I just made this simple Desmos model of a flower formed by rotating and translating multiple cardioids. Here is an example of what a flower with 6 petals looks like:
A "cardioid flower" with 6 petals
Here is the Desmos model:
I am currently trying to figure out how to find expressions for:
- The area of the flower formed
- The perimeter of the flower formed
However, I have no experience in doing anything like this and I am struggling to figure out where to start/what I need to learn even after spending a few days looking around online. Based on what I currently know, I am under the impression that this has something to do with integrals, so I've been trying to teach myself basic calculus. Could I please have guidance? I would really appreciate it!
Edit: here is the equation for the "base" cardioid. Copies of this cardioid are translated and rotated around the origin to form the flower. $$ \left(20\left(1-\sin\left(t\right)\ \right)\cos t,\ 20\left(1-\sin\left(t\right)\right)\sin t+10\right), 0\le t\le2\pi $$
For a flower with 6 petals, the original cusp point of the cardioid at (0, 10) is rotated by 60 degrees, and so is the cardioid itself. For example, here is the equation for the directly adjacent cardioid: $$ \left(20\left(1-\sin\left(t+\frac{\pi}{6}\right)\right)\cos t+5\sqrt{3}\ ,\ 20\left(1-\sin\left(t+\frac{\pi}{6}\right)\right)\sin t+10\right) $$
Here is the result:
This is repeated 6 times for a total of 6 petals.