Numeric solutions might still be not as bad as you think. Keep for example relaxation in mind, in case you are not satisfied with what analytic solutions you find.
Something like "bending" (with no particular mathematical notion such as it should be exactly a Bezier curve) can be implemented in a very straightforward way with a particle/spring system and 4-5 steps of relaxation.
Connect a number of points with fixed length constraints (there you have your constant arc length, each segment retains its length, thus the sum of all segments does as well). Then let the user drag the last point and run a few steps of relaxation. The other particles will follow by their constraints.
Now obviously, that alone will not cause the line to "bend", how to get that to happen? Apply a constant force orthogonal to the straight line's direction in the opposite direction the user is pulling on all particles (except the one the user is pulling). Now the position of the last particle is fixed by the user's pull, and it cannot go away any further as the sum of length constraints will allow it, and each particle in between will try to move towards the straight line, limited by its length constraint.
What you'll get is something that looks like a piece of rope that you lift on one end. A curve. And the nice thing is, you don't need to do any more calculations.