Consider a projective plane with an absolute quadric, so that it is a hyperbolic plane.
Given a curve I wonder how the tangent to a curve is defined in a plane with constant positive curvature.
I am unsure whether the tangent in a space of constant positive curvature should be defined as a straight line or somehow as a geodesic and how would I describe such a geodesic geometrically?
When possible, I understand synthetic explanations much better.
On
One usually defines tangent line $L=T_p(C)$ to a curve $C$ in a surface $S$ at a point $p$ as a 1-dimensional linear subspace of $T_p(S)$. This definition is intrinsic to the topology of the surface $S$. You can also (but this is nonstandard) define tangent to $C$ at $p$ as the unique maximal geodesic in $S$ tangent to the line $L$. This of course requires a Riemannian metric $g$. Constant curvature is irrelevant here as well as the specific realization of $(S,g)$.
Yet another model of the Hyperbolic plane that allows one to construct geodesics on the surface of the hyperboloid given two points on it. From an analytic point of view , it is not evident that the geodesic equations on the hyperboloid are satisfied, but I believe it can be shown that they are.
Again, this is not the Beltrami-Klein model suggested above, since the geodesics in this disk model are circles orthogonal to the unit circle. Yes, I know, a lot of models of the hyperbolic plane!