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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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10 votes
4 answers
807 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 ...
0 votes
0 answers
25 views

Archimedes' approximation of length of a curve

I have been told by a colleague that the following way of approximating the length of a curve is due to Archimedes (he heard of it somewhere in Greece) but we could't find any reference. Let me ...
0 votes
0 answers
10 views

Finding or constructing Archimedes spirals with/from parametric lengths

I'm using Desmos, and have already combed through this site not finding anything close to what I need, nor have the equations and modifications I have tried been of help. Desmos Trial by Combat I need ...
5 votes
2 answers
102 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 ...
0 votes
1 answer
1k views

Circumference of a circle with known cord length and circumcircular arclength

I have been trying to figure out a way to relate the length of a string that is curved into a circumcircle, along with the distance between the two ends of that string, to the arclength created by ...
0 votes
0 answers
28 views

Notation issues - Arc length over manifold

Hi I'm working on some notes where there's this little excursus on differential geometry, topic is arc length. In the first part arc length over $\Bbb R^n$ is defined using the limit of small segment ...
14 votes
2 answers
464 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 ...
18 votes
1 answer
565 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 ...
3 votes
1 answer
64 views

Limit with a geometric interpretation

Let $f:ℝ \to ℝ$ be a $C^∞$ curve. Determine the following limit; $$\lim_{x_1 \to x_2} \dfrac{ \int_{x_1}^{x_2} \sqrt{1+f'(x)^2} dx}{\sqrt{(x_2-x_1)^2+(f(x_2)-f(x_1))^2}}$$ My attempt: I recognized ...
1 vote
1 answer
139 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$

I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where $$f(n)=\prod\limits_{k=1}^n \frac{2n}{e}\left(\arcsin\left(\frac{k}{n}\right)-\arcsin\left(\frac{k-1}{n}\right)\right)$$ We have: $f(1)\...
3 votes
0 answers
55 views

How should I keep arc length equal between multiple points on a parametric curve?

I made this thing in desmos: https://www.desmos.com/calculator/na9sehjskk The distance between points changes depending on the speed of the points. Is there a way to keep the distance between them ...
-1 votes
1 answer
95 views

The length of the curve

Find the length of the curve: $$\theta = \frac{r}{2} \sqrt{r^2+2}+\ln \left(r+\sqrt{r^2+2}\right),\quad 0 \leq r \leq 2.$$ Is it possible to apply the formula for calculating the length of a curve in ...
0 votes
2 answers
110 views

Strict proof of infinitesimal equivalency between $\sin{x}$ and $x$

When I was teaching infinitesimal equivalency between $\sin(x)$ and $x$ ($x\rightarrow0$) for Calculus, I realized that it was not very easy to have a pure elementary proof for it without using the ...
0 votes
0 answers
45 views

Approximation Error on Arc Length of Quadratic Bezier curve

Given a quadratic Bezier curve defined by: $$ B(t) = (1-t)^2P_0 + 2t(1-t)P_1 + t^2P_2 $$ The arc length $ s(t) $ from $0$ to $ t $ is: $$ s(t) = \int_0^t |B'(τ)| dτ. $$ It's known that the arc length ...
-1 votes
1 answer
69 views

Calculating the arc length of a circular layer cut

As per the drawing attached, I am trying to get at the arc length of a circular layer. The problem constitutes itself as follows: Let there be a circle with a known radius $r=219$ Let there be a ...
1 vote
1 answer
93 views

Help with using "infinitesimal Riemann sums" to arrive at the formula for arclength

I am trying to arrive at the formula for arclength using infinitesimals. So far, I have a definition which says: $\displaystyle \mathrm{Re}\sum_{k=0}^{\omega}f(x_k)\Delta x:=\int_{a}^{b}f(x)\mathrm{d}...
0 votes
2 answers
54 views

Find loop passing through two points with length $L\pi$

Problem: Find a nice simple closed curve other than circle which passes through the points $(0,0)$ and $(1,0)$ on the Cartesian plane and whose length is $L\pi$. If the given condition is not the loop ...
3 votes
2 answers
87 views

What radius of circle has a circumference equally divided into 10 sections by a pentagram?

Given a regular pentagram whose outer vertices lie on a circle of radius 1, a circle interior to and sharing a center with the larger circle will intersect the pentagram in ten places, save for two ...
4 votes
1 answer
157 views

Finding length of function using improper integral

So I want to find the length of the function $y = a - 2\sqrt{ax} + x$ in the interval $(0, a)$, assuming $a > 0$. I found the derivative: $ y' = 1 - \frac {\sqrt{a}}{\sqrt{x}} $ Then using the ...
0 votes
3 answers
96 views

On the curve $y=\frac{\sin (\pi x)}{x^p},x>0$, for what values of $p$ does the product of all the arc lengths between neighboring roots exist?

