What is this shape $r=a+b\cos(n\theta)$ called?

512 Views Asked by At

The polar equation $r=a+b\cos(\theta)$ produces a limaçon and for different ratios of $a$ and $b$, more precisely $|\frac{a}{b}|$ it produces inner looper limaçons, cardiods, dimpled limaçons and convex limaçons.

So, now if we decide to stretch the the curve a little more $r=a+b\cos(n\theta)$, then we end up with so many graphs.

Is there a name for the general shape? for different values of $a$ and $b$ what are those called?

1

There are 1 best solutions below

1
On

They are called cyclic-harmonic curves. They are not roses but conchoids of roses.

https://www.mathcurve.com/courbes2d.gb/conchoidderosace/conchoidderosace.shtml

Depending on the ratio $a/b$, they are said curtate, cuspitate or prolate.

https://www.jstor.org/stable/1967850?seq=1#metadata_info_tab_contents