# How do these equations get derived?

Please look at this webpage for reference: http://mathworld.wolfram.com/Hypocycloid.html

Go to line 7 to 15 on that webpage. Line 7 and 8 show 2 parametric equations. These have been rewritten as in line 12 and 13. I understand how this has been done. But I want to know how these parametric equations (line 12 and 13) have been used to get the arc length and the area equations (line 14 and 15 respectively).

I have tried using the arc length formula:

I rearranged to get the radius in terms of S and theta. Then I substitute that into the area equation:

But that gives me the area in terms of theta. However, on the webpage, there is no angle or theta.

How should I do this? If there is something I am mission or these is another, better way to do this, please tell me.

Sorry for my incompetence and thanks in advance.

EDIT

By "LINE" I mean the numbers that are on the right side of the page in brackets

• May refer to my answer here – Ng Chung Tak Aug 1 '16 at 13:06
• @NgChungTak I don't think that answers my question. I already have the parametric equations of the hypocycloid. They are on line 7 and 8. I want to know how the equation in line 14 and 15 came to be. – Lakshya Goyal Aug 1 '16 at 13:18
• Line $7$ reads $$\theta=\frac{a-b}{b} \phi$$ which is equation $(2).$ Please specify which equations you don't understand instead of which lines, THAT'S HELP! – Ng Chung Tak Aug 1 '16 at 13:52
• @NgChungTak Look at my edit – Lakshya Goyal Aug 1 '16 at 14:02
• @NgChungTak Sorry for not making it clear before. – Lakshya Goyal Aug 2 '16 at 2:25

Note that hypocycloid is simply connected when $\displaystyle \frac{a}{b}=n=3,4,5, \ldots$

Now \begin{align*} x &= \frac{a}{n}[(n-1)\cos t+\cos (n-1)t] \\ y &= \frac{a}{n}[(n-1)\sin t-\sin (n-1)t] \\ x' &= \frac{a(n-1)}{n} [-\sin t-\sin (n-1)t] \\ y' &= \frac{a(n-1)}{n} [\cos t-\cos (n-1)t] \\ x'^2+y'^2 &= \frac{a^2(n-1)^2}{n^2} (2-2\cos nt)] \\ &= \frac{4a^2(n-1)^2}{n^2} \sin^2 \frac{nt}{2} \\ ds &= \frac{2a(n-1)}{n} \left| \sin \frac{nt}{2} \right| dt \\ P &= n\int_{0}^{2\pi/n} \frac{2a(n-1)}{n} \sin \frac{nt}{2} \, dt \\ &= 2a(n-1) \left[ -\frac{2}{n} \cos \frac{nt}{2} \right]_{0}^{2\pi/n} \\ &= \frac{8a(n-1)}{n} \\ A &= \oint_{C} x\, dy \\ &= \frac{a^2(n-1)}{n^2} \times \pi [(n-1)-1] \quad \quad \text{(see the note below)} \\ &= \frac{\pi a^2(n-1)(n-2)}{n^2} \end{align*}

Note that $$x\, dy = \frac{a^2(n-1)}{n^2} [(n-1)\cos^2 t+(2-n)\cos t \cos (n-1)t-\cos^2 (n-1)t] \, dt$$

By $$\int_{0}^{2\pi} \cos nt \, \cos mt \, dt = \pi \delta_{n,m}$$ only the square terms in $x\, dy$ survive after integration.

For the area of hypocycloid with other values of $n$, see another post here.

• Interesting answer. Besides, I had a look at your answer dated July 15th. May I ask you how you do you realize your animations ? – Jean Marie Aug 1 '16 at 15:46
• @JeanMarie, The animation was made by Geogebra which is a freeware and free to download. Thanks to your visit anyways. – Ng Chung Tak Aug 1 '16 at 15:53
• Thank you very much. I didn't know that Geogebra permit animations... – Jean Marie Aug 1 '16 at 15:57
• @NgChungTak It's a very good answer but can you explain what is "ds" and "P". Also, that integral sign with the circle and the letter C (for the area), what is that? Sorry, I don't know what that symbol means or what its called. – Lakshya Goyal Aug 2 '16 at 2:24
• @NgChungTak Also, why did you square and then add the x and y parametric equations? What is the reason for doing this? I looked online and some of them said do this: (Go to think link) http://latex.codecogs.com/gif.latex?A=\int_{0}^{2\pi}Y(\theta)(\frac{d}{d\theta}X(\theta))d\theta Does this work? – Lakshya Goyal Aug 2 '16 at 2:44