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Questions tagged [cycloid]

Use this tag for questions about the curve traced by a point on a circle as it rolls along a straight line.

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Geometric Proof for Cycloid Problems

Let a circle $C$ of radius $50$ units be placed initally at point $F$ on the ground which then be rotated by $30$ units, and shift into a new position at point $G$ on the ground. Let this new circle ...
Student's user avatar
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1 answer
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Proof of the Cycloid Parametric Equation

One of the steps of deriving the equations for the parametric curve of a cycloid is the following: Here we establish that the distance PT is equal to the distance OT, which then (alongside other ...
Agustin G.'s user avatar
2 votes
2 answers
80 views

Finding area under the cycloid without parametrizing [duplicate]

I tried to calculate the area under the cycloid without parametrizing the $x$ & $y$ coordinate in angle $\theta$. Let's say the radius of circle is $R$ and we are rolling $2\pi$ rad. If I assume ...
Mardia's user avatar
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3 votes
1 answer
113 views

What shape would the orbit of a free falling object inside a 'massive' planet be according to Newton?

I first posted this at the physics stack but was suggested to go here for real answers. Imagine a hypothetical spherical planet with a massive core but which is somehow internally traversable without ...
ajorna's user avatar
  • 31
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1 answer
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Why is the plot of $\left|x\right|^{\frac{2}{3}}+\left|y\right|^{\frac{2}{3}}=4^{\frac{2}{3}}$ an astroid.

I was exploring the implicit function $\left|x\right|^{n}+\left|y\right|^n = 1$ and noticed that when $n$ is between 0 and 1 the plot looks similar to an astroid (The resulting curve of tracing a ...
ME_MIDI's user avatar
  • 11
2 votes
1 answer
74 views

Cycloid of Ceva - going from polar to parametric curve

Ceva Cycloid polar coordinates form is: $$ r = 1 + 2\cos(2\phi) $$ I found that the relation between polar and Cartesian coordinates can be expressed: $$ x = r\cos\phi, y = r\sin\phi $$ I need to ...
luqo33's user avatar
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2 answers
97 views

Why they are subtracting $(-\sin(t/2),-\cos(t/2))$ on MIT calc III problem set.

Good day, so the question is from the MIT open courseware page and comes from here which is the "problem set 3". The problem that I don't understand is as follows: A circular disk of radius ...
Стойчо Робертов Русинов's user avatar
5 votes
0 answers
127 views

Why is $1- \cos ( x+ \sin (x+ \sin(x +\sin(x + \cdots))))$ the cycloid?

As many of you probably know, the cycloid is given by the parametric equation: \begin{equation} x= t-\sin t\tag{1} \label{eq:x} \end{equation} \begin{equation} y= 1- \cos t\tag{2}\label{eq:y}. \end{...
DerFinand's user avatar
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Hypocycloid - how to offset (not start at 0) Parametric Equations

I have been using the p5.js library to animate some hypocycloids of n=6 but all the examples I can find always have the hypocycloids starting from angle = 0 and rotating to TWO_PI. Instead of this I ...
DeafMap's user avatar
2 votes
1 answer
454 views

Alternative cartesian equation for epicycloid (quatrefoil)

In the problem: Eliminate $\theta$ from the system of equations. $$x\sin\theta-y\cos\theta=-\sin4\theta$$ $$x\cos\theta+y\sin\theta=\frac52-\frac32\cos4\theta$$ it is stated in a previous answer ...
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Resolve nested sin functions in function for cycloid

I'm looking for a function whos graph looks like a cycloid of a circle with the radius 1. Given the parametric equations for this cycloid: $$\begin{align}x& = t - \sin(t)\\y &= 1 - \cos(t)\end{...
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Is AffineTransform a reasonable way to draw "spirograph" patterns?

