Chapter 5

Observations to Keep Us Spinning

I hope you had as much as we did playing around in Desmos with all of these equations! Here are some of our favorite things we noticed:

Hypotrochoids

A Perfect Pronged Shape

Setting (the radius of inner wheel) to (the radius of the track divided by some integrer), yields a pronged shape.

A Pointed Image

Likewise, setting to , where (with and being relatively prime numbers), then the resulting shape will have petals.

Limacons-ish

Setting to where is an integer, (notice how this will make ), the resulting shape is a limacon-like shape with layers.

Epitrochoids

Cardiods

Setting to forms a cardiod:

Neproid

Setting to forms a neproid:

Flowers

Similarly, setting to where is an integer forms a flower with petals:

Reversed Limacons

Setting to where is an integer yields something that looks like a reversed limacon:

Have fun exploring more!