Questions tagged [curves]

For questions about or involving curves.

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24 views

Is there a function that is the envelope of the sum of ceilings of reciprocal functions

TL;DR: Given a sum of ceilings of reciprocal functions $$y_1 = T = \sum^{n-1}_i \Big\lceil \frac{p_i}{k} \Big\rceil$$ is there a corresponding form for a function that envelopes the $T$ on the left? ...
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26 views

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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Singularity of the curves in affine form vs in projective form

I have just started learning about elliptic curves and I have this thought about curves in affine form and projective form. Apologies in advance if the question sounds silly, admittedly I am not ...
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Does a linear homotopy between two simple closed curves cover the area between them?

Consider two closed simple curves parametrised by $\gamma_1,\gamma_2: [0,1]\rightarrow \mathbb{R}^2$ that are smooth functions. Both curves $\gamma_i$ encircle a bounded subset $A_i$ of $\mathbb{R}^2$,...
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Smoothness of reparameterization

Suppose we have a parametric curve in $\mathbb{R^2}$, i.e. $\phi: [a, b] \subset \mathbb{R} \to \mathbb{R}^2$, $\phi(s) = (\phi_1(s), \phi_2(s))$ for $s \in [a, b]$. Suppose there is a different ...
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40 views

Finding asymptotes for the given curve of two variables

I am looking for the asymptotes for the curve $x_1=\frac{C}{x_{1}^{2}-3x_1x_2+3x_{2}^{2}}$ where $C\in \mathbb{R}$ and $x_1$ and $x_2$ are the set variables. Oftentimes the asymptotes are found by ...
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39 views

Intersection of a sphere and a cylinder

Take the Viviani curve intersection of a sphere and a cylinder. Analytically explain what happens to the intersection curve if you keep the radius of the sphere constant, fix one side of the cylinder ...
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84 views

Why is it called an 'integral' curve?

The concept of a integral curve is relatively easy to understand as path through a vector field which is tangent to the field at each point. But why is it called an "integral" curve? It ...
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in the sphere or disc,there is no essential simple closed curves?

How we can show in the sphere or disc,there is no essential simple closed curves ? In the mapping class groups By Benson farb , definition of essential closed curve is : a closed curve is called ...
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57 views

Find the area inside the curve $r^2=2\cos(5\theta)$ and outside the unit circle.

I found the area of one full rose-petal($A_1$) and the area enclosed by the petal and the unit circle($A_2$), subtracted these from one another to get the area enclosed by the curve outside of the ...
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How can I generate this wave with a formula? [closed]

I am not a mathematician, so please forgive my lack of appropriate mathematical terms! I'd like to generate the below curve procedurally. I had a go, but I couldn't quite get there and I feel like I'...
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1answer
32 views

Curve Discussion with $ f(x) = \frac{1}{x^2+r} $ and $ r > 0 $ with r as a constant. Need guidance

As stated, i have $$ f(x) = \frac{1}{x^2+r} $$ with r being a constant and $ r > 0 $ I am familiar with curve discussion normally, but confused by the constant r. How do i properly calculate this, ...
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25 views

Arc length parameter relation with fourth derivative

Let non-planar curve $\gamma:I \rightarrow \mathbb{R^{3}}$ with arc length parameter $s$. Find $a,\,b,\,c$ such that $\gamma^{(4)}(s)=a\gamma^\prime(s)+b\gamma^{\prime\prime}(s)+c\gamma^{(3)}(s)$. I ...
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Find a tangent line of a given parabola perpendicular to another given line

The question from my textbook: Find the equation of the tangent line to the curve $y = x²-2x$ that is perpendicular to the line $x-2y = 1$ The derivative of the parabola: $2x-2$ It's normal slope of ...
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44 views

Estimating the length of a curve from below by a bound on its curvature.

I got stuck on a problem. Let $\alpha(s)$ be smooth simple closed curve with curvature $k(s)$ and $0 <k(s) \leq c $. Then show that the length of the curve is limited from below by $2\pi \frac{1}{...
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Let us consider a curve defined by [closed]

[enter image description here][1] [1]: https://i.stack.imgur.com/JrVSC.jpg**For now, I just need to know how to solve a**
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Why does the dimension of variety involve algebraic closures? [duplicate]

On page 4 of Arithmetic of Elliptic Curves, Silverman defines the dimension of a variety $V$ (defined over $K$) as the transcendence degree of the function field $\overline{K}(V)$ over $\overline{K}$. ...
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66 views

How to make a looping (Bézier?) curve?

