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I was reminded of this by our recent discussion of the old chestnut about possible shapes for utility hole covers. Perhaps this question is less familiar.

A sword can be made in any shape at all, but if you want to be able to put it into a scabbard, only certain shapes will do. For simplicity, let us neglect the width of the sword, and take it to be the image of a continuous mapping $[0,1]\to {\Bbb R}^3$. Clearly, a sword in the shape of a circular arc will fit into a similar scabbard, as long as the arc is less than about half of a full circle:

shamshir

(For non-mathematical reasons, such swords are rarely more than about $\frac1{36}$ of circle.)

If we let the curvature go to zero, we get the special case of a straight segment, which can be of any length:

rapier

It seems to me that if one had a sword which was a helical segment, it would go into a matching scabbard. The circular and straight swords are special cases of this, where the helical pitch is zero. (A cursory search for helical swords or scabbards turned up nothing—unsurprisingly, since a sword which has to be worked into the enemy like a corkscrew is a really stupid idea.)

Are there any other shapes of swords which will go into scabbards?

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I suppose that since the scabbard is wider at the opening and almost pointed at the other end there's some loose space to fit a non-exactly costant curvature blade. Anyway, thanks for teaching me a new English word today (scabbard) :) –  Andrea Mori Jul 30 '12 at 8:15
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1 Answer

up vote 12 down vote accepted

Clearly the curvature and torsion of the sword must be constant. A curve of nonzero constant curvature and torsion is a helix.

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Are you sure? ${}{}$ –  MJD Jul 30 '12 at 5:33
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Since the curvature and torsion are the two local invariants of a curve, each point of the sword that needs to "fit" a given point of the scabbard must match these invariants. For as much of the sword as must enter the scabbard (and as must of the scabbard as must be entered), these invariants must be constant. This would be true even with a short initial "holster" where the rest of the sword hung free. –  mjqxxxx Jul 30 '12 at 5:39
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