What sort of trace would arise if you plotted..?

Question

..the path followed by a - 'follower' - riding on a square cam against 'time'?
And on a cam the shape of an eccentric - or is it 'accentric' circle, where the wobble provides the 'change'?
Could you derive a formula for the relationship b/w the variables.. and would the formula be 'linear' type equation for ex.1 and a circle-type equation for ex.2 - at first sight?
Something is rhythmically changing in both instances,but what? ( I made up this question myself, so eagerly await your insightful responses..)

Answer 1

You don't say how fast the cam is rotating, so I'll just leave in terms of θ. You can define θ in terms of t. For instance, if it is rotating once per second, and t is in seconds, then θ = 2πt.

Let f(θ) be the distance from the center of the cam to where the follower touches the cam. Let s be the length of one side of the square cam. Define that at θ=0, the cam touches the midpoint of a side of the square.

Now, turn the cam a little bit (less than π/4) and draw a right triangle: make f(x) the hypotenuse, and one leg from the center to the edge of the cam. Then:

cos θ = (s/2) / f(x)
f(x) = s / (2 cos θ)

Likewise, between π/4 and π/2:

sin θ = (s/2) / f(θ)
f(θ) = s / (2 sin θ)

This repeats every π/2. So, we have a piecewise function:

f(θ) = { s / (2 cos θ) ....... πn/2 ≤ θ ≤ π(2n + 1)/4
.........{ s / (2 sin θ) ........ π(2n + 1)/4 < θ < π(n + 1)/2

For every integer n.