## MATH3050.06 Hints on the Bicycle Track.

I know a number of people looked at this and did not complete it.
I want to give a few hints. I will be handicapped by the fact that
my normal response is to draw some pictures but I am working the
web as text right now.
First you should consider applications of dragging and 'straight
lines' to decie which of the two paths will fit the rear wheel and which
will fit the front wheel. (If in doubt, just watch how the
front and rear wheels of a bus move when it turns.
Or you can get out
your bike, get the wheels muddy and ride through Vari Hall!)

Consider how the bicycle wheels sit along the tracks at any time:
they are tangent to the two curves. Now consider the frame of the
bike - it is a direct extension of the tangent vector for the back
wheel - so it is a straight line segment extending out from the
back wheel to the point just above where the front wheel is tangent
to the other curve. ** Draw this. **

With this in mind, given any position of the rear wheel, and a selected
direction of travel, you know where the front wheel is at the same time.
(Extend the tangent vector till it crosses the other curve.) If you move this
rear wheel / direction picture along one curve you see a series of measurements for the ** length** of the crossbar on the bike.
Is it constant? If it is not, you are looking at an impossible path for the bike.

Can you now find a direction (and a choice of back wheel / front wheel
path ) which works?

Notice that I do not need to examine the depth of the tracks
etc. It is pure geometry with the diagram you are given.

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