I have a parametric curve (in polar coordinates) that describes the trajectory of the center of a rolling ball. This ball (assimilated as a circle) rolls smoothly along a relief. I need to get an expression for the curve that describes the relief.
Ideally I'd like to have a general method to determinate this trajectory.
It that's not possible, here are the equations for the trajectory of the center of the ball:
r(t)=sqrt(A^2+B^2-2*A*B*cos(alpha(t)))
theta(t)=t-arcsin(B/r(t)*sin(alpha(t)))
with:
alpha(t)=Pi/12*(1+sin(t*12))+0.56
A=10.5
B=3.2
If the circle (roulante / rolling curve) rolls without slippage over a curve (base), then a point fixed on the roulante (the center in this case) will describe a trajectory which is a Roulette , better treated here, and in particular a General Trochoid.
So your problem is the reverse: given the roulette (trochoid), and the rolling circle and its center, find the base.
Now, the contact point of the rolling circle with the base is the instantaneous center of rotation. That means that it will lay on the normal to the trochoid and to the base. If the base is supposed not to intersect itself, and the radius ($\rho$) of the circle is constant, then $\rho$ shall always be less than the curvature radius of the trochoid, in the direction of the base curve, that could be internal or external.
If it is so, then it is a matter of finding the locus of the points at constant distance $\pm \rho$ from the roulette.
But note that your parametrization of the curve in $t$, considered as time, will not provide a constant tangential speed $ds / dt$. Then if you want the movement of the roulante to be also parametrized in time, that shall allow for a non-constant rotating speed.