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The paradox is given in the chap. 1 of the book titled : Mathematical Fallacies and Paradoxes, by Bryan Bunch; as given here.

The book explanation has no mathematical formulation, say if states the path in terms of cycloid (also shown in Fig. 1.5, on page #7); then there is no formula for the same.

Also, using that the nearly flattened ends of the curve for inner circle's point B could have been given a mathematical explanation.

Further, there should be some mathematical explanation & quantification for the reverse loop attained by the fastest point on the rim, as shown in Fig. 1.7 on page #9, as also shown below.

enter image description here

Also, I am not clear if the reverse path is restricted to only the fastest point, or decreases quantitatively with each inner point till the center.

jiten
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The figure for which you're seeking a mathematical formula is called a prolate cycloid. The Wolfram mathworld article just linked to gives the parametric equations of this curve for a rolling circle of radius $\ a\ $, with the point $\ A\ $ at a distance $\ b\ $ from the centre: $$ x = a\phi - b \sin\phi\\ y = a - b\cos\phi\ . $$ The parameter $\ \phi\ $, here, is the angle (in radians) through which the radius from the centre of the circle to the point $\ A\ $ has rotated, since that point was at its greatest negative extent in the $\ y\ $ direction. If you differentiate $\ x\ $ with respect to $\ \phi\ $: $$ \frac{dx}{d\phi} = a - b \cos\phi\ , $$ you can see that $\ \frac{dx}{d\phi} < 0\ $, whenever $\ cos\phi > \frac{a}{b}\ $. If $\ \phi_0\in \left(0,\frac{\pi}{2}\right)\ $ is the unique angle in the first quadrant satisfying $\ \cos\phi_0 = \frac{b}{a}\ $, the point $\ A\ $ will be moving backwards whenever $\ 2n\pi-\phi_0 < \phi< 2n\pi +\phi_0\ $ for some integer $\ n\ $.

lonza leggiera
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There are two motions involved.

One is the horizontal movement, another one is the clockwise rotation.

Let the initial position of the center of rotation be $(0, r)$, at time $t$, its location is $(vt, r)$. This point is not rotated.

Let $\gamma$ be a real number that will indicate the distance from the center for points that are initially at the same vertical position as the axis and let $\omega$ be the angular velocity. The locus can be written down as

$$(vt-\gamma r \sin (\omega t-\theta), r - \gamma r \cos (\omega t-\theta))$$

If we differentiate the first coordinate, we get

$$\frac{dx}{dt}=v-\gamma r \omega\cos (\omega t-\theta)=\omega \left(\frac{v}{\omega}-\gamma r \cos (\omega t-\theta)\right)=\omega y$$

Hence $\frac{dx}{dt}<0$ if and only if $y<0$.

That is any point that can go below the track will experience a negative velocity in its trajectory.

