# Questions tagged [arc-length]

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

757 questions
Filter by
Sorted by
Tagged with
186 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 ...
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 ...
• 1,456
13 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 ...
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 ...
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 ...
• 101
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 ...
• 1,214
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 ...
• 24.4k
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 ...
• 705
1 vote
143 views

• 494
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 ...
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 ...
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 ...
• 11.5k
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 ...
• 117
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 ...
• 2,681
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 \iota\left(\begin{pmatrix}x \\y\end{pmatrix}\right) = \...
• 456
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 ...
• 1,540
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: ...
• 1
49 views

• 5,180
22 views

• 5,504
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 ...
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 ...
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 ...
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} \ \...
• 163
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 ...
• 137
1 vote
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 ...
1 vote
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 ...
1 vote
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 : $$u_t+uu_x=0$$ And ...