Rotations Through Translations

Looking closely, you'll notice that while this shape rotates, it is made up of individual points that do not rotate, but rather each moves along its own line. So the shape is made up of a bunch of points; the shape rotates, but each points translates.

This picture was inspired by this link, shared by my friend at  @mathawakenings. I'm not sure who made that image first. I was intrigued by it, and several days later, attempted to make it myself.

If you are not convinced that each point only moves along a line, consider the following image, where each point is colored:

That was my attempt to imitate the original image. I realized later that mine is different than that original post in several ways, including the cool motion-blur effect on the dots. Maybe I'll figure that out later.

If you are still unconvinced that each point only moves along a line, try this last image:

It is easy to just follow the red dot and see it travels along a line.

Making the image wasn't too hard to do, though I didn't have immediate success. Apart from a few abject failures, this was the first "successful" animation I made:

It looked cool, though not exactly what I wanted. I went ahead and made a color version of it too:

Again, one can follow a single color and see that each point moves along a line. For the skeptical:

This image gives the impression that we have one circle rolling around inside a second larger circle. Picking one point on the interior circle and following its path creates a curve called a hypocycloid  For instance, with an outer circle with radius 2 and an interior circle with radius 0.3, we get:

As a kid, I had a Spirograph with which I tried making similar curves. (I found it both really cool and really frustrating for my young/little mind/hands.)

Is there a hypocycloid where each point on the interior circle just travels in a straight line? Indeed, there is: let the interior circle have half the radius of the outside circle:

Each of the two rotating shapes shown here (the circle and the original shape) are examples of rose curves. They can be created with a certain (integer) number of petals with the polar equation r=sin(nt). If n is odd, the curve has n petals; if n is even, the curve has 2n petals. (This makes it hard to get a curve with six petals.)

Here is a sneaky way of seeing that each point in the 3-petal curve above actually moves in a straight line. The curve is defined by r=sin(3t); each point on the curve corresponds to a specific value of t. We can rotate any given shape by adding to the angle (if r=f(t), then r=f(t+a) rotates the shape clockwise a radians/degrees). 

To rotate the 3-petal curve, add a to the angle: r=sin(3t+a). This also accomplishes something else: if we think of a specific t value, adding a to the angle changes how far (r) the point is from the origin. In polar terms, we are still "looking" in the "t" direction, but just at a different  length than before. Moving in the same direction, but at different distances, means one is moving along a line.

So we can move a shape by translating its points. I made rose curves with various number of petals, but I think the 3-curve hits the sweet spot for beauty and simplicity. 

Here is the challenge: can one create a shape that translates by each of its points rotating in a circle? I played with the notion for a while, but my brain began to hurt. I'd like to see an example, or a clever proof showing it can't be done.



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