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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?

Asha R
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    Are you looking for a single petal or a flower-like set? You may want to specify the angle between tangents at the "point" of the petal and the length/max width ratio, which would allow us to "target" the equation. – Oscar Lanzi Feb 05 '23 at 20:36
  • I'm looking for a single petal, but all I can find is the flower-like set you mentioned. What I'm trying to find is a general equation where the parameters you mentioned can be adjusted. – Asha R Feb 05 '23 at 20:52

2 Answers2

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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).

Giuseppe Negro
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    Well, not quite. The integer $n$ sometimes returns $2n$ petals – FShrike Feb 05 '23 at 20:31
  • @FShrike: Right. If $n$ is odd, it returns $n$ petals. But if $n$ is even it returns $2n$ petals. Prior to your comment I had casually checked casually with odd $n$ only, and I reached my wrong conclusion. Thanks – Giuseppe Negro Feb 05 '23 at 20:48
  • Thanks, but that gives a set of petals, what I'm looking for is a single petal. – Asha R Feb 05 '23 at 20:54
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    Oh, I see, you mean something shaped like the wing of a plane? Please have a look at the very last Java applet of https://complex-analysis.com/content/joukowsky_airfoil.html, play with the parameters a bit and see if you like those shapes. In that case we can try and derive an equation for those – Giuseppe Negro Feb 05 '23 at 21:07
  • Those were fun to play with, but still not quite the shape I'm looking for. Google beauty blender ellipse and you should be able to see the shape of that thing, kind of like an egg but with one pointy end. Is it possible that there's no equation for it at all? – Asha R Feb 05 '23 at 22:07
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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.

Oscar Lanzi
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