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Questions tagged [arc-length]

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

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63 votes
8 answers
6k views

Graphs for which a calculus student can reasonably compute the arclength

Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by $$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$ For many choices of $f$ this can be a ...
Mike Pierce's user avatar
40 votes
3 answers
2k views

Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
Keshav Srinivasan's user avatar
39 votes
10 answers
64k views

What is the length of a sine wave from $0$ to $2\pi$?

What is the length of a sine wave from $0$ to $2\pi$? Physically I would plot $$y=\sin(x),\quad 0\le x\le {2\pi}$$ and measure line length. I think part of the answer is to integrate this: $$ \int_0^{...
philcolbourn's user avatar
23 votes
2 answers
621 views

Another interesting property of $y=2^{n-1}\prod_{k=0}^n \left(x-\cos{\frac{k\pi}{n}}\right)$: product of arc lengths converges, but to what?

Here is the curve $y=2^{n-1}\prod\limits_{k=0}^n \left(x-\cos{\frac{k\pi}{n}}\right)$, shown with example $n=8$, together with the unit circle centred at the origin. Call the arc lengths between ...
Dan's user avatar
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20 votes
6 answers
4k views

Is the arc length always irrational between two rational points?

Recently I was wondering: Why does pi have an irrational value as it is simply the ratio of diameter to circumference of a circle? As the value of diameter is rational then the irrationality must come ...
Caelan Miron's user avatar
18 votes
1 answer
574 views

A remarkable fact about the unit circle; looking for a shape with an even more remarkable fact.

You may have heard of the following remarkable fact about the unit circle: If $n$ equally spaced points are drawn on a unit circle, and line segments are drawn from one of the points to each of the ...
Dan's user avatar
  • 24.5k
17 votes
9 answers
289k views

Calculate the radius of a circle given the chord length and height of a segment

I have a (circular) segment of known height and known chord length. Is is possible to determine the radius of the circle? Any help much appreciated.
user108308's user avatar
14 votes
2 answers
471 views

Conjectured connection between $e$ and $\pi$ in a semidisk.

A semidisk with diameter $\dfrac{e}{\pi}n$ is divided into $n$ regions of equal area by line segments from a diameter endpoint. Here is an example with $n=6$. Consider the $n$ arcs between ...
Dan's user avatar
  • 24.5k
13 votes
4 answers
2k views

Is this "derivation" of the path length formula actually correct?

Saw this in a physics lecture. This all assumes we have some function, $y=f(x)$. First he defined $$ds = \sqrt{dx^2+dy^2},$$ where the professor drew a picture and seemed to be using dx and dy to ...
Dargscisyhp's user avatar
13 votes
1 answer
824 views

Arc Length Integral of $x^x$ from 0 to 1 in closed form.

I was recently trying to compute the arc length of $x^x$ from $0$ to $1$ as follows: $$L=\int_0^1 \sqrt{1+\left(\frac{\text{d}}{\text{d}x}x^x\right)^2} \text{d}x=$$ $$\int_0^1\sqrt{1+x^{2x}(\ln x+1)^2}...
Тyma Gaidash's user avatar
13 votes
2 answers
902 views

An amazing property of the Catenary

I discovered that if we want an arc of catenary in the interval $[a,b]$ we solve $$\int_a^b \sqrt{\cosh '(x)^2+1} \, dx=\int_a^b \cosh x \, dx$$ which means that the "result" of the length ...
Raffaele's user avatar
  • 26.4k
11 votes
5 answers
1k views

Is it possible to find the coordinates of a point on the circumference of a circle, without using trigonometric functions?

I don't have a particularly good reason to want to do this, and I'm just asking out of curiosity. I am looking for the coordinates of point $\pmb B$, a point on the circumference of a circle. If I ...
Rein Ernst's user avatar
10 votes
4 answers
822 views

Approximating the length of a circular arc using geometrical construction. How does it work?

