Curve with longest arclength between two points Take 2 points in the $XY$ plane, WLOG make one point at the origin $(0,0)$ and the other, $(d, 0)$, lie on the $x$ axis. Picture drawing different curves between the two points and measuring the arclength.

What is the longest arclength you can have between those two points given you're always moving away from the first point and towards the second, that is your distance from the first is monotonically increasing?

Of course a line suffices, but isn't optimal. $\sin(x)$ would have more arclength but I'm not sure the maximum amplitude it could have before you started moving closer back to the original point at one point of the travel along the curve. So, what is the curve (not necessarily function) that would maximize this arclength where the distance from the origin is monotonically increasing and you end up at $(d, 0)$?
 A: tl; dr: Despite the constraint, there are smooth paths of arbitrarily long length.

Call the points $p_{0} = (0, 0)$ and $p_{1} = (1, 0)$, i.e., assume $d = 1$. Fix a positive integer $n$ and consider the circles of radius $i/n$ about the points $p_{0}$ and $p_{1}$ for $1 \leq i \leq n$. (The "upper half" of the case $n = 24$ is shown; $p_{0}$ and $p_{1}$ are the lower left and right corners.)

Let's first assume that moving away from the first point and towards the second means the distance to $p_{0}$ is non-decreasing and the distance to $p_{1}$ is non-increasing. Geometrically, this amounts to walking in an obvious sense along the grid shown. The zig-zag path along the bottom comprises $2n$ circular arcs joining the horizontal axis and the ellipse
$$
d(x, p_{0}) + d(x, p_{1}) = 1 + \tfrac{1}{n}.
$$
(Each upper endpoint of the arcs is, for some integer $i$ with $1 \leq i \leq n$, at distance $i/n$ from $p_{0}$ and distance $(n - i + 1)/n$ from $p_{1}$, so lies on the stated ellipse.)
Because this ellipse has minor axis $O(1/\sqrt{n})$, the zig-zag path contains (in addition to other arcs near the ends) $O(n)$ circular arcs of length $O(1/\sqrt{n})$. Consequently, its length is no smaller than $O(\sqrt{n})$, so can be made as large as we like. (Strictly we should estimate the constants of magnitude, but the diagram makes fairly clear these constants are bounded away from zero as $n \to \infty$.)
If we want the distances to be strictly monotone, and/or want the path to be smooth rather than piecewise-smooth, we can "approximately follow the grid" and round off corners while decreasing the length as little as we like.
