Questions tagged [arc-length]
For questions about/on finding the arc length of a curve/parametrized curve
721
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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
$$\sum_P\lVert df\rVert=\sum_{i=1}^m\lVert f(t_i)-...
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How to Find the height of the arc or distance between arc and straight line given both curves have exact same start and end points?
Im trying to figure out how to find the height of the arc or maybe the distance between arc and line given than both of these lines/curves have exact same start and end points...the only difference is ...
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26
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How to divide a catenary curve into parts of equal length?
I know the basic equation of a catenary is
y = a*cosh((x-x0)/a)+b
Length of a catenary curve is
L = a*sinh((x-x0)/a)
where x0 is a symmetry point or vertex or lowest x co-ordinate of a curve.
I can ...
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Can We Prove These Removed Areas Are an Arc?
[I've been delving into math during my free time and came across an intriguing problem involving the perimeter of a shaded region. This particular challenge is part of a module that focuses on arc ...
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6
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How to deal with total derivative in the arc length formula for a 2D function?
I am having difficulties with the following formula in order to compute the arc length of a function.
$$L = \int_{\theta_1}^{\theta_2} \sqrt{\left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^2+r^2} \ \...
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26
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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 ...
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How do I determine the plastic section modulus Z of an arc? [closed]
I am trying to find a simplified formula for determining the plastic modulus, Z, of an arc of a circle.
In my case, I have a circle that has a diameter of 20" and I am removing an arc length of 6&...
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59
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Arc Length of the parabola
Can You please tell me how to derive this formula?
$$L=\frac12\sqrt{b^2+16a^2} + \frac{b^2}8a \ln \left(\frac{4a+\sqrt{b^2+16a^2}}b\right)$$
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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 ...
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How to solve these complicated SA and arc length integrals?
My partner and I are working on a project for our multivariable calculus class where we have to solve the integrals to find the arc length and surface area of our three piecewise functions. We've used ...
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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 ...
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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 ...
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2
answers
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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). ...
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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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Deriving the formula for length of function curve.
In my textbook the first step in deriving the length of a function curve is to the the sum
$\lim_{n\to \infty}\sum_{i=0}^{n}\sqrt{(x_{i}-x_{i-1})^{2}+(f(x_{i})-f(x_{i-1}))^{2}}$
Now my text continues ...
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Calculate length of involute without calculus
Is it possible to calculate the length of an involute of a circle without using calculus, and how?
Say we’re interested in the involute length for $\theta$ between $0$ and $2 \pi$.
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59
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Arc length for continuous implicit differentiable functions
We have a continuous differentiable function defined as $$F(x,y)=0$$And I am looking for a formula for its arc length between $x$ values $a$ and $b$. Doing a quick search, I could only find formulae ...
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A person's quickest path between any two points on perimeter of elliptical lake never involves both swimming and running. Find maximum eccentricity.
I made up this question.
A police officer's job is to patrol the perimeter of an elliptical lake. They have a constant (unknown) swimming speed and a constant (unknown) running speed. Their quickest ...
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Two Distinct Arc Length Parameterisations
$f$ and $g$ are two arc length parameterisations of a regular simple curve $C$, then prove either $f(s) = g(e+s)$ or $f(s) = g(e-s)\; \forall s$ for some constant $e$.
Intuitively I can understand as ...
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A property of complete metric spaces makes them length (path or inner) metric spaces, Clarification of a proof
In the book "Metric Structures for Riemannian and Non-Riemannian Spaces", by Misha Gromov, I found a proof of the following statement (of Theorem 1.8. restated here more concentrated)
Let $(...
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Explanation for arc length in parametrized curve
let $\Delta s_i$ be a piece of arc length hence:
$$\Delta s_i = \int_{i-1}^i \sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}dt$$
Why is that the length of $\Delta s_i$
in 2d I know that the as $\Delta x$ -> ...
