Regular polygonal wheels. Side-view of unicycle. Can the trajectory of the center of the wheels be re-created using round wheels and distorted floors?

Question

Scenario 1

So imagine a rolling wheel, call it a unicycle if you'd wish.

Now the wheel should be considered as being one of the regular polygons which are the building blocks of the platonic solids, or could even be considered generally as being any regular polygon.

Now imagine that the floor is flat. Let the wheel roll. We imagine a mathematical continuous contact between the floor and the wheel: a perfect roll.

If you look at the side of the unicycle, and the unicycle would be continuously sputtering out ink from its sides, onto a blank page behind it (we imagine no other force, such as gravity, changing the direction of the ink sputtered out ... a simple projection), one would neatly see the trajectory of the center of the wheel while it had been rolling.

Now the question is the following.

Scenario 2

After having imagined all of these, for some interesting cases of regular polygons on a flat floor. Now imagine that the wheel would be perfectly round. Can you, or can you not create the same drawings (of scenario 1) on the paper, by varying the surface of the floor?

Are there any limitations to this re-creation of the drawings, compared to the previous scenario.

Note: any illustrations / visualizations would be appreciated.

Answer 1

When a convex polygon "rolls (without slipping)" along a flat floor, it "pivots" about a single vertex at a time. Consequently, each interior point traces a continuous path made of arcs of circles.

When a round wheel rolls along a floor made of (suitable) arcs of circles, its center also traces a path made of arcs of circles.

Particularly, the path traced by the center of a regular $n$-gon when rolling without slipping along a flat floor can be expressed as the path made by the center of a circular disk rolling without slipping along a floor made of arcs of circles.

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In more detail, let $n \geq 3$ be an integer, and let $P = P_{n}$ be a regular $n$-gon whose sides have length $2$. Each side of $P$ subtends an angle $\frac{2\pi}{n}$ at the center $C$, so

• The distance from $C$ to the midpoint of a side is $\cot \frac{\pi}{n}$;

• The distance from $C$ to each vertex is $\csc \frac{\pi}{n}$.

Place $P$ (green) in a Cartesian coordinate system with its center at $(0, \cot \frac{\pi}{n})$ and with the midpoint of one side at the origin. As $P$ rolls to the right (successively more blue), the center $C$ traces an arc of circle of center $(1, 0)$ and radius $\csc \frac{\pi}{n}$, with the arc subtending an angle $2\pi/n$ and carrying $C$ to $(2, \cot \frac{\pi}{n})$.

It's easy to check that when a circular disk rolls over a floor made of arcs of circles, its center also traces a continuous path made of arcs of circles. (The exact shape of the floor depends trivially on the radius of the rolling disk.)