For context, I'm not a math person. I'm a software person trying to do something that is pushing me to the limit. I did ok with unit-circle trig in school, but that's about as far as I got. Please &...
SaganRitual's user avatar
2 votes
1 answer
61 views

Solution Verfication (and question about): Path equation of a point on a rolling circle

I was given the task to come up with an equation $p(t)$ describing the path of a point on a circle rolling along the x-Axis (with $t$ describing time). The center of the circle has a constant velocity ...
Tahunie's user avatar
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Find a function that satisfies...

I almost solve one problem, the thing is that to finish it I need to give an example of a function $f(x) \in C^1$ such that: $$ \int_{1/4}^1 (f'(x))^2 dx \leq \int_{0}^{1/4} (f'(x))^2 dx $$ and $$f(0)=...
Math_D's user avatar
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Why Doesn't the Arc Length Formula of the Cycloid have π in it?

So basically what I was thinking is if a cycloid curve is made by a rolling circle then its length should include $\pi$ somehow. I understand it's not the same length as the circle itself ($2\pi r$), ...
izzat5233's user avatar
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28 views

Where does this expression for a Brachistochrone come from?

I came across this expression in Thomas' Calculus: early transcendentals. I understood the expressions describing the Brachistochrone curve up until this point where the derivation jumps from: $\frac{...
Ahmed's user avatar
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0 answers
310 views

Express the curve of the cycloid in the form of a function $y=ϕ(x)$

I have the following problem. If we express the curve of the cycloid in the form of a function $y=ϕ(x)$, show if it is possible to eliminate parameter $t$ in order to determine the cartesian ...
David Hume's user avatar
5 votes
1 answer
174 views

Is it possible to make straight line trajectory of a certain point on the unit circle rolling on the special curve?

I'm interested in cycloid and make another problem about it. If a unit circle rolls one lap on the straight line, the point on the circle draws a cycloid trajectory. Then what if a unit circle rolls ...
NeutrinoAnt's user avatar
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2 answers
287 views

Rotation of a wheel in the positive direction and clockwise?

The Question I’m riding my bike at a constant speed of 20 miles per hour on flat land when i ride over a dot of fresh yellow paint, thereby getting a yellow spot on my front wheel, which is 2 feet in ...
Tarek Chafei's user avatar
0 votes
1 answer
137 views

How to derive the cartesian equation of cycloid expressed for y? [closed]

I found an expression for x in wiki: https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates but I need it expressed for y. How can I do that?
sinusoida's user avatar
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1 answer
261 views

How to plot cycloid curve? [closed]

I have having trouble plotting the curve. It only plots half of the curve, what may be missing here? https://en.wikipedia.org/wiki/Cycloid#Equations x = r arccos (1 - y/r) - sqrt (2ry - y^2) Plot ...
charsee's user avatar
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1 answer
270 views

How can I find the formula for a cycloid with a given speed?

I know that for a cycloid with radius $R$ and time $t$, it can be defined as $x = R(t - \sin t)$ and $y = R(1 - \cos t)$. However what if it's not at unit speed and we have a speed $z$/sec such that ...
MathematicalWonders25's user avatar
3 votes
1 answer
459 views

How can I understand a hypocycloid as an ideal in a polynomial ring?

Today I was reading On teaching mathematics by V. I. Arnol’d and came across the following quote. "Rephrasing the famous words on the electron and atom, it can be said that a hypocycloid is as ...
Bumblebee's user avatar
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Cycloid variation using theta and radians

I was using matlab to plot these equations x=a*(theta+sin(theta)); y=a*(1+cos(theta)) which happen to be equation of Cycloid When I plotted it taking theta in radians enter image description here I ...
Newton Nadar's user avatar
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0 answers
29 views

Geometric transformation that sends a circle to a cycloid

Is there a one to one function from the plane to itself (geometric transformation) that sends a circle to its cycloid (i.e. obtained by letting the circle roll over the x-axis)? If so, can you suggest ...
MMM's user avatar
  • 473
0 votes
1 answer
60 views