Consider the curve $y=\frac{\sin (\pi x)}{x^p}, x>0$, shown here with $p=0.75$. It occurred to me that if $p$ is large enough, then the curve flattens quickly, so the arc lengths between ...
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^{...
0 votes
1 answer
52 views

Deriving the catenary from a hanging chain

Assume a heavy chain (constant mass per unit length) takes the shape of a plane curve $\mathcal C$ after being suspended by its two ends from the same height. Let $s$ be its arc length starting from ...
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},\...
0 votes
0 answers
20 views

Length of rectifiable curves in Finsler spaces

Let $U$ be an open set in $\mathbb{R}^n$, let $E$ be the set of norms on $\mathbb{R}^n$, and let $N: U\rightarrow E$ be a map such that $(x,v) \mapsto N(x)(v)$ is continuous. We define the length of a ...
0 votes
0 answers
41 views

Proving the curvature of a plane curve is equal to that of a space curve

Let $\gamma : (a,b) \rightarrow \mathbb{R}^2$ be a regular curve. Let $\iota : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the map \begin{equation}\iota\left(\begin{pmatrix}x \\y\end{pmatrix}\right) = \...
4 votes
1 answer
230 views

The simplest curve which is never straight and has a rational arc length.

This tweet claims to give an explanation for why one should expect the perimeter of a circle with a rational radius to be irrational. It doesn't strike me as that convincing (although feel free to ...
0 votes
4 answers
2k views

Arc length of the cardioid

Compute the length of the segment of the cardioid $(r, θ) = (1+ \cos(t), t) $ such that $ t \in [0, 2π].$ How do I find the arc length of the cardioid. I did $\mathbf{r}'=\langle -\sin(t),1\rangle$ ...
2 votes
1 answer
264 views

Finding the ideal B-spline through data points using Euler-Lagrange: is it just too hard to do?

I am not even sure I have a question anymore (I will just give up)... in the past month or so I have been researching cubic Bézier curves. The idea was to find a fit through data points, using ...
0 votes
0 answers
58 views

Calculating the arc of a curve

I need to find the arc length from the function $$ y^{2}= - 2.6\times x$$ Will the result change if I replace the function with? $$x = \frac{y^{2}}{-2.6}$$ I also ask you to check my answer, I did it: ...
0 votes
1 answer
37 views

Proving that $t \cos \frac{\pi}{2t}$ is nonrectifiable (Tom Apostol's Calculus vol. $1$, ex $14.13.22$)

The exercise is to show that $f(t) = t \cos \frac{\pi}{2t}$ is not rectifiable. To show that, Tom Apostol is guiding us to consider the partition $P = \{0, \frac{1}{2n}, \frac{1}{2n-1}, ..., \frac{1}{...
2 votes
0 answers
49 views

Distance Travelled by a Projectile

I wanted to come up with a formula for the total distance travelled by a projectile with some initial velocity $\langle v_x,v_y\rangle$ in $\mathbb R^2$. Its parametrization should be the following: $$...
2 votes
1 answer
54 views

Arclength parametrisation in 4D.

My question is to do with parametrisation of arclengths. As part of a course on mechanics (with mathematical focus), I have covered intrinsic coordinates in a plane curve. In class and homeworks etc, ...
2 votes
2 answers
593 views

Intuition behind equation for finding arc length in polar coordinate

I know how to derive the equation for finding arc length in polar coordinates but I don't understand this: Given a parametric equation let L be the length of the arc from point t = a to to t = b we ...
0 votes
0 answers
60 views

Arc length of a reparametrization differs by the sign of the derivative of the reparametrization map

I have the following problem largely figured out, and just want some pointers with the details that actually justify what is being done working fast and loose with differentials. Suppose $\gamma$ is a ...
0 votes
0 answers
18 views

The arclength of a rectifiable curve is continuous:

Let $(X,d)$ be a metric space and $\gamma:[a,b]\rightarrow X$ a curve. We define the length of $\gamma$ on $[a,b]$ as follows: \begin{equation} \ell_a^b(\gamma):=\sup\left\{\ell_a^b(\gamma,P)\mid ...
2 votes
0 answers
87 views

