Updated picture arc path around circle So that the object following the arc path always starts and ends on the intersecting point between arc path and circumference of containing circle.
I simply can't figure out the maths to rotate the arc path as it moves around the circle. As shown in my new awesome picture...
Thank you
I still think a hypocycloid could draw a picture like yours. It's just that the examples shown in various web pages don't make the rolling circle small enough.
To adjust where the bounces land in the hypocycloid, just measure the arc distance between the landing points. If the landing points are separated by an angle $\theta_0$ radians on a large circle of radius $R$, then the arc distance between the points is $R\theta_0,$ and you would want to make the small circle's radius $$r = \dfrac{R\theta_0}{2\pi}.$$
A different alternative would be to simulate the physics of a ball bouncing against the inside of a cylinder in a rotating frame of reference with no other forces applied. That is, if you were sitting inside the rotating cylinder rotating along with it, and watched a bouncing ball, this is what you might see.
Given a circle of radius $R$ that you want to bounce within, one way is to choose two points on the outer circle that are farther than you want the distance between bounces to be. Using polar coordinates (in radians), suppose you want the ball to hit the outer circle at angle $\theta_1$ (so in polar coordinates it hits at the point $(R,\theta_1)$) and then again at angle $\theta_2.$ Let $A$ be the point at polar coordinates $(R,k\theta_1)$ and $B$ be the point at polar coordinates $(R,k\theta_2)$.
Convert the polar coordinates to Cartesian $x,y$ coordinates. Develop a parametric equation in $x$ and $y$ coordinates that takes you from $A$ to $B$ at a constant speed. Convert the $x$ and $y$ coordinates of your equation to polar coordinates and then divide the angle in your polar coordinates by $k$.