I was going through my Engineering Drawing textbook and came upon this topic. Using only a compass and a straightedge, one can supposedly approximate the length of a given circular arc by following ...
Aayush Dhungana's user avatar
9 votes
4 answers
981 views

Integral inequality of length of curve

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function. Prove that for any $a.b\in \mathbb{R}$ $$\left (\int_a^b\sqrt{1+(f'(x))^2}\,dx\right)^2\ge (a-b)^2+(f(b)-f(a))^2$$. ...
RAM_3R's user avatar
  • 1,246
9 votes
5 answers
1k views

Given a circle of radius r, and two points ('X' and 'Z') on that circle, can some circumcircular arc "XYZ" be constructed of length r?

I am strictly an amateur, not a professional mathematician or some such. This question occurred to me while considering the fact that an angle of 1 radian centered on the center of a circle will ...
MTL-VRN's user avatar
  • 193
9 votes
1 answer
1k views

Curvature and torsion of a spherical curve

I'm trying to show that if $\alpha$ is a regular curve parametrized by arc lenght whose range lies on the unit sphere centered at the origin, then $\kappa (s) = \sqrt{1+j^2}$ and $\tau (s) = \dfrac{j'(...
pedro's user avatar
  • 141
9 votes
2 answers
1k views

Arc length of the squircle

The squircle is given by the equation $x^4+y^4=r^4$. Apparently, its circumference or arc length $c$ is given by $$c=-\frac{\sqrt[4]{3} r G_{5,5}^{5,5}\left(1\left| \begin{array}{c} \frac{1}{3},\...
Ben Longo's user avatar
  • 1,068
8 votes
3 answers
1k views

Why do I get two different answers when solving for arclength?

I am given that $\frac{dx}{dt}=8t\cos(t)$ and $\frac{dy}{dt}=8t\sin(t)$. I tried solving for the arclength from $t=0$ to $t=1.$ Method 1: $$\text{Arclength} = \int_{0}^{1} \sqrt{\left(\frac{dx}{dt}\...
Jay's user avatar
  • 305
8 votes
3 answers
2k views

Explicit nontrivial examples of arc length parametrization

For pedagogical purposes (and for some numerical experiments) I was looking for some nontrivial explicit (i.e. closed-form) examples of arclength parametrized curves. I know that arclength ...
Christian Bueno's user avatar
8 votes
5 answers
868 views

Is the arclength of $x\sin(\pi/x)$ in the interval $(0, 1)$ finite?

I'm having trouble determining the arclength of $x\sin(\pi/x)$ in the interval $(0, 1)$ (that is, determining if it converges or not). My first attempt was to write $$ \begin{aligned} \int_0^1\sqrt{1+(...
user3002473's user avatar
  • 8,974
7 votes
1 answer
246 views

What does a function of its own arc length look like?

Introduction What does a function of its own arc length look like? A strange question for sure, but first let me elaborate: Imagine a function that starts at the point $\left( 0, 0 \right)$. If we now ...
Kevin Dietrich's user avatar
7 votes
4 answers
1k views

Elementary ways to calculate the arc length of the Cantor function (and singular function in general)

Cantor's function: http://en.wikipedia.org/wiki/Cantor_function There is an elementary way to prove that the arc length of the Cantor function is 2? In this article (http://www.math.helsinki.fi/...
zairhenrique's user avatar
7 votes
1 answer
164 views

Length of a super-circle

In a youtube video titled "Generalizing the circumference" Prof. Michael Penn works out the integral $$ L(n)=4 \int_{0}^{\pi/2}\sqrt{n^2 \cos^{2n-2}(\theta) \sin^2(\theta)+n^2 \sin^{2n-2}(\...
Peder's user avatar
  • 2,192
7 votes
1 answer
879 views

Show the equivalence of arc length definitions

Definition 1: Let $r: [a,b] \to \Bbb R^d$ be a continuous differentiable function. Then the arc length is given by $$L(r) = \int_a^b || r'(t) || \, dt$$ Definition 2: Let $r: [a,b] \to \Bbb R^d$...
mdcq's user avatar
  • 1,678
7 votes
1 answer
1k views

Can anyone tell me why the arclength integral is a lower semicontinuous function on the set of continuously differentiable real-valued functions?

