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Ive been trying to derive the equation of a hypocycloid. I am confused with one thing, in the hypocycloid is there a define direction of rotation and revolution of the smaller circle? (by direction I mean anticlockwise and clockwise). Because this seems to affect the answer derived.

Thank you for your help!!

J. M. ain't a mathematician
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paul
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  • Welcome to Math.SE. Please note that this is a question and answer site, not a chat/discussion site. Please don't re-ask the same question: Hi, I have been trying to understand the derivation of a hypocycloid's parametric equation, but am stuck with one part. Instead, be patient, and respond to requests for clarification in the comments. :) – Andrew D. Hwang Jul 15 '16 at 10:35
  • I am sorry if I came as impatient. But I would like to point out that although they are the same topic, if you would read them closely enough, they aren't the same questions... I initially had put these questions as part of my first question, but as someone pointed out, they seemed "vague". So I decided to make 2 questions out of that. My sincere apologies for making this appear as a "chat/discussion" site. Thanks for your guidance. – paul Jul 15 '16 at 11:55
  • Ah, I did mis-understand your intent in this question. Does it help to note that a single curve can be parametrized in more than one way, particularly, can be parametrized in either or two directions? That is, if the small circle travels clockwise inside the larger circle, it rotates counterclockwise about its center; the resulting formulas for the hypocycloid differ, but the set of points traced out is the same. – Andrew D. Hwang Jul 15 '16 at 21:35
  • Oh okay, I get it now. Thank you so much :) – paul Jul 16 '16 at 07:42

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Usual convention: \begin{align*} w &= a e^{i\theta} \\ c &= (a-b) e^{i\theta} \\ \frac{z-c}{w-c} &= e^{-i\phi} \\ a\theta &= b\phi \\ z &= (a-b)e^{i\theta}+b e^{-i\left( \frac{a-b}{b}\right) \theta} \end{align*}

enter image description here

$w$: point of contact

$c$: centre of blue circle

$z$: locus of the initial point of contact, i.e. the hypocycloid

$\theta$: angle swept by the centre $c$ or the contact point $w$

$\phi$: angle rotated by the blue circle relative to $w$

enter image description here

Ng Chung Tak
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  • Hi, thanks a lot for responding :) If it would be possible for you, could you please clear some of my doubts regarding your explanation. I am a bit confused with the points - w, z, theeta and phi. Would it be possible for you to label them on the gif? That would be extremely helpful. And also, does the equation you provided have something to with physics? Thank you very much. – paul Jul 15 '16 at 12:12
  • See the updated answer. – Ng Chung Tak Jul 15 '16 at 13:12
  • Thank you so much! It really helped :) – paul Jul 15 '16 at 14:43