There is a magical land wherein lies a certain landmark. This landmark (called henceforth the Spire) has the curious property of distorting time about itself, according to the function [imath]\tau = C r^{-n} + 1[/imath], where C and n are constants (in practice, C ≈ 11.5 and n ≈ 0.486) and r is the distance away from the Spire (i.e. the length of a circular arc concentric with the planet upon which the Spire is located drawn from the point where the Spire touches the planet's surface to a line passing through the Earth's surface and the point in question... or the 'map distance'; not taking the z-coordinate into account), and [imath]\tau[/imath] is the coefficient of temporal distortion with respect to "normal" time - which itself obviously does not exist in this world, due to the asymptotic nature of the temporal distortion function.

In case I need to explain what is meant by 'temporal distortion',

**Spoiler:**

Now. Suppose you want to travel from point A to point B in this world. Due to the distortion field, you might not want to travel in a straight line, like you would were all else equal. If point B is directly opposite point A from the Spire, for example, then walking in a straight line would actually take forever. The heat death of the universe would occur before you reached the Spire itself (though, as the old joke goes, you'd probably get close enough for all practical purposes)!

My question is this. Assuming a constant rate of travel, how can I find the equation of the path which will deliver our wayward traveler to his destination in the shortest amount of time (with respect, obviously, to the baseline [imath]\tau = 1[/imath])?

I have discovered - barring any error on my part, and I'd appreciate a second opinion on this - that if one takes a path defined by the vector-valued function [imath]r(t) = x(t)\hat{i} + y(t)\hat{j}[/imath], the total time taken from t = a to t = b will be equal to

[math]\int_a^b \left[ C \left( \left[ x(t) \right]^2 + \left[ y(t) \right]^2 \right)^{ - \frac{n}{2} } + 1 \right ] \sqrt{ \left[ x'(t) \right]^2 + \left[ y'(t) \right]^2 } \,\mathrm{d}t[/math]

So I just need a function [imath]r(t)[/imath] that minimizes that.

I'd be grateful for any help, from full solutions to research recommendations.