Warning: the following post contains some intense mathematics, caution is advised.

In the previous post, I investigated a simple toy that generated an ellipse. However, how did I know it was an ellipse? I proved it, and here is how.

First, we need to model the two pivot points of the toy. They are both fixed on the x and y-axis. The toy also moves in a circular motion. The pivot on the y-axis is starts at 1 and the pivot on the x-axis is at 0. Hence, we will model the motion of the y pivot by the cosine function and the motion of the x pivot by the sine function. Taken together, this information gives us two points:

I am using t to denote how the position of the pivots change in time. Due to boundary complications, I will only model one half of a complete revolution. However, since the resulting shape is symmetrical, this is model sufficient to determine the curve the toy traces. Thus, 0 ≤ t ≤ π

The handle is attached to a rigid piece of wood. The rigid piece of wood can be modeled by a line segment. Both of the pivot points lie on this line segment. To determine the equation of the line segment, we must use the good old slope formula:

Then we can use the slope-point formula:

Alright! Now we are getting somewhere! The next step is to model the handle. For simplicity, we will model the handle as a point 1 unit away from the x-axis pivot. To determine this point, we can use the distance formula:

We can use the previous formula for the equation of the line to determine y_{2} in terms of x_{2}:

Thus:

Rearranging we find that:

Using well-known trigonometry identities, the above reduces to:

Ignoring the trivial solution, we find that x_{2} is equal to:

And solving for y_{2} yields:

Hence, the point in parametric form is:

We would like to identify what curve these parametric equations will trace. This is equivalent to solving the follow 2 equations in terms of x and y only:

By squaring both sides of both equations, we get the following:

Which is the equation of an ellipse. Hence, the handle of the toy traces a path equivalent to an ellipse. Below is an animation I constructed using the parametric equations.

https://www.desmos.com/calculator/sz3roeuszm

A copious amount of work? Maybe. But what a beautiful proof.

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