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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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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
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26 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 ...
Bremen000's user avatar
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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 ...
CryptoMynd'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
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
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0 answers
29 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 ...
Turquoise Tilt'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
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3 votes
1 answer
65 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 ...
Cognoscenti's user avatar
1 vote
1 answer
143 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)\...
Dan's user avatar
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14 votes
2 answers
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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
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3 votes
0 answers
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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 ...
aaaaaaaa1234564's user avatar
-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 ...
Gleb Cloudy's user avatar
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0 answers
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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 ...
cxh007's user avatar
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2 answers
112 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 ...
Keqin Liu 'Kevin''s user avatar
-1 votes
1 answer
70 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 ...
Champignon's user avatar
1 vote
1 answer
94 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}...
Alice's user avatar
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3 votes
2 answers
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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 ...
Anthony Khodanian's user avatar
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 ...
Average_C_Enjoyer's user avatar
0 votes
2 answers
55 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 ...
Bob Dobbs's user avatar
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1 answer
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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 ...
Bifton Mifts's user avatar
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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 ...
Plop's user avatar
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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) = \...
spooleey's user avatar
  • 456
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 ...
Davis Yoshida's user avatar
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: ...
Ice's user avatar
  • 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: $$...
Leonidas Lanier's user avatar
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1 answer
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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}{...
S11n's user avatar
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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 ...
Carlo Wood's user avatar
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, ...
J.D's user avatar
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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 ...
Raúl Filigrana Villalba's user avatar
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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 ...
Bifton Mifts's user avatar
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 ...
Ventalto's user avatar
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 ...
moon river's user avatar
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 ...
Robert Mdee's user avatar
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 $\text{out}(C)$, ...
porridgemathematics's user avatar
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)=...
Sergej Lav Bojanić 2022's user avatar
-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 ...
PHV's user avatar
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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 ...
zeellos's user avatar
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2 votes
0 answers
83 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 ...
SRobertJames's user avatar
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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\...
TheWanderer's user avatar
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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 ...
mr_e_man's user avatar
  • 5,504
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)-...
mr_e_man's user avatar
  • 5,504
0 votes
1 answer
311 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 ...
vitbladez's user avatar
0 votes
0 answers
32 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 ...
vbalaji21's user avatar
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 ...
Giannnis AntetoSubtome's user avatar
0 votes
0 answers
6 views

How to deal with total derivative in the arc length formula for a 2D function?

I am having difficulties with the following formula in order to compute the arc length of a function. $$L = \int_{\theta_1}^{\theta_2} \sqrt{\left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^2+r^2} \ \...
Balfar's user avatar
  • 163
0 votes
1 answer
28 views

sequence of abscissas x corresponding to equispaced points on any function

I am writing a program where I need to generate the sequence of points $x_1$ , $x_2$, $x_3$,$...$ such that the corresponding ($x_1$, $y_1$), ($x_2$, $y_2$), ($x_3$, $y_3$), $...$ on a generic ...
Jada's user avatar
  • 137
1 vote
1 answer
48 views

Can't spot my error in calculating 3D Parametric Arc Length

We're asked to find a function s(t), for the arc length of a curve centered at point t=0, as a function of t. The function is as follows... $\gamma (t) =e^t i + \sqrt{2} tj-e^{-t}k$ My work is as ...
Glen Gaige's user avatar
1 vote
0 answers
53 views

Polar curves with specified length function

Given a probability density function $f(x)$ with support $[0,2\pi]$, I'm interested in constructing a "roulette" in which the "winning angles" follow that distribution. The shape ...
SuspiciousGarbage's user avatar
1 vote
0 answers
50 views

Arc-length change of variable in the inviscid Burgers equation.

Being quite new to the world of PDEs, I would like your help regarding a specific change of variable. Namely, I consider the inviscid Burgers equation : \begin{equation} u_t+uu_x=0 \end{equation} And ...
Lokipic Alias's user avatar
1 vote
2 answers
291 views

how to measure the arc length?

We're told to measure angles in radians, θ = arc length/radius. Therefore, 1 radian occurs when the radius of the circle is equal to the arc length subtending the angle you're looking to measure (θ). ...
Heidi Landon's user avatar

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