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Wolfram Alpha mentions the following general formula for the area and perimeter of hypocycloids given the number of cusps, n.

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My issue is I've seen Wikipedia that references this page but mentions a slightly different formula.

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I've checked other references and they mention the same formulas that in Wikipedia. The derivation from the general formula was clearly shown in Wolfram Alpha. So why is there a contradiction?

I've tried applying the same formula for an astroid, where there are 3 cusps. I manage to get the answer.

I don't get why in deltoids the general formula does not work.

https://en.wikipedia.org/wiki/Deltoid_curve#cite_note-Weisstein-2

AndroidV11
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1 Answers1

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There's no contradiction, just a difference in notation. In the Wikipedia article, $a$ is used as the radius of the rolling circle, while in the Wolfram MathWorld article, $a$ is used as the radius of the outer circle. Using Wikipedia's convention, we have $3a$ as the radius of the outer circle, so the formula given on MathWorld for the arc length of the deltoid yields

\begin{equation*} s_{3} = \frac{8(3a)(3-1)}{3} = 16a, \end{equation*}

consistent with the Wikipedia article. For the area of the deltoid, the formula given on MathWorld yields

\begin{equation*} A_{3} = \frac{(3-1)(3-2)}{3^{2}}\pi(3a)^{2} = 2\pi a^{2}, \end{equation*}

again consistent with the Wikipedia article.

kandb
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  • Does that mean the deltoid equation is different for Wikipedia and Wolfram MathWorld? If yes, what are those equations. I'm just confused. Reading your answer makes me think that since a has different representations, the general equation for deltoid they represent are also not the same. – AndroidV11 Feb 13 '23 at 08:27
  • On the Wikipedia page, the equations of a hypocycloid are given as

    \begin{equation} x = (b-a)\cos{(t)} + a\cos{\left(\frac{b-a}{a}t\right)} \hspace{1pc} \mbox{ and }\hspace{1pc} y = (b-a)\sin{(t)} - a\sin{\left(\frac{b-a}{a}t\right)}. \end{equation}

    For a deltoid, $b = 3a$, so we have

    \begin{equation} x = 2a\cos{(t)} + a\cos{\left(2t\right)} \hspace{1pc} \mbox{ and }\hspace{1pc} y = 2a\sin{(t)} - a\sin{\left(2t\right)}. \end{equation}

    These are identical to the ones on the MathWorld article, because $b$ in the MathWorld article is identical to $a$ in the Wikipedia article.

     – kandb Feb 13 '23 at 08:37
  • @AndroidV11 there's really nothing different between the two articles. Somebody just made the decision to use $b = 3a$ in the Wikipedia article, and somebody else (I would think) made the decision to use $a = 3b$ in the MathWorld article. It is just a silly, simple, change in notation. Also, Wikipedia uses $t$ where MathWorld uses $\phi$, but again this is just a difference in notation. – kandb Feb 13 '23 at 08:44