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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23 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 ...
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60 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 ...
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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 ...
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20 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 ...
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31 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 ...
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20 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 ...
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16 views

Finding the length of an arch of a cycloid by integrating by integrating velocity

I am trying to find the length of an arch of cycloid by integrating the speed at which a point on the cycloid moves. I know for a cycloid that is drawn out by a circle of radius 1 and rotates at 1 ...
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1answer
17 views

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=$\theta$-...
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34 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 ...
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Radius of curvature for a cycloid given parametric equations

I have been given $x=a(\theta - \sin\theta)$ and $y=a(1-\cos\theta)$ I need to prove the radius of curvature is = $2\sqrt2a(1-\cos\theta)^{1/2}$ $x'=a(1-\cos\theta)$ and $y'=a\sin\theta.$
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What is the maximum eccentricity of a hypocycloid gear mechanism in terms of pin count and diameter?

A hypocycloid gear is a gear mechanism that allows for high gear ratios in a compact space. I've made an interactive toy that allows the user to easily try out different parameters, based on the math ...
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30 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 } \...
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170 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 ...
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106 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 ...
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1answer
80 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 ...
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70 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 ...
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205 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 ...
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24 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 ...
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51 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’...
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1answer
417 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 ...
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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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1answer
73 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....
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135 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 ...
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237 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 $\mathbb{R}^2$?
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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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92 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 ...
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1answer
180 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 ...
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2answers
2k 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 ...
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1answer
486 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}}=...
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259 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 ...
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21 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 ...
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1answer
151 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\...
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1answer
285 views

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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1k views

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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1answer
70 views

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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1k views

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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171 views

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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1answer
53 views

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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1answer
68 views

Help needed for the equation for an outline of hypo/epicycloids

I'm writing a piece of software that will manipulate 2 servos to trace (hopefully) the outline of hypo/epicycloids using mapped pairs as coordinates where: x = $ (R-r)\cos \ \theta +z \cos \theta \...
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30 views

Simplification of Derivation

I'm a high school student and I am writing a paper on cycloids. To solve for the tangent to the curve at any point I use pythagoras. I understand that - however I came across this simplification and I ...
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1answer
368 views

Proving the Tautochrone Property

The tautochrone property (meaning equal time) is one of the dynamic properties of an inverted cycloid. This means that if one puts two objects at different positions on a inverted cycloidial shaped ...
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372 views

Another method of finding area of hypocycloids

I was finding the are the of hypocycloids. Then it struck me that apart from integration, there could be another method of finding the area of the hypocycloid with different curves. But the problem is ...
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236 views

Approximate equation for tapered cycloid offset curve without cusps

Is it possible to create parametric equations to approximate a tapered cycloid offset curve without cusps, that does not require manual adjustment of values when the primary curve parameters are ...
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1answer
59 views

I need input and help understanding how the formula for x arises in a cycloid that is parameterized with theta with the cusp at the origin

Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. ...
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3answers
776 views

Converting a Parametric equation into a Cartesian one

I was working on converting an parametric equation into a Cartesian one and i cant seem to figure this one out. I was hoping you could help with that for this equation of a cycloid, Thanks $x = cos(t)...
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89 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [closed]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
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518 views

Path of a cycloid

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: ...
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145 views

Cycloid angular parameter solution to an ODE for density fluctuations

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: $$ ...