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.

39 questions
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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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Distance between two points at same angle in trochoid curve

Anyone please help me to find out the distance in following case. Refer to the attached image. Consider an arbitrary point P on the circumference of a circle of radius r (mm). The point makes an ...
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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 $\|\sin{x}\|$ 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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arc length of a curtate cycloid

is there an equation that expresses the arc length of a curtate cycloid (radius B) as a fraction of the arc length of a regular cycloid (radius A)?
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Homothetic Transformation (Cycloid)

I could not proof the statement. For each point $(x,y)$, $x$ is not equal $0$, we can choose $r$ uniquely so that this point will lie on the first arch of the corresponding cycloid starting at $(0,0)$...
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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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Cycloid varying speed

I was wondering, given that the cycloid minimises the brachistocrone problem, whether or not it would be possible to introduce a variation in the speed of the brachistocrone and parameterise it. Of ...
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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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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 ...
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How to change the parametric equations of the curtate cycloid to set the initial point

I know that the parametric equations of the curtate cycloid of radius b and fixed point at the distance $a<b$ from the center of the circle are $$x(t)=at-b\cdot\sin{t}$$ $$y(t)=a-b\cdot\cos{t}$$ ...
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Cycloid (Maths HL IA)

I have chosen to investigate the fact that cycloid is a quicker path than the straight line for my HL Maths IA. I did my own experiment and was advised to only explain up to 'timing the fall' of the ...
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Integrating Square Root of Rational Trigonometric Equation

Problem Show that $$\int_k^\pi \sqrt{\frac{1-\cos x}{\cos k-\cos x}} \, dx = \pi$$ for all $0\leq k<\pi$. Remark I was trying to prove the isochronous property of the cycloid curve and I ...
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Is it possible to express a cycloid in polar coordinates

The parametric equations for a cycloid of radius 1 centered at the pole are $$x(t) = t - \pi - \sin t \\ y(t) = \pm (1- \cos t)$$ where the plus sign is a cycloid above the x-axis and the minus sign ...
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Area under a cycloid

If the length of the cycloid is $4$ times the diameter of a rotating circle, then the area under the arch traced out by that cycloid is how many times the area of the rotating circle? I tried using ...
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How to solve system of equations containing trigonometry (in radians)?

I am researching about the brachistochrone curve, which is the inverse of the cycloid. The equation for the cycloid is : \begin{cases} x = b(t - \sin\;t) \\ y = b(1 - \cos\;t) \end{cases} Based on ...
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Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [closed]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
In this question, it's said that the path of a cycloid can be given as this parametric equation: \begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*} and is shown here: ...
I'm just reading over some Cosmology notes and there is a little ODE solve that I am not quite understanding. I have an equation of the form: $$\ddot{R}=-\frac{GM}{R^{2}}$$ Integrating gives:  ...