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Questions tagged [curves]

For questions about or involving curves.

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20 views

Form of function constant on a closed curve

Suppose that a continuous function $f : (0,2\pi)^2 \to \mathbb{R}$ is constant along the closed curve $$ \gamma(t) = (\theta_0 + t \mod{2\pi}, \phi_0 + q t \mod{2\pi}) $$ where $q \in \mathbb{Q} - \{0\...
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Adapted coordinates in the entire open set where a congruence is defined

Context: Null congruences that define Kundt spacetimes. However my question is a little bit more general, since the metric is not relevant. Consider an $n$-dimensional differentiable manifold $M$, an ...
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27 views

Computing a Basis of Holomorphic Differential Forms for a Given Curve

I find myself repeatedly having to ask this question, because no one seems to have answered it. I put this up on Math Overflow with a bounty, got a guy who said he would answer it, but—though he ...
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0answers
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What kind of Planar Quartic Curve might this be?

I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem. I have convinced myself that the solutions always lie on a quartic curve, which is ...
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0answers
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The limit of a function from different directions: rotate the function and take limit

Let $\{(x,y)\}_{x\in\mathbb R}$ be a continuous curve; what will be implied if it is given that the any rotated version of that curve have a limit? i.e. let $(s,t)=A(x,y)$ be a rotation around the ...
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1answer
41 views

Let $A,B$ jordan curves, if $A\subset B$, then $A=B$, true or false? [on hold]

intuitively I think it’s true, but I can’t get a proof, any suggestion or counterexample?
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0answers
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Calculating Offset Curves defined by NACA Formula

I need to be able to scale an aerofoil so that it creates a new offset aerofoil. I'm going to be using the notation as suggested to me in my previous question. Taking: $$ 0 \le t \le 1\\ \ \\u(t) =...
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1answer
41 views

Area between curves.

I have to calculate area bounded by curves : $(x^3+y^3)^2=x^2+y^2 $ for $ x,y \ge 0 $. I tried to use polar coordinates, but I have : $r^4(\cos^6\alpha +2\sin^3\alpha\cos^3\alpha + \sin^6\alpha)=1$
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1answer
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Smooth, approximately space-filling curves in high dimensions

I'm looking for smooth (infinitely differentiable everywhere) functions (curves) $\mathbb{R}\rightarrow\mathbb{R}^d$ that are approximately space-filling, i.e. scaling allows the curve to eventually ...
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1answer
18 views

Finding a unit speed parametrisation for $\alpha(t)= 0.5 (t, 1/t, \sqrt{2} \log(t))$

I'm trying to find a unit speed parametrisation for the curve $\alpha: (0, \infty) \to \mathbb{R}^3$ s.t $$\alpha(t)= 0.5 (t, 1/t, \sqrt{2} \log(t)).$$ However, $$s(t) = 0.5 (t - \frac{ 1}{ t} ),$$ ...
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1answer
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Reparametrization of a curve in opposite direction and find intersection of two particles travelling in opossite directions on the same curve.

Hi I need help with this problem: Given $$r(t)=(e^t,e^t\cos t, e^t \sin t),\quad t\in[0,2\pi]$$ It represents the trajectory of a particle $P$. Draw the curve. Reparametrize $r$ such ...
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1answer
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Help me please with an exercises [closed]

Help in mathematics calculate the points of the curve x2-xy + y2 = 27, in which the tangents are horizontal and vertical. Deriving and Prima = y-2x/2y-X and find the equations of the tangent and the ...
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2answers
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Resource for a library of closed curves.

I am working on a computation project and I need a bunch of closed curves to test my programs on. Does anybody know of a resource or library of such curves somewhere preferably online. I would like to ...
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Weak Tangent Problem - Reconciling Two Approaches

The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces. My question is regarding part (a). Let us first show $\alpha:I\rightarrow\mathbb{R}^3$ has weak tangent at $t_0=...
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Elliptic curves (Tate normal form?)