I want to display the path traced by a point, $p_1$ rotating around another, moving point, $p_2$. The point $p_2$ moves parallel to an axis on a plane, and the distance between the points is fixed. ...
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33 views

Hausdorff distance of inclusions vs. distance of parametrizations

This question is very closely related to this post. Two piecewise smooth parametrizations $\gamma_1,\gamma_2:[0,1]\rightarrow \mathbb{R}^2$ with $\gamma_1(0)=\gamma_1(1)$ and $\gamma_2(0)=\gamma_2(1)$ ...
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Need help with understanding this math part from the book "An Abundance Of Katherines" by John Green

I am reading the book "An Abundance Of Katherines" by John Green. There is a lot of math stuff in it which I am unable to comprehend. Somebody help me out in this regard. Here the ...
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83 views

What is the function of a curve like a "」"?

I'm doing curve fitting and the curve look like this. I've tried polynomial or exponetial functions, but none of them fit the curve very well. Can someone give some advice? Thanks in advance!
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Some help with the obtention of curve derivatives

Let $\alpha:I\subset\mathbb{R}^{3}$ be a regular curve (not nec. parametrized by arc length), $\beta: J\subset\mathbb{R}\longrightarrow\mathbb{R}^{3}$ a reparametrization of $\alpha(I)$ by arc length $...
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complex integral over a spiral

Good morning everyone, I am not sure how to solve the following integral, can anybody help me? $$\int_\gamma \frac{1}{z}dz$$ with $\gamma (t) = (t+1)e^{it}$ and $t\in [0, 2\pi]$ I split the curve in ...
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1answer
41 views

curve between 0:0 and 1:1 [duplicate]

I want to describe a single smooth curve between 0:0 and 1:1, where a curvature c describes the curve. When c=1/2, the 'curve' is just a straight line (f(x)=x) ...
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1answer
29 views

Computing the steepness of a curve on a surface

Given a curve $x(s),y(s)$ where $s$ is also the arc length. Let $u(x,y)$ be a function such that $u(x(s),y(s))$ forms a surface $D$. Now I want to compute the steepness $\theta$ of that curve on $D$ ...
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221 views

Prove ratio of curvature and torsion = -sin𝜃 d𝜃/ sinφ dφ where 𝜃 and φ are angles between a fixed vector and the tangent and binormal respectively

I'm trying to prove the following: If the tangent and binormal at a point of a curve make angles 𝜃 and φ respectively with a fixed direction, then prove that $\frac{\sin{\theta} \ d\theta}{\sin{\phi}...
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58 views

How can I determine wheter or not two curves are tangents?

Let $a_0,b_0 \in \mathbb{R}, a_0^2 + b_0^2 = 1$. How can I show that the curves $$\frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1$$ and $$x+y = 1$$ are tangents to eachother? I tried to find a point such ...
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Reconstructing curve from curvature as a function of arclength

Let's say we have the curve on the left, for which we can can calculate the curvature as a function of the arc-length using: $$\kappa = \frac{\frac{d^2y}{dx^2}}{(1 +(\frac{dy}{dx})^2)^{3/2}}$$ But ...
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1answer
42 views

Creating Bézier curves with a maximum curvature

I am creating $5$-point Bézier curves. Sometimes, these curves are too sharp, see for example the below curve: there is a sharp cusp on the left. Is there any way that I can generate Bézier (or ...
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44 views

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 ...
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Rational points over an extension of a finite field

Let $f(x,y)$ be an affine curve over a finite field $\mathbb{F}_q$. Assume that it has some rational points over $\mathbb{F}_{q^r}$, i.e. it has some $\mathbb{F}_{q^r}$-rational points for some ...
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when two curves are **transverse**?

in the "A Primer on Mapping Class Groups by Benson Farb and Dan Margalit" we define intersection number as follow : Let $\alpha$ and $\beta$ be a pair of transverse, oriented, simple closed ...
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how can calculate gradiant of curvature of surface

I am solving question 7.2 Zangwill electrodynamics and I need to prove that $$ \vec{\nabla \Psi} = \partial\Psi/\partial n \enspace \hat{n} + \partial\Psi/\partial\tau_{1} \enspace \hat{\tau_{2}} ...
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58 views

Could I understand the following results by interpreting curvature as acceleration?

