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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.

Here is a detail of what I am graphically expecting. In red is the trajectory of the center of the ball and in black is the relief I need to find

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
Anne Aunyme
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  • The relief has probably some curvatures. if the radius of roller is less than the radius of curve on the relief the required trajectory is similar to relief, all you have to do is to substitute $R$ by $R-r$, where R is a parameter of trajectory and $r$ is the radius of roller. – sirous Mar 20 '20 at 15:11
  • @sirious Are you sure about that? It looks like an approximation to me. – Anne Aunyme Mar 20 '20 at 15:16
  • In polar coordinate we just have $R, \alpha$ as parameters. R varies with $\alpha$. So if you convert your relations to polar it will be much easier.When r=0, we get equation for trajectory the center of roller moves on. If you google cam and follower which is a mechanical mechanism you will get good idea, – sirous Mar 20 '20 at 17:55
  • @sirous Here the polar coordinates are r and theta. Both varies with t. the radius of the ball (let's call it r_b to avoid confusions) is not zero and of course the problem would be much easier if it was.But it isn't. – Anne Aunyme Mar 21 '20 at 11:54

1 Answers1

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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.

G Cab
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  • @AnneAunyme I checked your last edit, and my answer is exactly responding to that: you have a roulette /trochoid and you are looking for the corresponding base : please have a look to the references I indicated and then let me know if and which is the mis-matching with what you need. – G Cab Mar 21 '20 at 16:57
  • It seems like you are talking about my issue, and that actually for this specific case finding the base given the roulette is the same as finding the roulette given the base, but your links are very complex and are about the general problem where I only need to have a solution for a specific case. – Anne Aunyme Mar 25 '20 at 23:12
  • @AnneAunyme: so you are just looking for a curve at a constant distance from the given one. Then just determine the unit normal vector $\bf n$ to your curve at each point and then add $\pm \rho \bf n$ to the ${\bf r}=(x,y)$ describing your curve. – G Cab Mar 26 '20 at 00:23
  • nice! this is what i was looking for – Anne Aunyme Mar 27 '20 at 00:51