion of space situated above the fundamental plane.

2. Plane -> Sphere cutting orthogonally the fundamental plane.

3. Line -> Circle cutting orthogonally the fundamental plane.

4. Sphere -> Sphere.

5. Circle -> Circle.

6. Angle -> Angle.

7. Distance between two points -> Logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cutting it orthogonally. Etc. Etc.

Does he talk about Poincare's disk here? I cannot understand the meaning "Sphere cutting orthogonally the fundamental plane". If possible, can you please illustrate this?

In addition, he says an theorem of Lobatschewsky's geometry: "The sum of the angles of a triangle is less than two right angles," can be translated thus: "If a curvilinear triangle has for its sides arcs of circles which if produced would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles."

I would appreciate if you explain the above meaning more easily.

Thank you for your help.

Answer 1

Does he talk about Poincare's disk here?

No, with all those references to a “fundamental plane” this has to be the half-space model.

I cannot understand the meaning "Sphere cutting orthogonally the fundamental plane".

The wording is a bit hard to understand as it refers to three dimensions. Go for 2d only to get started. You may want to read the Wikipedia article on that, and its illustrations.

In 2d the “fundamental plane” is simply the x axis (or the real axis if you use complex numbers to represent points). The “sphere” is just a circle and it cuts the x axis at a right angle if and only if the center of the circle is on that axis. (You should include vertical lines as special cases of such orthogonal circles, else your are missing some important corner cases.)

I would appreciate if you explain the above meaning more easily.

If you draw a hyperbolic triangle then in general you get circular arcs instead of straight line segments as the edges. According to the translation, these arcs are parts of circles where the whole circle cuts the axis orthogonally. And now you measure the angles between these circular arcs (which is the same as the angle between the tangents at the point of intersection). That's all.