I saw these two theorems in the book "Riemannian Manifolds" by John M. Lee: Thrm 1.5: Two smooth unit speed plane curves are congruent iff their curvatures are always the same. Thrm 1.6: The ...
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How can I prove that this defintion of a winding number is valid?

I have read a definition of a winding number on wikipedia and it involves finding the continuous polar parametrization of the curve, but then the question arises, why does such a parametriation always ...
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Obtain z value from x and y input

I wonder if there are tools, that can generate an equation from my (x, y, z) values. this is a sample from the data that I am using: ...
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Do superellipses provide examples of submanifolds of $\Bbb{R}^2$ that are not smooth?

Consider the curve (a kind of Lamé curve or superellipse (https://en.wikipedia.org/wiki/Superellipse)) in $\mathbb{R}^2$ defined by the equation \begin{equation} |x|^n + |y|^n = 1, \end{equation} ...
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Interaction between two viscoelastic elements

Consider two cortices formed from two different kinds of viscoelastic filaments 1 and 2. Cortex 1 is inside cortex 2. Both cortices are said to have resistance to bending and extension. So both can we ...
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orientation of a curve

In the eponymous wikipedia article we read the following definition of curve orientation: In the case of a planar simple closed curve (that is, a curve in the plane whose starting point is also the ...
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50 views

Total Curvature for a curve in a metric space

Is there a theorem involving total curvature and some notion of index number for a curve in a metric space, as there is for planar curves? (i.e. total curvature is an integer multiple of $2\pi$.) I'm ...
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why Alexander method gives us a finite combinatorial problem?

The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
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83 views

Exercise 6.19 of Baby Rudin

Let $\gamma_1$ be a curve in $\mathbb{R}^k$, defined on $[a, b]$; let $\phi$ be a continuous 1-1 mapping of $[c, d]$ onto $[a, b]$, such that $\phi(c) = a$; and define $\gamma_2(s) = \gamma_1(\phi(s))$...
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Defining a differentiable curve's torsion

In exercise 10, page 26 of Manfredo's Differentiable Geometry of Curves and Surfaces, we are asked to look at the following differentiable curve: $$\alpha(t)=\left\{\begin{array}{ll} (t,0,e^{-\frac{1}{...
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Simple, open curve that can be parameterized with its length

I am looking for a "simple" curve in $\mathbb{R}^3$ that can be easily parametrized with its length. When I say simple, I mean it needs a small number of points to be uniquely determined and,...
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82 views

What parametrization should I use to evaluate $\int_{\phi}x^{4/3} + y^{4/3}$, where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$?

I´ve recently tried calculating this: $$\int_{\phi}x^{4/3} + y^{4/3}$$ where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$. And I couldn´t think of any parametrization or substitution that would ...
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38 views

Is still true that if $\gamma: I \rightarrow \mathbb{R} ^n $ is a curve of class $C ^2$, and if $\tau (s)=0$, then $\gamma$ lies in a plane?

I alredy make the proof when $n=3$ and i use the Frenet-Serret formulas, but i´m not sure how to do in $\mathbb{R} ^n$ Class of $C^2$ means twice continuously differentiable, and $\tau$ is the torsion ...
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Spider-Man: No Way Home and Archimedean spiral. POSSIBLE SPOILERS [closed]

In the movie while Spider-Man fights Strange it is mentioned that the dimension he is in is an Archimedean spiral, Spider-Man says "Square the radius. Divide by $\pi$. Plot points along the curve&...
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1answer
21 views

area of one rose petal using iterated integral but the outer integral is of radius

I have a Rose leaf, described by the equation r=a*sin(3θ), from θ=0 to $\frac{\pi}{6}$. I need to make an iterated integral of this area. It's easy when the outer ...
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19 views

Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisy

Fitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate. In my setup, I have 3D spacial observations of ...

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