A Doubt regarding Cycloid

If we consider a cycloid made by a wheel. Then will the cycloid intersect the wheel when the wheel touches the topmost point of the cycloid? Thus will the radius of curvature be same to that of the ...
Arnav Mahajan's user avatar
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30 views

Rearranging an equation to solve for a variable

In the equation below, I need to solve for $t$. Can anyone assist with rearranging the equation? My algebra skills have wilted over the years as I spent more time in front of engineering / CAD / GIS ...
Scott's user avatar
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1 answer
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Vector analysis| Line Integral | Cycloid path

I am given the following to be evaluated along a cycloid from (0,0) to ($\pi$,2) ->now the integral is this: $$\int_C{(6xy-y^2)}dx+(3x^2-2xy)dy$$ ->and the path being a cycloid is this: x=$\theta$-...
F.N.'s user avatar
  • 145
0 votes
1 answer
145 views

Given a segment length, how do I calculate coordinates of a cycloid?

Given the parametric equations for finding the rectangular coordinates of a cycloid, $x = r(t - \sin{t})$ $y = r(1 - \cos{t})$ where $r$ is the radius and $t$ is the angular displacement. Using ...
Scott's user avatar
  • 3
0 votes
1 answer
44 views

Curve length of cycloid and Helix $\gamma(t) = r \begin{pmatrix} r \cos t \\ r \sin t \\ ht \end{pmatrix}$

I have to calculate the curve length of the (a) $ \text{ cycloids } \gamma:[0,2\pi] \to \mathbb{R^2}$ $$\gamma(t) = r \begin{pmatrix} t-\sin t \\ 1 - \cos t \end{pmatrix}$$ (b) $\text{Helix } \...
user1234567890's user avatar
0 votes
2 answers
190 views

Mathematical explanation for rim point moving backwards.

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 ...
jiten's user avatar
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1 vote
1 answer
431 views

Proof that the tautochrone is a cycloid

In the Wikipedia article about the tautochrone curve, there is a proof of the fact that the tautochrone curve must be a cycloid. The proof starts with the following statement: One way the curve can ...
valerio's user avatar
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1 vote
1 answer
239 views

Cut off cycloid at an angle for Mountain Bike Jump / Ramp

So I'm making an MTB jump to double as a ramp for my 6-year-old sister to carry her bike up this step in our front yard. I want to totally over engineer it and decided I wanted it to have a lip angle ...
Fwappy_Potato's user avatar
1 vote
0 answers
125 views

Differential equation involving brachistochrone

I have that: $$ f(x)=e^{\Psi'(x)} $$ So I took the natural log of both sides: $$ \ln(f(x))=\Psi'(x) $$ Then I integrated both sides: $$\int \ln(f(x))dx =\Psi(x).$$ Here $f(x)$ is required to be ...
zeta space's user avatar
1 vote
1 answer
836 views

The prolate cycloid

A cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid ...
Amanuel Getachew's user avatar
0 votes
1 answer
75 views

Writing y value of Curtate Trochoid in the function of x?

The parametric equations of a trochoid are $x = Rt-d\sin(t)$ $y = R-d\cos(t)$ For $d < R$, there should be only one corresponding y value for every $x$ value. So can we express this equation as ...
MMM's user avatar
  • 3
2 votes
1 answer
254 views

Is $\vert\sin{x}\vert$ a cycloid?

Forgive this seemingly basic question; I recently found out about cycloids and cannot find any answers on the web. My guess is that it’s not, due to some part of the definition of a cycloid, but I can’...
kusa's user avatar
  • 191
1 vote
1 answer
942 views

What are the practical applications of the Astroid curve?

The astroid curve is a fascinating and famous curve — but why do we care? Several famous mathematicians and physics worked on it, like Roemer, Bernoulli, and Leibnitz, but why? Is it simply for ...
Brendan McDonnell's user avatar
0 votes
1 answer
117 views

How can I transform one equation about cycloidal cams to another via trigonometry? Does the author makes any assumptions on this?

