The Plot of a Leaf

Motivation

Recently, when I was doing some searches for some syntax in the help pages of my Computer Algebra System (CAS), accidentally, I found this parametric curve in polar coordinates

\begin{align} r &= \left(\frac{100}{100+\left(\theta-\frac{\pi}{2}\right)^8}\right) \cdot \left(2-\sin(7\theta)-\frac{1}{2}\cos(30\theta)\right) \\ \theta &= \theta \end{align} \qquad \qquad -\frac{\pi}{2} \le \theta \le \frac{3\pi}{2}

which displays the logo of my CAS. I guess that you can say what is my CAS now! :D

It is a curve like a leaf as the following picture shows. As I am a little skeptical, I didn't think that it came out of no where! :)

Question

I was just asking my self that how they came up with this equation? Do you have any ideas?

It might help to take a look at some Cartesian plots. Here is $y=2-\sin(7t)-\frac{1}{2}\cos(30t)$. You can see this creates smaller waves superimposed on larger waves by adding a periodic function of small amplitude and small period to a periodic function of larger amplitude and larger period.

Here is $y=\frac{100}{100+(t-\frac{\pi}{2})^8}$. What this is doing is creating a function that's symmetric about $\frac{\pi}{2}$, close to $1$ for most $t$-values in the middle of the interval $(-\frac{\pi}{2},\frac{3\pi}{2})$, but tapers off near $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$. If we plot this as a polar function, you see roughly an outline of where the leaf will go. When this is multiplied by the other factor plotted above, the second factor adds oscillations to this outline to give it a leaf shape. The first factor has the effect of dampening the oscillations near $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$, which you can see here. This is what produces the small leafy bits near the origin.

As for how to find the exact coefficients that make this graph look like a maple leaf, that takes some playing around. Notice that the $\sin(7t)$ is the "dominant" periodic term, which I use to mean it has the larger amplitude, and it goes through $7$ oscillations on the interval $(-\frac{\pi}{2}, \frac{3\pi}{2})$. This is what produces the $7$ "leaflets" - one centered on the $y$-axis, two pointing in the second and third quadrants, two pointing below the $x$-axis, and two small ones near the origin. Altering the coefficient $\frac{1}{2}$ in front of the cosine would result in spikier or smoother leaves. Altering the coefficient $30$ would lead to more or fewer "bumps" on each leaflet. Altering the first term would change the shape of the "outline" we plotted earlier. I'll also note that the sine and cosine functions were chosen to have a local maximum at $\frac{\pi}{2}$, and they are both symmetric about $\frac{\pi}{2}$, which is what gives the maple leaf a nice symmetric shape with a point at the very top.

To summarize, I think the process of finding this equation went something like this:

• Determine the number of leaflets desired, and use a sine or cosine function that oscillates that many times over the desired range of $\theta$-values.
• Add another periodic function with a smaller period and amplitude to introduces smaller oscillations on the leaflets. Add a constant so the function is always positive.
• Find a function that is close to $1$ on the range of $\theta$-values but tapers off toward the ends so that the polar plot roughly matches the desired shape of the leaf.
• Multiply the two functions, and adjust constants to get the desired look.
• (+1) Nice observations. Thanks for the good explanation. :) – H. R. Dec 25 '15 at 9:48