- 1,043
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Roots of an arc-length system
Related:
Calculating the height of a circular segment at all points provided only chord and arc lengths - but this doesn't care about height, and doesn't have the arc-angle as a given
Finding out an ...
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$\ell(f)>\int_{a}^{b}||f'(t)||dt$
We know that if $f:[a,b] \rightarrow \mathbb{R}^{n}$ is a $C^1$ path then
$$\ell(f)=\int_{a}^{b}\|f'(t)\|dt.$$
Moreover, in the proof of this result we use explicitly the continuity of $f'.$
I'm ...
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1
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Catenary hanging chain problem
A chain hangs in the shape of a catenary with equation $y=\cosh(x)$ for $x\in[-a,a]$. If the length of the chain is $20$, how far apart are the endpoints of the chain?
I am familiar with the arc ...
- 129
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29
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Involute of circle: Get point by distance
Given is an involute of a circle. The basic circle radius $a$ and a point $A$ on the involute defined by the involute angle $\phi$ and the radius of the involute $\rho$ (and thus the arc length $l$) ...
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If i want to calculate the length of a hanging chain given three heights, how do i iteratively find the correct tension factor?
I work in lighting. We rig bistro bulb strings a lot, and I'm trying to write a script that will calculate the arc length of cable we need to bring to different installs.
The givens I have to work ...
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1
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83
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Surface area generated by revolving the astroid about the y-axis(Without parametrization)
Find the area of the surface generated by revolving the
astroid about the y-axis
an astroid is defined implicitly by:
$$x^{2/3}+y^{2/3}=a^{2/3}$$
now:
$$y = (a^{2/3}-x^{2/3})^{3/2}$$
$$y' = -\frac{\...
- 1,043
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Calculating arc length of an infinite(?) curve
English is not my first language so please excuse me for possibly not expressing myself clearly.
I am a computer science student and I've just finished calculus this semester.
I was having fun with ...
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Calculate the length of a curve [duplicate]
I'm trying to learn this out of my own genuine curiosity.
If $f(x)$ gives you the rate of change, then $\displaystyle\int{f(x)}$ gives you area under the curve
How would I calculate the length of the ...
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2
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Deriving the formula for arc length in $\mathbb{R}^2$
I've seen the proof the arc length formula in $\mathbb{R}^2$ in both polar and standard cartesian coordinates before, but I saw this interesting formula while piecing through my brother's calculus ...
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How to derive arc length formula in three dimensions?
Usually textbooks show the formula in 2D and attention is given as to how the mean value theorem allows the introduction of a derivative into the equation.
Thus far I have not been able to find a ...
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Geometric proof that $\tan \alpha \geq \alpha$ for any acute angle $\alpha$
For any acute angle $\alpha$, then $\tan \alpha \geq \alpha$. Is it possible to prove this geometrically?
My work so far:
A rigorous proof requires defining a radian measure of angles: this can be ...
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3
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Find the arc length of $f(x)=4x^\frac{1}{2}+9$
Question:
If the arc length formula of a function $f$ on an interval $[a,b]$ is given by $L_a^b=\int_a^b \sqrt{1+[f'(x)]^2} \ dx$. Find the arc length of $f(x)=4x^\frac{1}{2}+9$ on $[0,1]$.
We have $...
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The inverse of an arc length parametrization of $\partial U$ is Lipschitz
Let $U$ be a simply connected open bounded subset of $\mathbb{C}\cong\mathbb{R}^{2}$ with smooth boundary (thus the boundary $\partial U$ is diffeomorphic to a circle). Let $L\geq0$ denote the total ...
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Bounds of arc length of a curve
The problem I came across asks the following:
Find the arc length of a curve in its parametric form: $$x=a(2\cos{t}-\cos{2t}),$$ $$y=a(2\sin{t}-\sin{2t})$$
To solve this we can use the formula $\...
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Is the curve $t \mapsto t \sin \frac 1 t$ rectifiable?
