Questions tagged [arc-length]

For questions about/on finding the arc length of a curve/parametrized curve

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27 views

Characterization of arclength as unique function on continuous curves that satisfy certain conditions (resolution of "$\pi=4$ paradox")

I was again thinking about the famous $\pi=4$ paradox, and this question in particular: How to convince a layperson that the $\pi = 4$ proof is wrong?, about why the standard sup over polygonal ...
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22 views

Is there a more clean way to derive parametric arc length?

This is one possible way to derive the formula for arc length: Assume we have a parametric curve $f(t): \mathbb{R} \rightarrow \mathbb{R}^n$ We can sample the curve at regular intervals $t_i$ ...
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7 views

Orthogonality from length constraint

I want to ask you for help with a question related with a definition from optics. There the refractive index of a medium is given by \begin{equation} n(\vec{q}, \dot{\vec{q}}) \end{equation} and \...
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44 views

Are there curves similar to Bézier curves, but with a fixed length?

Bézier curves have some nice properties, such as starting at $P_0$ and ending at $P_n$ (for an $n$-degree Bézier curve). I am looking for a class of (curvy) curves, but with the additional property ...
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39 views

Measure polynomial arc-length between given range

Is there a way to measure the arc-length of a curve created by a 4-degrees polynomial, between a given range? For example, I want to measure the length of the polynomial $-\frac{1}{200}x^3+x+1$ ...
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22 views

References for minimizers of energy in metric spaces

As the title suggests, do you know some literature, books, articles or notes, that treats the connection between geodesic in a metric space and the energy associated to the curves? In particular, I am ...
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32 views

Calculate movement along circle arc

I want to simulate the turn of an aircraft, and therefore need to calculate its position on a circle arc within a cartesian coordinate system. I need to calculate its position every 5s. What I ...
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1answer
47 views

Geometric intuition behind pseudo-distances

Could somebody please offer some intuition into how pseudo-distances work? Something like what the geometric interpretation is, and how they differ from distances, would be appreciated. Background: I ...
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57 views

Selecting a random point on a function.

Say there is a continuous function $f(x)$ which does not have any undefined points in the interval $[a,b[$. I want to randomly select a point $(x,y)$ in that interval on the line formed by that ...
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1answer
30 views

How can I find $\Delta_1$ for the first of two reversing curves?

This is a problem from the design of roadways. I know someone has the answer, but I haven't been able to work it out myself. The sketch below shows a single curve made up of two asymmetrical ...
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38 views

How can I find the arclength of this complicated function?

I want to integrate $$\int_0^a \sqrt{1+\left(\frac{d\delta(x)}{dx}\right)^2} dx$$ Where $$ \delta(x) = C_1 \sqrt{a^2-x^2} \left(C_2 \left(\frac{x}{a}\right)^3+C_3 \left(\frac{x}{a}\right)^2+C_4(\frac{...
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59 views

Is there at least a continuous curve connecting any two points in a length metric space?

These are three questions about metric spaces with the intrinsic metric on them. Let $(M,d)$ be a length metric space, i.e. its metric is the intrinsic metric. Can such a space have a point $p$ whose ...
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62 views

Is the following function twice differentiable?

Let $\alpha : I \rightarrow \mathbb{R}^3$ be a twice differentiable curve such that $\alpha ' $ and $\alpha ' ' $ are nonzero everywhere. Let $$s(t):=\int_{t_0}^{t}|\alpha ' (u)|du$$ and let $\beta := ...
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87 views

Integrating the square root of a trigonometric polynomial [duplicate]

I am trying to find a solution to the following integral. $$ \int \sqrt {17+8\cos(3t)+9\cos^2(3t)} dt$$ I have run into exactly the same problem as this previous unanswered question. This integral ...
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67 views

Find the arc length $\int_ 0 ^{4\pi} r(t)=3\cos ti+4\sin tkj+tk$

So my components are $$f(x)= 3\cos t, \ \text g(y)=4\sin t ,\ \text h(z)= t$$ $$f'(x)=-3\sin t, \ \text g'(y)=4\cos t,\ \text h'(z)=1$$ I found the derivatives of each component and plugin to the arc ...
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1answer
88 views

Length of the curve $\displaystyle 3ay^2=x(x-a)^2$

I'm working on finding out the length of the curve $$ 3ay^2=x(x-a)^2 \tag{1} $$ I ran into a small problem, but was able to end up with an answer that looks right but I'm not entirely sure about it. ...
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1answer
54 views

The Length Of A Contour

I have just begun a course in complex analysis and have been presented with this definition: $$\gamma(t):[a,b]\to\Bbb{C}\text{ be differentiable. The length of the curve }\\ \\\{\gamma(t):a\leq t\leq ...
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67 views

How can I find the arc length of some interval on this parametric object?