I basically have two question, the other question can be found below. Let $E/k$ be an elliptic curve with $P\in E(k)$ a point of order $\geq 4$. Show that $E$ can be described by \begin{equation*} y^...
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1answer
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Length of curve - Calculus

I need to find the LENGTH of the curve with x,y,z components listed below: $ x(t) = t\sin(2t)$ $ y(t) = t\cos(2t)$ $z(t) = (4/3)t\sqrt(t) = (4/3)(t^{1.5})$ from $t= 0$ to $t=2\pi$ can anyone ...
2
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1answer
29 views

Define a parametrization of the curve $C$

Consider the curve $C\in\mathbb R^3$ defined in Cartesian coordinates, by the equations $$x=\sqrt{4-y^2-z^2}\qquad y=z+1$$ from $P_0=(\sqrt3,1,0)$ to $P_1=\left(\frac{\sqrt6}2,\frac32,\frac12\...
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1answer
43 views

Application of winding number and the roots of complex polynomial from a non simple closed cuvre

There is a formula for the simple closed curve $\gamma(t)$ and complex polynomial $p(z)$. The winding number of $p(\gamma)$ around (0,0) is the sum of roots counting multiplicity of $p(z)$ within the ...
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2answers
41 views

Implicit form for a logarithmic spiral

I was wondering if there is an implicit form for a logarithmic spiral. For example, if $$ x=e^{-t}\cos(t)\\y=e^{-t}\sin(t)$$ we can write $x^2+y^2=e^{-2t}$ and $y/x=\tan(t)$ which yields $$x^2+y^2=...
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1answer
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Find three points of order two on elliptic curve.

Let $C$ be the cubic curve defined by $y^2z = x^3 -xz^2$ where $O = (0:1:0)$ is an inflection point. Find three points of order two in the group $(C, O, +)$. I know that $2\cdot P = O$ if and only if ...
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1answer
38 views

Minimum distance of the curve $xyz^2=2$ from its origin

Consider a $3D$ figure represented by $xyz^2=2$. Then what is its minimum distance from its origin? What I try: Given $xyz^2=2$ and I have to find minimum of $x^2+y^2+z^2$ Let $x^2+y^2+z^2=k^2$ ...
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0answers
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Formula for the osculating conic of a plane curve

A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the ...
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1answer
37 views

Linear map preserving area?

In Sean Dineen's book for multivariate calculus and geometry, there is an exercise which states: If $T:\mathbb{R}^n \to \mathbb{R}^n$ is a linear mapping such that $$\|T(x)\|= \|x \| \quad \forall ...
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2answers
525 views

What formula could mimic the following curve?

For the purpose of deforming a 3D mesh, I am looking for a formula to generate a curve I could evaluate like the following: Its shape would be more or less a simplified version of wind waves over an ...
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0answers
22 views

Prove that the curve $\alpha(t)$ is tangent to the $x$ axis.

I have the curve $\alpha(t):(-1,\infty) \rightarrow R^2$ given by $\alpha(t)= ((\frac{nat}{1+t^3}), (\frac{nat^2}{1+t^3}))$ with $n$ a natural an $a$ a constant both of them fixed. I need to prove ...
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0answers
18 views

Unfold a loop by perturbing a small amount in the direction of tangent vector

I am interested in designing an animation of unfolding a loop. I am trying to unfold it by taking a tangent at each point and then adding a small amount of $\delta$ in the direction of the unit ...
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0answers
21 views

Geodesic Co-ordinate Patches, proving $u^1$ curves are geodesic.

Suppose $\textbf{x}$ is a co-ordinate patch such that $g_{11} = 1$ and $g_{12} = 0$ at all instances, prove that the $u^1$ curves are geodesics. I can see that $u^1$ curves are unit speed, but I'm ...
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2answers
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About the slope of a parametric curve at a point. (James Stewart Calculus 8th Edition.)

I am reading "Calculus 8th Edition" by James Stewart. Suppose $f$ and $g$ are differentiable functions and we want to find the tangent line at a point on the parametric curve $x=f(t),y=g(t)$, ...
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2answers
121 views

Shortest distance between two points on the surface of a closed cylinder

What is the shortest distance between two points on the surface of a closed cylinder? I understand simple euclidean distance will work if both points are on curved surface, but I am looking for a ...
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Prove all the integral curves of a vector fld given by $\phi_1(x,y,z) = c_1$ $ \phi_2(x,y,z) = c_2$, where $\phi_1, \phi_2$ are first integrals of $V$

In a PDE book, I've come across the following theorem; Let $\phi_1$ and $\phi_2$ be two functionally independent (i.e $\nabla\phi_1 \times \nabla \phi_2 \ne 0$ on the given domain) first ...
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1answer
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Curves intersecting level surfaces at right angles

I've been trying to solve a question in my textbook for a little while now, and I can't seem to get the reasoning behind the answer. I'll quote the question, followed by my attempted solution, then ...
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1answer
22 views

Scale invariant curvature (plane curve)

Is there a form of curvature for a plane curve which is invariant under uniform scaling? Ideally, I am looking for a way to characterize the effective 'local eccentricity' of a plane curve so that [...
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2answers
49 views

