Questions tagged [arc-length]

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

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2answers
36 views

Find the arclength from 0 to 1 on the function $y =\arcsin(e^{-x})$.

I need to find the arclength from 0 to 1 on the function $y = \arcsin(e^{-x})$. I know that $$y' = \frac{-e^{-x}}{\sqrt{1-e^{-2x}}}$$ By applying arc length formula I get this nasty integral: $$\int_{...
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1answer
42 views

Arc length (line integral of a scalar field)

The given arc I tried and got 0.. The integral I have tried: $\cos\left(t\right)\sin\left(t\right)\left(\cos^2\left(t\right)-\sin^2\left(t\right)\right)-2\cos^3\left(t\right)\sin\left(t\right)+t$ The ...
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23 views

Bounding length of a curve by a triangular hull

Let $\gamma: [0, 1] \to \mathbb{R}^2$ be a curve (continuous, at least piecewise differentiable?) with end points $\gamma(0) = (0, 0)$, $\gamma(1) = (1, 1)$. I'm interested under which conditions I ...
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1answer
47 views

Calculating the arc length of a difficult radical function

I am fairly new to calculus and self learning integration from home has been challenging so I'm sorry if I make any mistakes. I want to work out the arc length of: $y = \sqrt{7.2 (x-\frac {1}{7}}) - 2....
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1answer
47 views

Find function given arc length [closed]

I'm creating a program that has two points and a cable hanging between them. I feel like modeling the cable using a catenary would be too hard, so I just simplified it to a parabola. However, I'm ...
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2answers
44 views

How do I prove both arcs are equal? [closed]

As in the following image, the segments AD, DB, BE and EC make the same angle (x) relative to the diameter of the circle QP. How can I prove the arcs L1 (AB) and L2 (BC) are equal?
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A Question on the Rate of Change of the Arc-Length

Main Question Consider some curve $y(x)$, going from a point $(x_0,y_0)$ to a point $(x_1,y_1)$. Let $L$ be the length of the curve, and the function $F$ be the rate of change of the length of this ...
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1answer
69 views

How to find equation of a circle tangential to two straight lines?

The image given above has two lines, AB and AC that are tangential to the circle with radius r. Points y1,y2 and the slope m of line AB are known. The graph represents a linear increase of speed (y ...
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2answers
52 views

Show that length of sine is equal to length of cosine on the same interval.

Let $$f(x)=\sin(x)\\\ g(x)=\cos(x)$$ Let $L_1$ be $$\int_0^{2\pi}\sqrt{1+\cos^2(x)}\space dx$$ And $L_2$ $$\int_0^{2\pi}\sqrt{1+\sin^2(x)}\space dx$$ I.e. L is a length of sine/cosine during it's ...
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1answer
110 views

Calculating the arc length of a radical function

I am very new to calculus and StackExchange so I'm sorry if I make any mistakes. I want to work out the arc length of: $y = \sqrt{5x} - 2.023, [0.075, 0.58]$. I have used the definition of a definite ...
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1answer
43 views

Find arc midpoint in 3D given start, end, center, normal and rotation direction

I have an circular arc on a plane in 3D space where I have the start point (x, y, z), end point (x, y, z), center point (x, y, z) as well as the normal to plane the arc is on and the direction of ...
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6answers
244 views

Does this math problem have a solution or not?

While programming I have faced a math problem that needs to be solved before I can move on. Maybe you need more input data to be able to solve it, if that's the case, just let me know and I will ...
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2answers
39 views

Negative arc length value. True or not?

I saw a post about this and someone said the arc length is an integral of a positive function, so it is positive. But by solving this exercise I found the arclength as a negative value. The arc ...
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1answer
39 views

Why is the derivative of the binormal vector parallel to the normal vector?

In my notes, it says to consider an arc length parameterization r(s). Then we can show that B' points in the direction of N such that we can write B'=-𝜏N. I understand why ||B'||=𝜏 but am unsure how ...
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2answers
35 views

Intuition behind equation for finding arc length in polar coordinate

I know how to derive the equation for finding arc length in polar coordinates but I don't understand this: Given a parametric equation let L be the length of the arc from point t = a to to t = b we ...
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1answer
15 views

Find the arclength curve of $r(t)=i+3t^2j+t^3k$ for $0\leq t\leq \sqrt{12}$

I asked a question similar to this one, but I'm still confused on how to integrate this. I have $r'(t)=\langle 0,6t,3t^2\rangle$. and so this gives you the integral from $0$ to $\sqrt{12}$ of $\sqrt{...
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1answer
14 views

Reparameterize the curve $r(t)=\langle e^t\sin t,e^t\cos t,5e^t \rangle$ in terms of the arclength parameter, s with $(0,1,5)$ as the base point.

