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.

0
votes
1answer
31 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 ...
1
vote
1answer
25 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 ...
1
vote
0answers
45 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
19 views

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 ...
0
votes
1answer
12 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 ...
0
votes
1answer
24 views

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’...
0
votes
0answers
30 views

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)?
0
votes
0answers
22 views

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 $(0,0)$...
1
vote
1answer
158 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 ...
0
votes
1answer
50 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 ...
3
votes
1answer
45 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....
0
votes
0answers
41 views

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 ...
0
votes
0answers
78 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 ...
0
votes
1answer
138 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 $\mathbb{R}^2$?
0
votes
0answers
65 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 $\omega,g$ ...
1
vote
1answer
77 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 ...
0
votes
1answer
134 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 ...
3
votes
2answers
1k 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 ...
0
votes
1answer
394 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}}=...
2
votes
0answers
140 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 ...
0
votes
0answers
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 ...
0
votes
1answer
144 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\...
0
votes
1answer
206 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 $$x(t)=at-b\cdot\sin{t}$$ $$y(t)=a-b\cdot\cos{t}$$ ...
1
vote
0answers
924 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 ...
5
votes
1answer
68 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 ...
1
vote
1answer
870 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 ...
0
votes
0answers
148 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 ...
3
votes
1answer
48 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 ...
1
vote
1answer
58 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 \...
0
votes
0answers
27 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 ...
1
vote
1answer
294 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 ...
2
votes
2answers
250 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 ...
3
votes
0answers
204 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 ...
0
votes
1answer
54 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. ...
1
vote
3answers
716 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)...
0
votes
2answers
59 views

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

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
0
votes
2answers
413 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: ...
0
votes
0answers
134 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: $$ ...