# 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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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: $$x= t-\sin t\tag{1} \label{eq:x}$$ y= 1- \cos t\tag{2}\label{eq:y}. \end{...
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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 ...
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### 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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### 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 ($2\pi r$), ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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’...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 $\mathbb{R}^2$?
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### 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 $\omega,g$ ...
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### 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 ...
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### 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 ...
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