I posted the question stating that it was upper semicontinuous, but that was definitely wrong. I am trying to prove lower semicontinuity.
user11314's user avatar
  • 141
7 votes
0 answers
106 views

A person's quickest path between any two points on perimeter of elliptical lake never involves both swimming and running. Find maximum eccentricity.

I made up this question. A police officer's job is to patrol the perimeter of an elliptical lake. They have a constant (unknown) swimming speed and a constant (unknown) running speed. Their quickest ...
Dan's user avatar
  • 24.5k
7 votes
0 answers
242 views

perimeter of ellipse

As it is well-known, there is no formula for expression of perimeter of the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1$, as an elementary function of $a$ and $b$. I am interested to find an exact ...
XIII's user avatar
  • 413
6 votes
1 answer
2k views

Distance between two points on a surface with metric $g_{mn}$

Say you have a two dimensional surface with a metric tensor $g_{mn}$ on which there are two points with coordinates $(\theta_1, \phi_1)$ and $(\theta_2,\phi_2)$, how would you calculate the distance ...
Beta Decay's user avatar
6 votes
4 answers
3k views

Shortest distance between two points via calculus of variations

This problem might be trivial but when solving it using calculus of variations it's not so stupid. Suppose we have a fixed boundary condition $f(a) = f(b) = 0$ and we want to find the shortest ...
user148848's user avatar
6 votes
1 answer
710 views

"Increasingify" a function / Total variation of a function

Let $f : [a,b] \rightarrow \mathbb{R}$ be a $C^1$ function such that $f$ is monotonic on each $[t_k, t_{k+1}]$, with $a = t_0 < t_1 < ... < t_N = b$. Let g be the increasing-ified version of ...
Basj's user avatar
  • 1,551
6 votes
1 answer
303 views

The length of Christmas Tree Lights wrapping a conical tree from bottom to top with uniform vertical spacing.

Our $20$-year old artificial Christmas tree has built-in lights, but the top and bottom thirds are burnt out. So I decided to buy several (much cheaper) new "tree light" strings (during a ...
Jim Clark's user avatar
  • 212
5 votes
3 answers
507 views

calculating the arc length of a function

The function is: $$ f(x) =2\ln(4-x^2)$$ and the length of the arc is from $-1$ to $1$. So i know the formula for calculating the arc length is$$\int_{a}^{b}\sqrt{1+(f'(x))^2} dx$$ the derivative of ...
Wouter Lommerse's user avatar
5 votes
5 answers
1k views

Understanding the formula for curvature

In this khan academy article, they discuss how can define curvature as, $$ \bigg\| \frac{dT}{dS} \bigg\| = \kappa$$ In the post they write, "However, we don't want differences in the rate at ...
Babu's user avatar
  • 12k
5 votes
1 answer
235 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 ...
Noah's user avatar
  • 59
5 votes
3 answers
3k views

Integrate the function $\int \sqrt{2+ e^{2t} + e^{-2t} } \; dt$

I'm unsure how to solve this arc length. The original problem says to find the arc length of $r(t) = \langle \sqrt{2}t, e^t, e^{-t} \rangle $ deriving I arrive at $r'(t) = \langle \sqrt{2}, e^t, -...
alphanumeric0's user avatar
5 votes
2 answers
103 views

How do I find the arc length of $y=1-e^{-x}$ from $0 \leq x \leq 2$?

I can set up the integral $\int_{0}^{2} \sqrt{1+e^{-2x}}\,dx$ by taking the derivative of y and by using the arc length formula. I'm really stuck on how to evaluate this integral. I've tried to follow ...
Brian Lilley's user avatar
5 votes
1 answer
3k views

Curvature Formula Proof By Definition

Question: Use Definition 3.2 to prove Theorem 3.4. Definition 3.2 “The signed curvature $k(s)$ of a plane curve $ \alpha: I \rightarrow \mathbb{R^2}, \alpha(u)=(x(u),y(u))$ is defined by $t’(s)=k(s)...
B2K's user avatar
  • 415
5 votes
1 answer
635 views

Can the strict triangle inequality hold in a length space?

