Polar equation for a k-leaf rose: is it possible to define an inner radius?

Question

Is it possible to define a polar equation for a k-leaf rose with an inner radius for a k-leaf rose (as in this image)? I'm familiar with the general equation for a k-leaf rose $$r = \cos(k*\theta)$$ and the corresponding Cartesian equations $$x = \cos(k*\theta) * \cos(\theta)$$ and $$y = \cos(k*\theta) * \sin(\theta)$$ However, I've been unable to use these to come up with an equation that produces a rose with a hollow center. I'm very curious to know if it's possible. Thanks in advance, any insights will be greatly appreciated!

Answer 1

You have two issues: cosine goes to zero, so the inner radius is not there, and cosine goes negative so adding a bit doesn't do what you want. If you square the cosine you can add a bit. This doesn't look like your figure, but it does have an inner radius

If you increase the denominator it looks like this

You can play with the parameters a bit