I'm trying to make a nomogram for finding the maximum pressure angle on cycloidal cams with radial followers. See image for the nomogram. I've obtained the paper from E.C. Varnum where he first ...
Artur Avelar's user avatar
3 votes
1 answer
164 views

Shape drawn by cycloids

So, I'm not a maths wizard. My knowledge of it runs up to what you'd expect to find in your common core algebra 2 class. I'm trying to describe a shape. I've seen it somewhere, can't say when or where....
BenjaminF's user avatar
  • 141
1 vote
0 answers
292 views

An ellipse rolling inside a circle

Is there a name to the curve created when you roll an ellipse inside a circle? Like for example, if it was a circle rolling inside a circle, the name of the curve could be prolate cycloid or curtate ...
artour's user avatar
  • 11
0 votes
1 answer
345 views

What is parametric equations of a locus of a fixed point of a circle rolling along a ellipse in $\mathbb{R}^2$? [duplicate]

I have learnt about cycloids and have a related question: What is parametric equation of a locus of a fixed point of a circle rolling along an ellipse in $\mathbb{R}^2$?
Daniel K's user avatar
0 votes
0 answers
118 views

Is this curve a cycloid? How to prove or disprove this preposition?

I have this complicated function given by: $$y(x)=c-\sqrt{(g^2-4\omega^4x^2)}-g\log(\sqrt{(g^2-4\omega^4x^2)}+g$$ This looks sort of like a cycloid for different values of the constants $\omega,g$ ...
Anonymous's user avatar
  • 467
1 vote
1 answer
134 views

Area of intersection between Cycloid and square

I understand how to compute the area under the whole cycloid. But how do you compute a partial area. Let say, we want to know the area of the intersection between a cycloid generated by a circle of ...
Adrien's user avatar
  • 11
0 votes
1 answer
425 views

Is a cycloid arch the most stable of arches?

I figured I could find this information pretty readily on the internet, but AFAICT it's not there, at least not in any obvious form. So, I have a vague recollection of years ago being taught that the ...
bob.sacamento's user avatar
4 votes
3 answers
3k views

Brachistochrone - Solution of a Cycloid - Parametric Equations

I am trying to understand the math behind the Brachistochrone. I could understand all the technical intricacies of the mathematical treatment of the topic found at Wolfram-Mathworld|Brachistochrone ...
Pragyaditya Das's user avatar
0 votes
1 answer
638 views

Solving the cycloid equation

The equations are: $x=r(t-\sin{t})$ $y=r(1-\cos{t})$ Lets say that $(x,y)=(1,2)$ for a point. How can I find the radius of the cycloid? I can't solve the mathematics equation: $$\frac{1}{t-\sin{t}}=...
Electrician's user avatar
2 votes
0 answers
2k views

How to make a sharp 5-pointed astroid in parametric coordinates?

A long time ago I discovered the 4-pointed astroid. I was able to make it sharp by changing the exponent to any odd number greater than 3. It gets sharper and sharper with 5, 7, 9, etc., but I find 5 ...
DrZ214's user avatar
  • 1,411
0 votes
0 answers
26 views

Is Any Coordinate (x(t), dx(t)/dt) Cycloid?

Today I had learned cycloid which well parametarized with t as (t-sint, 1-cost) What looks peculiar with this notation is that coordinate y is differentiated form of x. Thus I had questioned myself ...
Beverlie's user avatar
  • 2,645
0 votes
1 answer
201 views

Fitting a prolate cycloid between two points (to a certain length)

I need to draw a prolate cycloid such that it fits a certain length l, and has an integer number of wavelengths. I have these equations for the prolate cycloid: $$x = h\cos(t+\phi)\cos\theta+at\sin\...
Youssef Moawad's user avatar