At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in hyperbolic space.

If $L$ is a hyperbolic line not passing through $p$, what does its preimage in the flat space look like?


First consider the case $n = 2$. View the hyperbolic plane $H$ as the isometrically embedded upper nappe of the unit hyperboloid in the Minkowski space $\mathbf{R}^{2,1}$: $$ H = \{(x, y, z) : x^2 + y^2 - z^2 = -1, z > 0\}. $$ For simplicity, assume $p = (0, 0, 1)$. The exponential map at $p$ is easily checked to be $$ (r\cos\theta, r\sin\theta) \mapsto (\sinh r \cos\theta, \sinh r \sin\theta, \cosh r). $$ In this model, a hyperbolic line $\ell$ is the intersection of $H$ with the (Minkowski-)orthogonal complement of a spacelike vector (a real plane). Up to rotation about the $z$-axis, this plane has equation $ax - cz = 0$, with $0 < c < a$, and in polar coordinates in $T_p H$, $\ell$ has equation $$ a\sinh r \cos\theta = c\cosh r $$ or, as a polar graph, $$ r = \frac{1}{2} \log \frac{\frac{a}{c} \cos\theta + 1}{\frac{a}{c} \cos\theta - 1}. $$ (This description of $H$ follows Patrick Ryan's Euclidean and Non-Euclidean Geometry. The hyperboloid model is surprisingly amenable to calculations of this type, not that the question can't be answered in the Poincaré disk model as well.)

Hyperbolic lines in exponential coordinates

Since a (hyperbolic) line and a point lie in a (hyperbolic) plane, and since the preceding geometric assumptions can be accomplished by hyperbolic isometries, the above description is essentially general.


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