# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...