Questions tagged [arc-length]

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

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Characterization of arclength as unique function on continuous curves that satisfy certain conditions (resolution of "$\pi=4$ paradox")

I was again thinking about the famous $\pi=4$ paradox, and this question in particular: How to convince a layperson that the $\pi = 4$ proof is wrong?, about why the standard sup over polygonal ...
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Is there a more clean way to derive parametric arc length?

This is one possible way to derive the formula for arc length: Assume we have a parametric curve $f(t): \mathbb{R} \rightarrow \mathbb{R}^n$ We can sample the curve at regular intervals $t_i$ ...
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Orthogonality from length constraint

I want to ask you for help with a question related with a definition from optics. There the refractive index of a medium is given by \begin{equation} n(\vec{q}, \dot{\vec{q}}) \end{equation} and \...
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Are there curves similar to Bézier curves, but with a fixed length?

Bézier curves have some nice properties, such as starting at $P_0$ and ending at $P_n$ (for an $n$-degree Bézier curve). I am looking for a class of (curvy) curves, but with the additional property ...
39 views

Measure polynomial arc-length between given range

Is there a way to measure the arc-length of a curve created by a 4-degrees polynomial, between a given range? For example, I want to measure the length of the polynomial $-\frac{1}{200}x^3+x+1$ ...
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References for minimizers of energy in metric spaces

As the title suggests, do you know some literature, books, articles or notes, that treats the connection between geodesic in a metric space and the energy associated to the curves? In particular, I am ...
32 views

Calculate movement along circle arc

I want to simulate the turn of an aircraft, and therefore need to calculate its position on a circle arc within a cartesian coordinate system. I need to calculate its position every 5s. What I ...
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Geometric intuition behind pseudo-distances

Could somebody please offer some intuition into how pseudo-distances work? Something like what the geometric interpretation is, and how they differ from distances, would be appreciated. Background: I ...
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Selecting a random point on a function.

Say there is a continuous function $f(x)$ which does not have any undefined points in the interval $[a,b[$. I want to randomly select a point $(x,y)$ in that interval on the line formed by that ...
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How can I find $\Delta_1$ for the first of two reversing curves?

This is a problem from the design of roadways. I know someone has the answer, but I haven't been able to work it out myself. The sketch below shows a single curve made up of two asymmetrical ...
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How can I find the arc length of some interval on this parametric object?

I want to find the arclength some interval bounded by points $P$ and $Q$ on a parametric object. This parametric object is defined by the 2D stereographic function: $f(x, y) = (tx, 1 + t(y-1))$ . If ...
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Calculating ellipse arc length via elliptic integral

I'm looking for an exact equation to calculate the arc length of an ellipse. I've found https://keisan.casio.com/exec/system/1343722259 to correctly calculate the length, but cannot calculate the ...
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Square and Quarter Circle [closed]

$ABCD$ is a square of side $18$ cm. $F$ is a point inside the square, such that $BCF$ forms an equilateral triangle. $CFA$ is a quarter circle with centre $B$. $E$ is the point on $AB$ such that the ...
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how to solve the integral I get on the length of an arc?

I am to determine the arc length of a function given in parametric form of the form $$x = 50 (1 - \cos(t)) + 50 (2 - t) \sin(t)$$ $$y = 50 \sin(t) + 50 (2 - t) \sin(t)$$ I must determine the arc ...
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Find the entire arc length of the curve $r=2acos^3(θ/3)$

Question: Find the entire arc length of the curve $r=2a \cos^3(\frac{\theta}{3})$ My attempt: Given, $r = 2a \cos^3(\frac{\theta}{3})$ Using chain rule while differentiating with respect to $\theta$,...
Calculate the length of the curve: $y = \frac{1}{x}$ between points $(1,1)$ and $(2, \frac{1}{2})$. What I tried: $$\int_a^b\sqrt{(x')^2+(y')^2} dt$$ $$r(t) =(t,1/t)$$ $$\int_1^2\sqrt{(1)^2+\left(\... 0answers 37 views Arc length of curve x = 2a\sin^2{t}, y = 2a\cos{t} 0 \leq t \leq 2\pi. Easier solution? I decided to solve it for [0, \pi/2] as a first thing. (2a\sin^2{t})' = 2a\sin{2t} and (2a\cos{t})' = -2a\sin{t}. So$$l = \int_0^{\pi/2}{{2a\sqrt{4\sin^2{t}\cos^2{t} + \sin^2{t}}\,dt} = 2a\...
Let $f:[a,b]\longrightarrow\mathbb{R}^n$ be a rectifiable path, $P=\{a=t_0<...<t_m=b\}$ a partition of $[a,b]$. I want to prove that $l(f)=\sum_{k=1}^m l(f_{|[t_{k-1},t_k]})$ (length of the path ...