Solution to The October 1, 2005 PoW

You are a bounty hunter tracking a fugitive on a bicycle. (Gas prices have hit everyone.) He may be headed south to the border or west to California. (But we know he is not going east or north.) You come upon the following two pairs of bicycle tracks through a slightly muddy part of a field. Is it possible to determine which pair of tracks you should follow?
Specifically, can you determine whether the top set is a bicycle traveling east to west or west to east, and whether the second set is a bicycle traveling north to south or south to north?

bike

The bicycle is going west to east. Note that the tangent at the rear tire hits the front tire's curve at a constant distance. If you assumed the bicycle went east to west, the tangent at the rear tire would intersect the front tire's curve at varying lengths.
Maple code: bicycle2.mws . Open URL -> http://math.asu.edu/~rich/puzzles/bicycle2.mws

And
bike

Rotate the following image 90 degrees clockwise to get the above tire tracks. (Again, only one possibility has the tangent at the rear tire hitting the front tire's curve at a constant distance.) Therefore the bicycle is headed south!
So follow that bicycle.
Maple code: bicycle.mws . Open URL -> http://math.asu.edu/~rich/puzzles/bicycle.mws

Call the point of contact of the rear tire with the ground B, and the point of contact of the front tire with the gound A. As the bicycle moves, points A and B move along their respective curves. Now picture a line segment from B to A. This line segment will always be tangent to the curve of the rear tire at B. See the above animations.