So first, $r^\prime(t)= \langle e^t\sin t+e^t\cos t, e^t\cos t-e^t\sin t,5e^t \rangle$ Then I took the magnitude of $r^\prime(t)$ which is $\sqrt{(e^t\sin t+e^t\cos t)^2+(e^t\cos t-e^t\sin t)^2+(5e^t)...
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1answer
22 views

Find the arclength of the curve defined by $r(t)=i+9t^2j+t^3k$ for $0 \leq t \leq \sqrt28$.

First I found $r'(t)=\langle 1,18t^2,3t^2\rangle$ and so the magnitude of $r'(t)= \sqrt{1+(18t)^2+(3t^2)^2}$ thus the integral from $0$ to $\sqrt{28}$ of $\sqrt{1+324t^2+9t^4} dt$. When I plugged $\...
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1answer
43 views

Question about the definition of the arc length of the graph of a function

We defined the arc length of a function as $$L_I(f):=\int\limits_a^b\sqrt{1+(f'(x))^2}dx$$ for $I=[a,b]$ and $f\in C^1(I)$. We arrived at this formula by approximating the graph of $f$ by a series ...
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Intersection of oblate spheroid $\frac{x^2+y^2}{R_e^2}+\frac{z^2}{R_p^2} =1$ and plane $n_xx+n_yy+n_zz=0$

To calculate the distance between two points on Earth, I used 3 different approaches. For small distances, I used the Euclidean distance. For medium distances, I used the arc length on the circle ...
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0answers
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Finding steering angle of a car given distance travelled, and how far the car ends up offset from a straight line?

Problem: I have a semi-autonomous vehicle that needs it's steering adjusted. I can attempt to send it in a "straight line" of distance $(l)$ -- but unless it's steering is perfect, it will veer from ...
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1answer
39 views

How do I determine the curvature of an arc length parameterized curve in the $xy$-plane? [closed]

I have a 2D curve in the $xy$-plane, which was arc length parameterized numerically, and fitted by cubic splines for both $x$ and $y$. If one of the segments of the cubic spline is: \begin{align} x&...
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1answer
63 views

How is it that integrals end with dx?

It is commonly taught that to integrate a function f(x), with respect to x, from x = a to x = b, one calculates: $$\int_a^b f(x) \ dx$$ We add dx at the end of the integral to show that we are ...
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16 views

How to find the arc centre and radius given the arc start point and arc end point and arc direction?

I know the arc start point and the end point. It is separated by a height/distance as shown in the figure.The blue line end points are the arc start and end points respectively. How can i drawn a arc ...
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203 views

Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration?

Let $Z (t)$ be the Hardy Z function. Then define \begin{equation} Y (t) = \tanh (\ln (1 + Z (t)^2)) \end{equation} Let us define the length $L_n$ of the curve which intersects the $n$-th Zero of $Z (...
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59 views

How do you solve a differential equation with a definite integral inside of it?

$$ 80=\int_0^b \sqrt{1+f'(x)^2}dx $$ ($b$ is a nonzero constant) Context: I was trying to do the amazon cable question (my friend sent it to me, I know the answer is zero, but I wanted to know how to ...
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1answer
41 views

Finding Arc Lengths in a 3D cartesian coordinate system from any 2 arbitrary points

So I'm familiar with the parametric form of the arc length integral that follows a given path at any point $t$, but what do you do when you have a 3D equation that does not follow any specific path $t$...
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Finding the length of an arch of a cycloid by integrating by integrating velocity

I am trying to find the length of an arch of cycloid by integrating the speed at which a point on the cycloid moves. I know for a cycloid that is drawn out by a circle of radius 1 and rotates at 1 ...
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27 views

Analysing the length of unit speed curve undergoing deformation by vector field.

Given a unit speed curve $\gamma:[a,b] \to \mathbb{R}^3$ and a (smooth) vector field $V(t)$ that is perpendicular to $\dot{\gamma}$ for any $t \in [a,b]$ and has length $|V| = 1$. Define $\gamma_s : [...
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1answer
95 views

Example of non-rectifiable curve with finite arc length integral

If $y=f(x)$ is a real valued function for $x\in[0,1]$ is the graph of a rectifiable curve, then the integral $\int_0^1 \sqrt{1+f'(x)^2}dx$ converges and is said to be it's arc length. According to my ...
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2answers
58 views

Estimating the arc length of a sine wave using this polynomial formula?