Siong Thye Goh
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  • Please help with my post concerning how the ratio $\frac{f'}{f} = \triangle x$ in Newton-raphson method at : https://math.stackexchange.com/q/3289605/424260. I seek how the ratio becomes equal to $\triangle x$, something unable to get from the answers given. – jiten Jul 11 '19 at 06:26
  • i have to handle multiple meetings until evening tonight. maybe tonight. – Siong Thye Goh Jul 11 '19 at 06:33
  • sorry, should have stated $\frac{f}{f'}= \triangle x$ in my first comment. – jiten Jul 11 '19 at 06:55
  • Doubt regarding applicability of Newton-Raphson (N-R) method (& other methods) only if eqn. of curve is known. Say, N-R method needs that curve & its desired properties (of being continuous everywhere, & derivative nowhere equal zero) be known. Else, no progress can be made. Say, for N-R method, the eqn.: $,f_{n+1}=f(x_n)+f'(x_n)(x_{n+1}-x_n)=0$ needs computn. of $f'(x_n)$, else no computation of $x_n,, n\ge 2$ is possible. Is there a method that doesn't need an elaborate polynomial eqn. I feel computing the polynomial eqn. is a tall order in itself, which can be flawed too in hypothesizing. – jiten Jul 11 '19 at 19:23
  • If you feel a chat is possible on the last comment topic, it can be done at : https://chat.stackexchange.com/rooms/95949/polynomial-composition. (I want t make a memory ref., that this chat-room was created for another contributor's question answered by you -- at : https://math.stackexchange.com/a/2873836/424260). – jiten Jul 11 '19 at 19:36
  • Is N-R method applicable for non-polynomial functions. It seems that if the two props. are satisfied by the fn. at hand, then yes. Say, should satisfy for expo. fn., as satisfies both properties over the entire real line. But what about the logarithmic fn. I mean what is the guarantee of not getting negative value of $x_{n+1}$ during computation, i.e. the region where the fn. does not satisfy the first property of being continuous everywhere. – jiten Jul 12 '19 at 07:22
  • In numerical optimization, if certain form is hard to compute, we think of substitutes. the criteria of polynomial is not needed. there are sufficient conditions for N-R to work. – Siong Thye Goh Jul 12 '19 at 08:00
  • I hope that you mean the two conditions (of being continuous everywhere, & derivative nowhere equal zero) as sufficient for N-R to work. But, I wanted the applicability of method to logarithmic fn., as can yield as $x_{n+1}$ a -ve value. You stated in that context of substitutes- I can think of only trigonometrical substitution - though do not know restrictions imposed by them still. But, seems a different sort of substitution would work for logarithmic fns., say modulus value to transform -ve domain value to +ve one. But it will effectively restart (by providing new initial value) the comptn. – jiten Jul 12 '19 at 08:36
  • avoid long comment threads. I am at chat. – Siong Thye Goh Jul 12 '19 at 08:39
  • sorry, had some work; so left the system for that. I am at chat now & will ensure am at it till 4-5 hours at least, so as to not miss you. I hope the chat room is at: https://chat.stackexchange.com/rooms/95949/polynomial-composition. – jiten Jul 12 '19 at 09:56
  • I am sorry for not being there in the chat earlier. – jiten Jul 12 '19 at 13:09
  • Please be at new chatrom https://chat.stackexchange.com/rooms/96713/newton-raphson-after-95949-polynomial-composition. Sorry, for a big gap, but had exigencies. I hope that the new chat would help resolve my lack of understanding. – jiten Jul 28 '19 at 02:51
  • Please help with my post concerning irrational conjugate roots at : https://math.stackexchange.com/q/3306564/424260. – jiten Jul 28 '19 at 15:03
  • it has been a very busy day. my comment so far is the same as the other commenters. – Siong Thye Goh Jul 29 '19 at 08:33
  • I request some sort of answer for my last post. No answer, just a comment; with no further one to remove doubts to that. This way, very difficult to feel addtl. inputs provided. Such inputs can only get by more details. The blankness in terms of getting a good response (for this question on mse, as stated in my last comment) hurts all who are not learned ones. Better one not ask this way, i.e never ask. – jiten Aug 01 '19 at 02:44
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    Another busy day for me. I glance through the post, i m not the right person to consult. Also, if a post consists of multiple questions, the first moment i can't answer any part of the questions, I stop reading the whole post and ignore the whole post. Neither should I read any comments actually. – Siong Thye Goh Aug 01 '19 at 03:18
  • I have added a comment to my last post concerned; to elucidate my need in a more clear way. Also, request you to help with my post of today at: https://math.stackexchange.com/q/3310200/424260. – jiten Aug 01 '19 at 10:03
  • Please help with the two questions (posts) stated in my last comment & earlier. The second post (of today) seems too simple, yet is a riddle & is just based on book's proof based on division algo. – jiten Aug 01 '19 at 14:46
  • Please help me. – jiten Aug 01 '19 at 16:08
  • I would be highly thankful if you could please comment (or, if possible provide Answer) on my post at: https://math.stackexchange.com/q/2523189/424260 . It has got no answer still, and is relevant to me for a similar problem. – jiten Feb 23 '20 at 08:09