Let $a\le 0\le b $ and let $ \gamma : \left[ a,b \right] \to \mathbb{R}$ be defined by
$\gamma(t)= t\text{sin}\frac{1}{t}$ for $ t\neq 0 $ and $\gamma(t)=0$ for $t=0$. I want to investigate that if ...
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Arc length of $(y-\sqrt[3]{x^2})^2+x^2 = 1$.
I am trying to compute arc length the area inside of closed curve $$\left(y-\sqrt[3]{x^2}\right)^2+x^2 = 1$$
It is known for the heart shape plotted at the end of the post.
I know how to compute the ...
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Path Integral: James J. Callahan's Advanced Calculus - A Geometric View
Page $22: 1.24$. Determine the work done by the force field F in moving a particle along the oriented curve $\overrightarrow{C}$, where: c. F = (y,x), $\overrightarrow{C}$: any path from $(5,2)$ to $(...
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Working on orbital mechanics...need to find the length of arc d along the Earth's surface. Any help or advice would be greatly appreciated!
I know the radius of the Earth and the radius of the orbit (r). I need to find the length of arc (d) along the surface of the Earth given the graphical information presented. Thanks for any help you ...
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Ellipse - Correct angle calculation
I need to calculate a point on ellipse based on elliptic rotation angle.
Though I am using the same formula as for circle (a = horizontal radius, b = vertical radius):
$$ x = a * cos(\alpha) $$
$$ y = ...
- 111
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Family of curves sharing the same length
Do you have an example of a family of curves $C$ that share the same length $L$?
By family, I mean a set of curves that can be expressed in a generic form - using one or multiple parameters.
Put ...
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Arc length in ML estimation
There are a lot of proofs in Complex Analysis where we use the following „Estimation lemma” (ML inequality):
$$\left|\int_{\Gamma}f\left(z\right)dz\right|\leq M\cdot l\left(\Gamma\right),$$
where $M=\...
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80
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Calculating the length of a rope hanging from one point
If you have a rope hanging from one point in the air, how can you calculate the length of it (without measuring it). I don’t really know anything that complicated about maths, but I’m curious if it is ...
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Length space and continuity assumption on curves
I think the definition of length metric works without assuming curves are continuous.
Let $X$ be a subset of $\mathbb{R}^n$ and $x, y\in X$.
Def 1. For a function $f:[0, 1]\rightarrow X$, $L(f)$ is ...
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Apostol, Vol I Ch. 14: When defining an arc length function $s(t)=\Lambda(a,t)$ why is it necessary to specify $s(a)=0$?
In Apostol's Calculus, Ch. 14 "Vector-Valued Functions" section 14.12, he derives the result that the function representing the arc length of a parametric curve is given by the integral of ...
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2
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Calculate the length of the arc of the curve with an integral not involving a square root.
Let $A$ and $B$ be positive constants. If $0 < a < b$, find a simple condition relating $A$ and $B$ that makes it possible to calculate the length of the arc of the curve
$$
y = Ax^4 + \frac{B}{...
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2
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Surface area - even dimensions
Consider the surface:
$$ (\log x_1)^2+(\log x_2)^2+\cdot\cdot\cdot +(\log x_n)^2=R $$
For $R=1$ and in even dimensions $n=2,4,6, \cdot\cdot\cdot$ we have the volumes:
$$V=\bigg( \frac{\pi^2 I_1(\sqrt{...
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When do we use the negative square root in the arc length formula
So I am learning the arc length of a function right now and encountered an example from my textbook as follows:
Question: fnd the arc length function for the curve $y=x^2- \frac{ln(x)}{8}$ taking $P_{...
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136
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Height of circle bulge along given arc
I am trying to determine line of sight along the surface of the earth between 2 points. Earth's curvature is a consideration. 2 points on the earth $A$ and $B$ are distance $s$ apart. At intervals ...
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Parametrization by arc length of Descartes Folium
I have just learned what parametrization by arc length is and trying to gain some intuition on it I have tried to parametrize some curves. Lines, circles and helixes are easy using the following ...
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