Parameterize Grim Reaper Curve by arc length

I have to solve the next problem for a course at university: Parametrise the grim's reaper curve $y = -\ln(\cos x)$ with $x \in \bigl(-\frac{\pi}{2}, \frac{\pi}{2}\bigr)$ by the arc length using the ...
1 vote
0 answers
30 views

finding the formula for the arc length of any curve problems with the dx

So I am reading the proof for the integral formula for the length of any curve from a to b. What I don't understand: I understand how delta x is extracted out from the expression algebraically. But ...
0 votes
0 answers
51 views

The length of an arc through parametric equations

Prove that the length l of an arc given by the parametric equations $x = \theta$ and $y = (\sec\theta)^2 $ from $\theta = 0$ to $\theta = \frac{\pi}{4}$ is given by $l = \ln(1 + \sqrt{2})$. I have ...
1 vote
0 answers
21 views

Minimum spherical length of Jordan curve separating pairs of points

Let $C \subset \mathbb{C}$ be a (locally rectifiable) Jordan curve separating the pairs of points $a_1,b_1$ and $a_2,b_2$ (i.e. $a_1,b_1$ lies in $\text{ins}(C)$ and $a_2,b_2$ lies in $\text{out}(C)$, ...
0 votes
1 answer
70 views

Length between two points on an arbitrary surface [closed]

Given a surface represented by the function $f(x,y):R^2 \rightarrow R$. I am trying to write a program that computes the distance between two points that belongs to that surface. For example, say, the ...
1 vote
0 answers
34 views

Arc length parametrization (from my exam) [closed]

[a)]picture of exam question (txt in Serbian)1]I have 2 long problem, and i must to solve this for exam from calculs 4: a)How to finde arc length of this function with vector- parametric rep. s(t)=...
-1 votes
2 answers
94 views

Can a fractal have an infinite area if it's bounded by a box? [closed]

Can a fractal/or any 2d shape have an infinite area if it's bounded by a box with finite area? And oppositely, if it can be bounded by a circle/line, how can it's "arc length"/perimeter be ...
2 votes
0 answers
79 views

Given its arc length, what is the shape of a parabola?

A chain of length $k$ is hung between two fixed walls of $2x$ distance apart. How low does the chain hang (at its lowest point)? Assume the chain makes a parabola (not a catenary). This is the ...
2 votes
0 answers
50 views

Recovering the first fundamental form from lengths of curves on surfaces

It is well-known that the length of a curve $\gamma: I=[a,b] \to S$, where $S \subseteq \mathbb{R}^3$ is a surface, may be computed by the first fundamental form as follows: $$ L(\gamma)=\int_{a}^b\...
2 votes
1 answer
22 views

When defining arclength of a curve, does it matter whether the partition norm approaches $0$ in the domain (interval) or the codomain (space)?

Suppose $f:[0,1]\to\mathbb R^d$ is continuous, and $P=[t_0,t_1,t_2,\cdots,t_{n-1},t_n]$ is a partition, so $0=t_0<t_1<\cdots<t_n=1$. Define $$\sum_P\lVert df\rVert=\sum_{1\leq i\leq n}\lVert ...
7 votes
1 answer
876 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 $r: [a,b] \to \Bbb R^d$...
0 votes
0 answers
30 views

Prove equivalence of two definitions of arclength (for non-differentiable curves) [duplicate]

Suppose $f:[0,1]\to\mathbb R^n$ is continuous. For a partition $P=\{t_0=0,t_1,t_2,\cdots,t_{m-1},t_m=1\}$, with norm $|P|=\max_i(t_i-t_{i-1})$, define $$\sum_P\lVert df\rVert=\sum_{i=1}^m\lVert f(t_i)-...
0 votes
0 answers
31 views

How to divide a catenary curve into parts of equal length?

I know the basic equation of a catenary is y = a*cosh((x-x0)/a)+b Length of a catenary curve is L = a*sinh((x-x0)/a) where x0 is a symmetry point or vertex or lowest x co-ordinate of a curve. I can ...
0 votes
1 answer
293 views

How to Find the height of the arc or distance between arc and straight line given both curves have exact same start and end points?

Im trying to figure out how to find the height of the arc or maybe the distance between arc and line given than both of these lines/curves have exact same start and end points...the only difference is ...
0 votes
0 answers
29 views

Can We Prove These Removed Areas Are an Arc?

[I've been delving into math during my free time and came across an intriguing problem involving the perimeter of a shaded region. This particular challenge is part of a module that focuses on arc ...

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