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Questions tagged [curves]

For questions about or involving curves.

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2answers
348 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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0answers
13 views

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
31 views

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
26 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
52 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
96 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
219 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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64 views

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
49 views

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
45 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
48 views

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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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
183 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
51 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
35 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
191 views

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
44 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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240 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
73 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, \, ...
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0answers
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If $x \in (\varphi(0) +\langle \varphi'(0)\rangle^\bot)\cap B_\delta(\varphi(0))$, do we have $|x-\varphi(0)|\le C\cdot |x-\varphi|$?

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

Surface area of ellipsoid created by rotation of parametric curve

I have a parametric curve (elipse) defined as follows $$\begin{aligned} x(t) &= \cos(t)\\ y(t) &= 2 \sin(t)\end{aligned}$$ and need to calculate the surface area of the ellipsoid produced by ...
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0answers
18 views

Total curvature of sine graph

We have to compute the total curvature of $\gamma(t)=(t,\sin(t))$ in the intervall $\lbrack 0,2\pi\rbrack$. We defined the total curvature of $\gamma$, parametrized over the interval $\lbrack a,b\...
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0answers
61 views

How to derive $(x,y)$ of clothoid curve from $rl=A^2$

Clothoid curve is described by $rl=A^2$, where $r$ is turning radius and $l$ is length from the origin. (This definition is very intuitive to understand for me) Also it's known that the coordinate $(...
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1answer
73 views

Shortest distance between any two curves

In general, if given equation of any two curves, how to find the shortest distance? According to me, finding common normal won't work as it isn't necessary for both of them to have one like in case ...
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0answers
22 views

How does one represent generalized polynomials in Conformal Geometric Algebra C(4,1)

I am interesting in representing arbitrary curves using conformal geometric algebra. I have a special interest in spatial loops. Also, I would like to represent characteristic polynomials of matrices ...
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2answers
24 views

Rewriting $|z+2i|=\sqrt{2}|z+1|$ in $x+yi$ form with $x,y\in{}\mathbb{R}$

According to my prof, the above equation is equivalent to $$x^2+(y+2)^2=2((x+1)^2+y^2)$$ I don't see how he could come to this conclusion, any insight/help would be highly appreciated. Edit: It's ...
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0answers
17 views

Eliminating minor crests/troughs in a tidal curve

I have some tidal data, I only have the turning points of the curve. I want to eliminate the minor troughs/crests so I only get the high and low tide without the minor variations. The curve Is ...
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1answer
20 views

Frenet Frame along a curve and Riemannian Curvature on $S^2$

I would appreciate some help showing the following statement. Let $\omega: [0,1] \rightarrow S^2$ be a smooth curve with velocity vector $V = \omega'$, speed $v = |V|$ and Frenet frame $\{T,N\}$. ...
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0answers
38 views

How can i show a curve is smooth

we have a curve:$$x=f_1(t),y=f_2(t)$$$$t\in I[a,b]$$How can i show that this curve is smooth?$$$$ So far what I read is that the curve must have vertical tangent and perpendicular tangent. But I don’...
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1answer
33 views

Calculate: $\int\int_D e^{-(x^2/a^2 + y^2/b^2)}dxdy$

Calculate: $\int\int_D e^{-(x^2/a^2 + y^2/b^2)}dxdy$ on the exterior of the $x^2/a^2+y^2/b^2=1$, how do I get the exterior of this? I know the parametrization is: $$x=ar\cos t$$ $$y=br\sin t$$ and ...
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0answers
91 views

Calculate the circulation of the field $V(x,y) = \frac {3x^3+xy^2}{\sqrt{x^2+y^2}}i+\frac {3y^3+x^2y}{\sqrt{x^2+y^2}}j$

Calculate the circulation of the field $V(x,y) = \frac {3x^3+xy^2}{\sqrt{x^2+y^2}}i+\frac {3y^3+x^2y}{\sqrt{x^2+y^2}}j$ on the curve $r=a(1+cost), t\in [0,\frac {\pi}2]$. I know the formula for the ...
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1answer
14 views

