# Questions tagged [arc-length]

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

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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) = \...
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### 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 ...
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### 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 $\mathbf{r}'=\langle -\sin(t),1\rangle$ ...
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### 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 ...
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### 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: ...
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### 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 ...