Are curves just portions of connected circles of different radii? [closed]

I just had the question in my mind, can any curve be reduced to different portions of circumferences of circles with different radii, for example the curve of $\sin$ and $\cos$, and the curve of $y = ...
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1answer
42 views

Smoothing of a step function using smoothstep. (Curve fitting)

I was trying to smoothen the step function (zero when $x$ is less than $2/3$ and equal to $1$ when $x$ is greater then $5/6$) as in the picture below. Trying to fit $f$ in between $2/3$ and $5/6$ ...
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1answer
204 views

Strangely but closely related parametrized curves

Compare the following two parametrized curves for $k \in \mathbb{N}^+$: $$x_r(t) = \cos(t)(1 + r\sin(kt))$$ $$y_r(t) = \sin(t)(1 + r\sin(kt))$$ with $0 \leq t < 2\pi$ and $0 \leq r \leq 1$ (being ...
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Calculate circumsphere of truncated icosahedron with algebraically profiled faces

I have a truncated icosahedron with theoretical side lengths of $a$ units (if it were a perfect, flat shape). However, each face is deformed, as when viewed from the side instead of being a flat line ...
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2answers
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If the tangents are parallel at each point for two curves, then so do their principal normal and binormal vectors

In the book of Differential Geometry by Kreyszig, at page 103, it is asked that Problem 13.1: Given two twisted curves which are in a one-to-one correspondence so that at corresponding points ...
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1answer
34 views

Is it just repeat control points and knots if we want to redraw the curve repeatedly

Suppose I have a cubic B-spline curve, it has $57$ knots vectors and $53$ control points. The knot vector is like $(0,0,0,0,1,2,...,50,50,50,50)$ The curve is like this If we want to generate the ...
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Parametrization of special family of tori knots

Finding the parametric equations of an (a-c)tori knot knowing that one turn has the following parametric equation: $$\alpha(t)=\begin{pmatrix} x=\Big(R_1+R_2\cos(t)\Big)\cos\biggr(c\arctan\Big(\dfrac{...
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1answer
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If there are any curve databases with structured data

I found these curve lists: http://www.lmfdb.org/EllipticCurve/ https://en.wikipedia.org/wiki/List_of_curves http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html http://old.nationalcurvebank....
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1answer
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How to show that the curves $r=a(1+cosθ)$ and $r=b(1-cosθ)$ cut orthogonally?

I tried to do the math by multiplying derivatives with respect to r and theta of both equation and then adding it. But I am not getting zero as expected. I think my method is wrong then.
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0answers
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Relation between tame symbol and residue on a curve

For an discrete valuation field $K$ we can define the tame symbol: $$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$ Consider ...
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2answers
153 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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0answers
48 views

How to translate estimated model parameters when fitting centered and scaled data?

I routinely use a non-linear curve fitting tool to fit data according to a user prescribed model / function. One piece of advice that I commonly see around non-linear curve fitting is about data ...
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0answers
31 views

Describing curves of complex valued functions

I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions: 1.$f(z)=exp(z^2)$ 2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$ My thoughts: I can write down what the ...
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3answers
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A continuous curve intersects its 90 degrees rotated copy?

This is almost the same problem as in this question. However, the OP there was looking for a solution where we could assume any number of things, while I want to stick with just the given assumption (...
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2answers
40 views

Can i find a 3D function given some points?

is it possible to find a 3D function given a set of data points? i tried plane-fitting it did not work, too chaotic for a plane. I am trying to find a 3D equation that cover most of points, how can ...
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3answers
208 views

Will a path between $(x, y)$ and $(-x, -y)$ always intersect a 90 degree rotated copy?

Suppose we have a path between two points $(x, y)$ and $(-x, -y)$. If we rotate it by 90 degrees around the origin, will the copy intersect the original? (You can add any number of assumptions to ...
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1answer
46 views

Curvature inequality involving a Curve within a disk

If a closed plane curve $C$ is contained inside a disk of radius $r$, prove that there exists a point $p \in C$ such that the curvature k of C at p satisfies $\lvert k\rvert \ge$ $1/r$. I understand ...
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1answer
56 views

Elementary reference for existence of a tubular neighborhood of a $C^2$-curve (or elementary proof)?

Consider a 1-periodic function $\varphi \in C^2(\mathbb R;\mathbb R^n)$ with non-vanishing derivative and such that $\varphi|_{[0,1)}$ is a bijection. Let $$ N:=\{\varphi(0)+z\,|\,z\in\mathbb R^n, \, ...