Questions tagged [arc-length]
For questions about/on finding the arc length of a curve/parametrized curve
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Arc length derivation [closed]
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Why does the hypothetical X value Xi that would equal the avg slope of a section get replaced by X? I don't remember using the mean value theorem in the definition of the ...
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How to prove that $ \underset{\varDelta x\rightarrow 0}{\lim}\frac{\left| MN \right|}{\overset{\frown}{MN}} $
In the derivation of the arc differential formula, why is the limit of the ratio of the length of the line segment between two points to the length of the arc considered to be 1: $$
\underset{\...
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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 ...
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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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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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How do I find the arc length of $y=1-e^{-x}$ from $0 \leq x \leq 2$? [closed]
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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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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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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3
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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}}$$
My attempt:
I recognized ...
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Evaluate $\lim_{n\to\infty}\prod_{k=1}^n \frac{2n}{e}(\arcsin(\frac{k}{n})-\arcsin(\frac{k-1}{n}))$
I'm trying to evaluate $L=\lim\limits_{n\to\infty}f(n)$ where
$$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)$$
We have:
$f(1)\...
- 25.8k
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Conjectured connection between $e$ and $\pi$ in a semidisk.
A semidisk with diameter $\dfrac{e}{\pi}n$ is divided into $n$ regions of equal area by line segments from a diameter endpoint.
Here is an example with $n=6$.
Consider the $n$ arcs between ...
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How should I keep arc length equal between multiple points on a parametric curve?
I made this thing in desmos:
https://www.desmos.com/calculator/na9sehjskk
The distance between points changes depending on the speed of the points. Is there a way to keep the distance between them ...
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The length of the curve
Find the length of the curve:
$$\theta = \frac{r}{2} \sqrt{r^2+2}+\ln \left(r+\sqrt{r^2+2}\right),\quad 0 \leq r \leq 2.$$
Is it possible to apply the formula for calculating the length of a curve in ...
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Approximation Error on Arc Length of Quadratic Bezier curve
Given a quadratic Bezier curve defined by:
$$ 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 ...
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