The equation of a petal shape

Question

I'm modelling a beauty blender that has a petal shape (ellipse with one pointed end), but since it is my first time doing such thing, I'm stuck. I need an equation that describe the petal shape and adjust it to fit my beauty blender, then integrating it to calculate the surface area and the volume. However, I can't find the equation online. Is there an equation like this out there for my shape?

Answer 1

FULL FLOWER. Try the equation $r=A\cos(n\theta)$ in the polar coordinates $(r, \theta)$. The integer $n$ gives you the number of petals: if $n$ is odd you get $n$ petals, whereas if $n$ is even you get $2n$ petals. The parameter $A$ controls the size of the flower.

POINTY ELLIPSE (beauty blender ellipse) A very similar shape has been discussed in this answer and in this other answer to the same question. (Incidentally, the latter uses exactly the equations of the Joukowski airfoil, the one which I proposed in comments).

Answer 2

One approach is to use the polar equation for the full flower but restrict $\theta$ so as to render only one petal.

Specifically,

$r=a\cos(b\theta);|\theta|\le\pi/(2b);a,b>0.$

Note that $b$ need not be a rational number in this rendering, so you don't need the petal to fit into an an actual flower.