I need to calculate the arc length of a half period of a sine wave with a given frequency and amplitude. I found this article which summarizes a polynomial method for getting a very close ...
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0answers
35 views

Smooth curve with given arc-length between two points (shape of paper strip)

I'm trying to find the rest shape of a paper strip with fixed endpoints (without gravity). Of course, the arc length of the resulting curve is given by the length of the strip, also the position and ...
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1answer
54 views

Finding average with respect to the arc length

The Problem There's an exercise in the MIT OCW 18.01SC course: What is the average distance from the $x$-axis of a point chosen at random on the cardioid $r = a (1 - \cos (\theta))$, if the point ...
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1answer
33 views

Arc length of a curve bounding from below a Fourier series

Assume $\gamma $ is a $C^1$ closed curve in the complex plane whose length is $2\pi$,and consider all its possible regular parametrization throught a parameter $t \in [0,2\pi]$ Let $\gamma(t)$ one of ...
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1answer
21 views

Does the ratio of infinitesimal arc lengths near a point on a smooth curve have an interpretation

Recently I was playing around with some calculus and stumbled upon a curious idea. Perhaps someone can help me understand this. Consider a smooth curve $f(x)$. Take a point $x=a$ and consider the arc ...
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1answer
24 views

Confusion with reparameterisation definition

Let $\gamma:I\rightarrow\mathbb{R}^n$ be a parameterisation of some regular curve $C$. The definition I am working with states that a parameterisation $\tilde{\gamma}:J\rightarrow\mathbb{R}^n$ of a ...
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1answer
26 views

Finding the horizontal extent of a sinusoid

I need some help with a calculus problem. Suppose you have a straight line of length L, and you squeeze it into a sinusoid with m...
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1answer
66 views

Curvature Formula Proof By Definition

Question: Use Definition 3.2 to prove Theorem 3.4. Definition 3.2 “The signed curvature $k(s)$ of a plane curve $ \alpha: I \rightarrow \mathbb{R^2}, \alpha(u)=(x(u),y(u))$ is defined by $t’(s)=k(s)...
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Angular gradient of an ellipse

This is my first post, I am not advanced in math so please be patient with me, I hope I can explain myself :) I am using a software called Substance Designer. It has a tool called pixel processor ...
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31 views

A limited elliptical arc length

I'm trying to calculate the length of an elliptical arc using Excel and later C++. The range is not $[0, \pi/2]$, and not $[0, \pi]$, and not $[0 ,2\pi]$, but $[0, \pi/3]$. Nowhere can I find a ...
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3answers
49 views

How would u relate arc length to chord length?

What is relationship of arc length of 2 points of a circle and if you connect a chord to the same points of arc. In the book it says. Points A and B of an arc is 2x length So the chord length is 2 ...
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46 views

Length of a Shadow Cast on a Sphere

I'm doing a project for a classical mechanics class about shadows cast on a sphere. Though the topic is physics-ish, the derivation is entirely math and geometry. Below is my derivation, but I'd ...
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1answer
28 views

Arc length of $|cos\theta|$ from $\theta=\frac{\pi}{6}$ to $\theta=\frac{\pi}{3}$

An insect is moving along the curve $r=|cos\theta|$ such that $\theta =\frac{\pi t}{6}$, where $t$ is time measured in seconds. What is the distance travelled by the insect in the time interval ...
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2answers
74 views

Parameterize and find the arc length

Parameterize the following curves in $\mathbb{R}^3$ and find their arc length: the intersection of the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$ the intersection of the sphere $x^2+y^2+z^...
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0answers
47 views

Arclength of intersection of cylinder and plane

The question is as follows: Parameterize the intersection of the cylinder $x^2 + y^2 = 1$ and the plane $x+y+z = 1$ in $\mathbb{R}^3$ and find its arclength. I have come as far as parameterizing ...
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When is the reparameterization by arc length 'very' impossible?

We can reparameterize a curve $r : [0,1]\to\mathbb R^n$ by its arc length $L(t)$ defined by $$s=L(t)=\int_0^t|r'(u)|du$$ If $L(t)$ has an inverse, letting $t=L^{-1}(s)$, we have $$\frac {d} {ds} r(L^{-...
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24 views

How to use Crofton's Formula (Proving Arc of Great Circle Smallest Distance Between Points)

Problem. The shortest distance between points on the unit sphere is an arc of a great circle connecting them. Attempt: I followed the hints my professor gave and looked back at the proof of the ...
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1answer
49 views

Finding derivative of arc length of curve

I’m trying to find the arc length of a curve and I only wanted to know if step 1 is correct so far then if I have trouble I’ll ask for help with my solution method Find the derivate $$r ( t ) = \...
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27 views

Three definitions of arc length

Let $\gamma:[0,1]\to M$ be a curve on a metric space $(M,d)$. There are three ways to define the arc length of $\gamma$. $$\ell(\gamma):=\sup\bigg\{\sum_{i=0}^kd(\gamma(t_{i+1}),\gamma(t_i)):0=t_0&...
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2answers
152 views

Length of Piecewise Linear Curve avoiding N points

Let $y_0,y_1,x_1,\dots,x_N$ be distinct points in $\mathbb{R}^n$. Clearly there exists a piecewise linear curve $\gamma:[0,1]\rightarrow \mathbb{R}^n$ joining $y_0$ to $y_1$ for which $$ \min_{t \in [...

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