A straight line is the shortest path between two points. Discussing curved space we would better call them geodesic lines to avoid confusion. I want to discuss these geodesic lines for surfaces of a sphere, elliptic space and hyperbolic space. Further we shall see how they are defined and that there is some resemblence between these spaces.
Geodesic lines on the surface of a sphere are great circles. An example is the equator of the earth. In general a great circle is the intersection between a plane going through the center of a sphere and the surface of the sphere. Thus we get a unique great circle for any two points on the surface of the sphere because these two points together with the center of the sphere define the plane that contains the great circle. This shows that two points determine a unique geodesic line in spherical geometry and that the geodesic line is not the same as in Euclidean space. Quite in general we expect that two points give a unique geodesic line. Its shape depends on the geometry of space.
We get an elliptic space from the stereoscopic projection of the surface of a sphere to an Euclidean drawing plane. Great circles on the surface of the sphere appear as circles too in the drawing plane. Thus the geodesic line between two points is now a circle going through these points. But what is its radius? We could find it by projecting the points back to the surface of the sphere, getting the great circle going through these points and projecting the great circle back to the drawing plane. But this is unnecessarily complicated and we better find a simple equation that directly defines us the radius of the circle in the drawing plane.
Note that the center of the stereoscopic projection is the north pole of the sphere and that the projection plane goes through the south pole. The projection plane is parallel to the equatorial great circle. The image of the equator is trivially a circle of twice the radius of the sphere. Any other great circle intersects the equator at two opposite points. Thus geodesic lines in the elliptic plane too intersect projected equator at opposite points. They look like this:
The red circle is the projection if the equator and the brown dot is the projection of the center of the sphere. The blue circles are geodesic lines that have their center on the black line. Note the yellow dots. As discussed, all these geodesic lines go through this point. If R is the radius of the red circle, r is the radius of the circle representing the geodesic line and d is the distance of its center from the projection of the center of the sphere, then we get
d² + R² = r².
This equation uniquely defines the geodesic line connecting two points in elliptic space. It is essential for creating images in elliptic space and relating them to images on the surface of a sphere.
For the Poincaré disc representation of hyperbolic space we get similar results. Geodesic lines are now circles that intersect the boundary of the disc at right angles:
The red line is boundary of the Poincaré disc and the brown dot is its center. The blue circles are geodesic lines that go through the yellow dot. If r is the radius of a geodesic and d the distance between its center and the center of the disc, then
d² – R² = r²,
where R is the radius of the disc. Again, this equation determines the radius of a circle going through two points and makes that geodesic lines are unique. We need it to create kaleidoscopic images in hyperbolic space. Obviously, there is some resemblance between the stereographic projection of a sphere and the Poincaré disc. But do not go to far on that.