Sunday, April 8, 2012

Ratio of chord length to arch length

Here is another "math-y" little thing from work, that reminds me of the "associated function" of Bill Pohl:
Given two points on a curve in space, we can compare the length of the straight line ("chord") between the points to the length of the curve ("arc") between the points. For a fixed chord length, the ratio of chordlength to arclength has a minimum value. Consider this minimum value as a function of the given chord length. So
F(d) = minimum { d/ archlength }
where the ratio is minimized over all pairs of points d apart. Equivalently
F( d ) = d/ max{ archlength }
calculated over all pairs of points on the curve, d-apart from each other.

For the circle,
For the square of side length 1,
F(d) is sqrt(2)/2 for d <1and jumps to 1/3 at d=1, and descends from there back to sqrt(2)/2.
A lot like the associated function of the square.

Lots of obvious math questions around the relation of this F( ) to the associated function, around the relation of this F() to the isoperimetric inequalities, around geodesics. My interest, is related to measuring the possibility of taking a short-cut, all in a "lava lamp" sort of way - meaning when the curve gets too close to itself, change it to take a short cut.