I want to find the arclength some interval bounded by points $P$ and $Q$ on a parametric object. This parametric object is defined by the 2D stereographic function: $f(x, y) = (tx, 1 + t(y-1))$ . If ...
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2answers
65 views

Calculating ellipse arc length via elliptic integral

I'm looking for an exact equation to calculate the arc length of an ellipse. I've found https://keisan.casio.com/exec/system/1343722259 to correctly calculate the length, but cannot calculate the ...
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1answer
75 views

Is the total variation function Lipschitz?

Let $f:\mathbb{R} \to \mathbb{R}$ be a function. Define the total variation function $V_f:[0,T] \times [0,T] \to \mathbb{R}$ over an interval $[s,s’]$ as usual: $$V_f(s,s’) = \sup_{D \subset [s,s’]} \...
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79 views

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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1answer
71 views

How can the formula for arc length work (for an ellipse)?

I want to apply the formula for arc length to an ellipse in polar coordinates to find its perimeter $$s=\int_{\theta_1}^{\theta_2}\sqrt{(dr/d\theta)^2 + r^2}$$ I'm looking to numerically integrate ...
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21 views

Apparently non uniform substitution in arc length (re)parametrization ( OpenStax Calculus III)

In the OpenStax Calculus III book ( page 287-286) , an example is given of arc length parametrization. The original function is : $ r(t)= 4\cos(t)$i$+4\sin(t)$j. The arc length function is : $s(t)= 4t$...
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the distance between 2 points on a curve C=xy

So I am no mathematician and have a curve the equation of which is $y=c/x$, or $c= xy$ where $c$ is a constant. I want to calculate the length of the curve between two points but my maths is not ...
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1answer
73 views

Why is it useful to use dS in the formula for arc length?

When discussing finding the arc length of some curve $C$ defined by parametric equations $x = f(\theta)$, $y = g(\theta)$, my professor said the following: Given $f', g'$ are continuous on $[\alpha, \...
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65 views

Arc Length for Parametric Equations $x={{\left( \cos u \right)}^{4}}$ and $y={{\left( \sin u \right)}^{4}}$

We know $-1\le \cos u\le 1$, $0\le {{\left( \cos u \right)}^{4}}\le 1$, therefore, $0\le u\le \frac{\pi }{2}$. The length itself can be calculated as $\begin{align} & \int\limits_{0}^{\frac{\pi }...
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2answers
44 views

why is the length of arc in polar coordinate like this?

I learned $L=\int_a^b{\sqrt{r^2+\frac{dr}{d\theta}^2}d\theta}.$ But, Why do I calculate like $L=\int_a^b{rd\theta}$ ? In my opinion, considering riemann sum like $\lim_{n\to\infty}{\sum_{i=1}^{i=n}{r(\...
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1answer
83 views

Arc Radius Calculation from 2 points

I've seen other questions on here and tried to follow them, but I was hoping somebody could help me understand where I'm going wrong in my solution, and point me in the right direction. I've got the ...
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1answer
111 views

Does there exist a rectifiable curve with infinite indefinite integral?

Since $\gamma\in C^1$ implies that $\gamma$ is rectifiable and that $\int_{a}^{b}\left|\gamma'(t)\right|dt=\Lambda(\gamma)<\infty$, I was wondering if there would be any counterexample in a more ...
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1answer
121 views

Curve with longest arclength between two points

Take 2 points in the $XY$ plane, WLOG make one point at the origin $(0,0)$ and the other, $(d, 0)$, lie on the $x$ axis. Picture drawing different curves between the two points and measuring the ...
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82 views

Parametric equation of an elliptical arc

Edit: My code had an error which is why I could not obtain the correct parametric plot. Hence, there is nothing wrong with the resource I used from W3 to create the ellipse plot. I would like some ...
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1answer
70 views

Find the length of the curve $y=\sinh(x)$, $0\leq x\leq1$ [closed]

How to calculate $L= \int\limits_0^1\sqrt{1+\cosh^2(x)}dx$? I tried substituting $t=e^x$, but it did not help.
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1answer
35 views

Find the function $f(x)$ so that, for $x\ge 0$, its arc length in the interval $[0, x]$ is $2x+f(x)$.

Find the function $f(x)$ so that, for $x\ge 0$, its arc length in the interval $[0, x]$ is $2x+f(x)$. I tried to use the formula $$y = \int_0^x \sqrt{1+y'^2} dx$$ with $y = f(x) + 2x$ and $y' = f '(x)...
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93 views

Arc length of $\cos(\ln(x)$) and $\sin(\ln(x))$ in closed form.

In this post of mine on the arc length of $cx^n$ I found a general formula for the arc length of the link name above. This formula cannot cancel the radicals and radicand as this may get rid of its ...
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2answers
110 views

Arc length of $x^n$ found using Hypergeometric function and series. Alternate representations and solution verification needed.

Take a look at the over 111000 values for the hypergeometric function. This shows that many results can be derived from using this genius function. Now for the arc length formula derivation. An arc ...
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227 views

Function equal to its own arclength.