Let $(M, d)$ be a metric space where (0) for relevance: set $M$ contains at least three distinct elements and distance function $d : M \times M \to \mathbb R$ explicitly satisfies (1) non-...
user12262's user avatar
  • 491
5 votes
1 answer
94 views

$\ell(f)>\int_{a}^{b}||f'(t)||dt$

We know that if $f:[a,b] \rightarrow \mathbb{R}^{n}$ is a $C^1$ path then $$\ell(f)=\int_{a}^{b}\|f'(t)\|dt.$$ Moreover, in the proof of this result we use explicitly the continuity of $f'.$ I'm ...
Math's user avatar
  • 2,379
5 votes
1 answer
120 views

Prove convergence of formula for length of parameterized curve

In class, our professor was discussing the arc length of a parameterized curve $x=f(t), y= g(t)$. In his derivation, he reached the sum - $$\sum_{i=1}^n L_i = \sum_{i=1}^n \sqrt{(\Delta x_i)^2 + (\...
Ishan Deo's user avatar
  • 3,604
5 votes
2 answers
617 views

Interpolation method that gives the least arc lenght of the curve.

this question may be a bit sloppy on my part but I will make it anyway, I recently have been fascinated by the idea that the surface of a soap bubble film restricted to a boundary will be so as to ...
Felipe Dilho's user avatar
5 votes
1 answer
2k views

How to determine X,Y position from point P based on time, velocity and rate of turn?

We need to figure out an estimated position along an arc using a set of known variables. The starting position on the arc would be called point P, here are the known/measured variables: P = initial ...
wayofthefuture's user avatar
5 votes
2 answers
2k views

Lower semicontinuity of length of graph: $L(g)\le\liminf_{n\to\infty}L(f_n)$

I am interested in the following claim:$\newcommand{\intrv}[2]{[#1,#2]}\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}$ Let $g,f_1,f_2,\dots$ be continuous functions on $\intrv ab$ such that $g=\...
Martin Sleziak's user avatar
5 votes
1 answer
359 views

"Unrolling" a 3d wedge

I have a parametric equation that describes a particular intersection in 3d space. I'd like to flatten this by deforming the object without stretching its length, as if this were the outline of a ...
OmnipotentEntity's user avatar
5 votes
0 answers
220 views

Find point of $n$-sphere after moving on geodesic by given angle

Let $\mathbb{S}^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\colon\lVert\mathbf{x}\rVert^2=1\}$ be the unit n-Sphere and $\mathbf{x}\in\mathbb{S}^n$. Given a vector $\mathbf{a}\in\mathbb{R}^{n+1}$ and an angle $\...
nullgeppetto's user avatar
  • 3,076
5 votes
0 answers
303 views

Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration?

Let $Z (t)$ be the Hardy Z function. Then define \begin{equation} Y (t) = \tanh (\ln (1 + Z (t)^2)) \end{equation} Let us define the length $L_n$ of the curve which intersects the $n$-th Zero of $Z (...
crow's user avatar
  • 1
5 votes
0 answers
88 views

Supremum of arc lengths of graphs of power towers

Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only: $$x,\;x^x,\,x^{x^x},\;\...
Vladimir Reshetnikov's user avatar
4 votes
2 answers
174 views

Calculus 2 test question that even the professor could not solve (Parametric Arc length)

everyone got this question wrong from my calculus 3 midterm. " find the arc length of the curve on the given interval: $x=\sqrt(t), \space y=8t-6,\space on\space 0 \leq t \leq 3$" I set the ...
Colton Matschke's user avatar
4 votes
1 answer
194 views

Calculating the arc length of a curve... Which formula?

Let $S$ be a surface parameterized by variables $u,v$ and $\alpha(t)=(u(t),v(t))$ be a curve on the surface. I am of the understanding that we can find the arc length of $\alpha$ by integrating it's ...
PhysicsIsHard's user avatar
4 votes
3 answers
8k views

Finding the arc length of the parabola $y=x^2 \; from \; (0,0)\;to\;(1,1)$

As the title says, I need to find the arc length of that. This is what I have so far (I'm mostly stuck on the integration part): $${dy\over dx}=2x \Rightarrow L=\int_0^1 \sqrt{1+(2x)^2}dx$$ Substitute ...
JustHeavy's user avatar
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