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I take a walk each morning along the sides of a square; each side is one mile. I start at one corner and walk at a constant speed.

As I start on the walk, an unfriendly wasp always starts at the center of the square and starts chasing me, always flying directly in the direction from the wasp to myself. She must be flying a bit faster than I walk, since precisely when I complete the walk (having returned to the starting corner) the wasp meets me and greets me with an unfriendly sting.

How far has the wasp flown in her chase?

By supplying the wasp with a FitBit (or perhaps by numerically integrating the equations of motion) I can tell you that the answer is roughly 4.029 miles. But I would like to have either a closed form expression for the distance travelled, or failing that, a perturbative solution that will tell me how much the distance exceeds the 4 mile perimeter of the square, to at least first order in some small quantity.

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    You can split this into several copies of the pursuit curve problem, and [this](http://mathworld.wolfram.com/PursuitCurve.html) link shows that a closed form expression does exist, possibly involving the Lambert-W function but nothing worse (equations 15 and 23, in a different coordinate system for the same problem). – 2017-02-21
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    the version I have seen is $n$ wasps beginning at the vertices of a regular $n$-gon, each flying at the next wasp ahead, all converging in the center in finite time. As I recall, the answer is extremely simple for triangles or squares. Not sure what changes when the wasp has different speed from you. As far as pets, you might consider switching to a dog. – 2017-02-21
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    About three quarters of the time, the wasp is very close to the sides of the square – 2017-02-22
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    @Will Jagy: Yes, I am familiar with the four flies chasing each other problem, in which the answer is exactly equal to the side of the square. However, in this problem the target is not adjusting its trajectory in (indirect) response to the chaser, so it is a bit deeper. – 2017-02-22

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