Geodesics that paint curves on the projective space bundle

Let $M$ be a geodesically complete connected riemannian manifold.

Let $p \in M$ be a point and $c: \mathbb{R} \to M$ an arbitrary geodesic that doesn't intersect p. Our aim is to find a "nice" map sending points on the curve $c$ to tangent lines at $p$.

Pick a point $q$ on the curve $c$ and connect it with $p$ via a geodesic curve (this is possible by completeness of $M$). Here is the relavent picture: Every such geodesic corresponds to a unique tangent line through $p$. Ideally we want to have a cannonical choice of such geodesic for every point of $c$. Assuming there is such a choice we could obtain a function $f: c(\mathbb{R}) \to \mathbb{P}T_pM$. where $\mathbb{P}T_pM$ is the tangent projective space or equivalently the fibre at $p$ of the $(1,dim(M))-$grassman bundle over $M$.

Now my question has two parts:

1) Given a initial point $q$ and an initial choice of geodesic connecting $p$ and $q$, is there then a cannonical choice for every other point on $c$? What i have in mind is a choice that "minimizes" in some suitable sense the variation of the geodesics with respect to variation of the initial point $q$. (I'm not so familiar with jacobi fields but i have a strong sense that this is what's missing here).

2) Under what conditions on $M$ is the map $f$ (defined above) smooth (resp. continuous) as a curve in $\mathbb{RP}^{n-1}$ where $n=dim(M)$. By which i mean the composition $f \circ c : \mathbb{R} \to \mathbb{P}T_pM \cong \mathbb{R P}^{n-1}$ is smooth (resp. continuous).

Here's the picture I'd like to have in mind in this context. It might be wrong though, so you better take it with a pint of salt. Note: The picture is far from being literal. $M$ is drawn as a surface so the fiber should be a projective line while i drew a projective plane.

I do not know what you mean by "nice", but here is a construction. Let $q=c(0)$. Then $c$ lifts to a parameterized straight line $\tilde {c}$ in the tangent space $T_qM$, such that $c(t)=\exp_q(\tilde{c}(t))$. The point $p$ lifts (noncanonically, unless $\exp_q$ is a diffeomorphism) to a point $\tilde{p}\in T_q(M)$. Now, connect $\tilde{p}$ to $\tilde{c}(t)$ by straight line segments in $T_qM$. These segments project to the required curves $a(s,t)$ in $M$ connecting $p$ to $c(t)$. The velocity vectors $$\frac{\partial }{\partial s} a(s,t)|_{s=0}$$
define the lines in $T_pM$ that you are asking for and, hence the map to $PT_pM$.

Note that the curves $a(s,t)$ above are not geodesics (in general). In order to use geodesics you have to assume that for the geodesics $\gamma(s,t)$ connecting $p$ to $c(t)$, the points $p, c(t)$ are not conjugate (otherwise you cannot make a smooth choice of $\gamma(s,t)$). For instance, if you assume that $M$ has nonpositive curvature, you can use geodesics $\gamma(s,t)$ for the construction. The clean way to do so is to lift $c$ to the tangent space $T_pM$ via the exponential map $\exp_p$ (since it is a covering map by Cartan-Hadamard theorem) and then use straight line segments to connect via the same procedure I described above.

Not sure if i understood all of your requirements well.To the extent it is clear to me, a geodesic flow field of the type you suggest is certainly possible in reply to first part of your question. The following from Wolfram Alpha demonstration needs only a minor modification to be suitable to your description.

From the link suggested you can pick one out of three geodesic regimes for returning geodesics behavior at hyperbolic points (saddle points, Gauss curvature negative) with low $\alpha$ and high initial height.

A minimum of two geodesics possible, infinitely many also by rotation.

In this option geodesics go to a minimum radius as per Clairaut's Law on surfaces of revolution and cannot proceed beyond the tangential circle of no return by virtue of invariant minimum radius in-built.They are tangential to a circle of radius $r \sin \alpha$ after boundary ring is touched before returning.

The pattern can be made ( by design) to return, or go to infinity on the other side of the surface which can be open/closed. CNC filament winding is performed on vessels of light weight design.

FilamentShell

The other two options ( non-returning and asymptotic minimum radius ) are not suitable.

The start point can be converted to line ( as a line to line map ) or its points can be picked/isolated.

I can prepare to upload another image to required dimensions if requirement of second part of your question is more clear with suitable examples.

HyperboloidGeodesics