Knowing that distance in polar coordinates is minimized by metric $ ds^2 = (r^2 + r^{\prime 2})\,d\theta^2 $ as representing a straight line geodesic in the plane, I was curious to know about the metric:
$$ ds^2 = (r^2 - r^{\prime 2}) \, d\theta^2 \tag{0}$$
$$ s = \int \sqrt{ r^2 - r^{\prime 2}} d\theta \tag{1}$$
(Primed with respect to $\theta$).
With Euler-Lagrange equation in Calculus of variations we find solutions
$$ F = \sqrt{ r^2 - r^{\prime 2}} \tag{2}$$
$$ F - r^{\prime}\,\partial F/ \partial r^{\prime} = const \tag{3}$$
$$ \frac{r^2}{\sqrt{ r^2 - r^{\prime 2}}} = const. \tag{4} $$
$$ r^{\prime \prime } = 2 r^{\prime 2}/r - r \tag{5}$$
Please note that by a simple change of sign of above, the straight line can be obtained by integration with constants $ (p,\alpha) $ in polar form as:
$$ r^{\prime \prime } = 2 r^{\prime 2}/r + r, \quad p = r \cos(\theta- \alpha) \tag{6}$$
Now (5) integrates to :
$$ 1/r = e^\theta /2a + e^ {-\theta}/2b \tag{7}$$
$(a,b) $ are arbitrary constants. When $a=b,\, r = a\, sech \,\theta $
This is a spiral falling to the origin, the projection is exactly the same as the polar projection of the central (Beltrami) pseudosphere asymptotic line or geodesic.
So the metric is fundamentally and qualitatively a hyperbolic metric. It remains to be shown that the lines could be hyperbolcally geodesic as well in $\mathbb R^3 $ in this particular case.
An integrand of differential equation gives plot of the curve with B.C. $ r(0) =2, r^{\prime }(0) = 0 $ with plot:
The condition $ a=b$ implies hyperbolic geodesics on Beltrami pseudosphere. When $ a\ne b$ typically meridian resembles a hypo pseudosphere containing the curious metric. $ \phi =\psi$ (slope and inclination to meridian) for the former and $ \phi +\psi = \pi/2 $ for meridian shown.
I expected in a not too wild and off-tangent imagination to mirror a skewed Pythagorean relation ( hyperbolic?) $ c^2 = (a^2-b^2) $ of equation tagged (0) geometrical relation being valid and should show up somewhere, but I cannot recognize it anywhere.
Sorry for the subjective nature of query, I am posting it despite whatever vagueness that is going with it ... and I do appreciate not down-voting as stimulus to such research inquiries..