Does there exist a differentiable function $f(x)$ such that $$f(x) = \int_0^x \sqrt{1+[f'(t)]^2}dt?$$ I'm quite interested in finding if there is such a function, as its value at any point $t$ would ...
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271 views

Why is arc length independent of parametrization? [duplicate]

For example, say a curve $C$ can be parametrized as $x_1(t)$, $y_1(t)$ over an interval $A$ and $x_2(t), y_2(t)$ over an interval $B$. Why is the arc length computed using the first parameterization ...
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1answer
57 views

Find arc-length of particle moving on constantly changing radius circle

I want to create function (integral) of particle moving on changing radius circle in constant speed. I have the following things I know: angle corresponding to the arc that the particle moved (...
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15 views

How is the arc length computed for (non-smooth) polar curves?

this is my first time raising a question here so I apologize for not following any conventions. I am only aware of the formula for computing arc lengths for smooth curves: I know that a polar curve $...
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2answers
71 views

If $f_n$ uniformly converges to $f$, the arc length of graph of $f_n$ converges to the arc lenth of graph of $f$?

If $f$ and $f_n$ are continuously differentiable functions on $[0,1]$, and $f_n$ uniformly converges to $f$, the the arc length of graph of $f_n$ converges to the arc lenth of graph of $f$ ? I feel ...
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1answer
45 views

Definition of the length of a path $[0,1] \to (X, d)$ with $(X, d)$ a metric space [closed]

If $(X, d)$ is a metric space, and $p \colon [0,1] \to X$ is a path (continous function). Then, can we define the length of $p$? Should we assume somehow differentiable? The only I know needs $X = \...
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1answer
44 views

Understanding why $|\beta'(s)|=|\alpha'(t)\cdot (dt/ds)| = 1$ in curve reparameterization by arc length

I am reading Do Carmo's Differential Geometry of Curves and Surfaces, and I am stuck understanding why $|\beta'(s)|=|\alpha'(t)\cdot (dt/ds)| = 1$ is true. For context, here is the paragraph where it ...
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1answer
77 views

Is this proof of the arc length formula valid?

The author uses only the definition of the derivative to prove the arc length formula. He does not invoke the Mean Value Theorem. Is this a valid proof? He claims that the difference quotient over ...
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71 views

$\operatorname{Length}(\gamma)\neq \int_0^1\|\dot{\gamma}(t)\|\,dt$ possible?

Let $\gamma:[0,1]\to\mathbb{R}^d$ be a rectifiable curve and define the length of the curve as $$\operatorname{Length}(\gamma)=\sup\left\{\sum\limits_{k=0}^{n-1}\|\gamma(t_{k+1})-\gamma(t_k)\|,n\ge 1, ...
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1answer
126 views

Square and Quarter Circle [closed]

$ABCD$ is a square of side $18$ cm. $F$ is a point inside the square, such that $BCF$ forms an equilateral triangle. $CFA$ is a quarter circle with centre $B$. $E$ is the point on $AB$ such that the ...
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55 views

how to solve the integral I get on the length of an arc?

I am to determine the arc length of a function given in parametric form of the form $$x = 50 (1 - \cos(t)) + 50 (2 - t) \sin(t)$$ $$y = 50 \sin(t) + 50 (2 - t) \sin(t)$$ I must determine the arc ...
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68 views

Find the entire arc length of the curve $r=2acos^3(θ/3)$

Question: Find the entire arc length of the curve $r=2a \cos^3(\frac{\theta}{3})$ My attempt: Given, $r = 2a \cos^3(\frac{\theta}{3})$ Using chain rule while differentiating with respect to $\theta$,...
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1answer
95 views

How to calculate the length of a curve between two points

Calculate the length of the curve: $y = \frac{1}{x}$ between points $(1,1)$ and $(2, \frac{1}{2})$. What I tried: $$ \int_a^b\sqrt{(x')^2+(y')^2} dt$$ $$r(t) =(t,1/t) $$ $$\int_1^2\sqrt{(1)^2+\left(\...
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37 views

Arc length of curve $x = 2a\sin^2{t}$, $y = 2a\cos{t}$ $0 \leq t \leq 2\pi$. Easier solution?

I decided to solve it for $[0, \pi/2]$ as a first thing. $(2a\sin^2{t})' = 2a\sin{2t}$ and $(2a\cos{t})' = -2a\sin{t}$. So $$l = \int_0^{\pi/2}{{2a\sqrt{4\sin^2{t}\cos^2{t} + \sin^2{t}}\,dt} = 2a\...
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1answer
41 views

Length of a rectifiable path is the sum of the restricted paths

Let $f:[a,b]\longrightarrow\mathbb{R}^n$ be a rectifiable path, $P=\{a=t_0<...<t_m=b\}$ a partition of $[a,b]$. I want to prove that $l(f)=\sum_{k=1}^m l(f_{|[t_{k-1},t_k]})$ (length of the path ...

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