$\int_C ydx+z^2dy+xdz$ on a specific curve

Evaluate $$\int_C ydx+z^2dy+xdz$$ on a specific curve, the intersection between $z=\sqrt{x^2+y^2}$ and $z=6-(x^2+y^2)$. I don't know how to parametrize this, also it seems wrong to me if I ...
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1answer
33 views

how do i show that a curves have no double point or have only 1 double point. (Double point mean a point in which the curves cut it self)

I have 2 problems in the theme double point that I didn't understand. 1)How can I show that a curve has no double point at all? 2)How can I show that a curve has only 1 double point( or more if needed)...
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1answer
25 views

Differential map of velocity vector

This is a very basic differential geometry question (please be patient, I am learning) I am given the definition of the differential map of $\phi:M \to N$ as $$d\phi_p(v)(g)=v(g\circ\phi)$$ where $v\...
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0answers
76 views

Calculate $\int_C \frac {x^3dy-y^3dx}{x^{5/3}+y^{5/3}}$ where $C:x^{2/3}+y^{2/3}=a^{2/3}$ in the first quadrant

Calculate $\displaystyle\int_C \frac {x^3dy-y^3dx}{x^{5/3}+y^{5/3}}$ where $C:x^{2/3}+y^{2/3}=a^{2/3}$ in the first quadrant. My attempt: Parametrization: $$x(t)=a\cos^3t$$ $$y(t)=a\sin^3t$$ Then ...
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3answers
50 views

Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$

Compute $\int_C ze^{\sqrt{x^2+y^2}} \mathrm ds$ where $$C:x^2+y^2+z^2=a^2, x+ y=0, a \gt 0$$ At first I thought to parametrize this as: $x=a \cos t , y=a \sin t, z =0$, but then the integral ...
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2answers
38 views

Compute $\int_C x\sqrt{x^2-y^2} \mathrm ds$

Compute $$\int_C x\sqrt{x^2-y^2} \mathrm ds$$ where $C:(x^2+y^2)^2=a^2(x^2-y^2)$ with $x\geq 0$. My attempt: For this I tried the parametrization: $x=r\cos t,y=r\sin t$, but it doesn't seem to ...
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2answers
45 views

Integral on a curve $\int_C (x^2+y^2)ds$

Calculate $\int_C(x^2+y^2)ds$ where $C$ is the segment $[AB]$, $A(a,a)$, $B(b,b), b > 0$. How could I parametrize this curve and can you also explain your thinking process while doing so? UPDATE: ...
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1answer
129 views

'Spiky Periodic Things' - Do these objects have a name, and is there a method for finding the boundary curves?

This question was originally about evaluating the sum $\sum_{n=0}^\infty e^{nix}$, but I figured out the answer about half way through writing it. So instead, I decided to ask a slightly different ...
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1answer
27 views

Function $ f(x,y)=x+\frac{y^3}{3} $ cut the xy-plane in a cutting-curve $h$. Find the tangent fuction of h in point$ P(9,-3)$.

$$ f(x,y)=x+\frac{y^3}{3} , D(f)=\text{(x,y)}\in R |(x^2 +y^2\le2) $$ So what i thinking here is find the function h, then find the tangent. But i dont know how to do or is there another way? Thank ...
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1answer
29 views

Geometric way to view affine connection and parallel transport.

Given a parametrized curve $\gamma$ on a manifold $M$ with metric $g$ and some affine connection on it, we can transport vectors of tangent spaces $T_{p} M$ and $T_{q}M$ to each other (when $p, q \in \...
2
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2answers
49 views

A curve is defined by the parametric equations $x=2t+\frac{1}{t^2},\; y=2t-\frac{1}{t^2}$. Find the Cartesian equation.

A curve is defined by the parametric equations $$x=2t+\frac{1}{t^2}$$ $$y=2t-\frac{1}{t^2}$$ Show that the curve has the Cartesian equation $(x-y)(x+y)^2=k$ So I understand I need to eliminate ...
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2answers
41 views

Adding changes or derivatives between two points

I've playing with the limit definition of derivative and I've to somewhat confusing conclusions. To clarify, I'm from an Engineering background so I don't think that an instantaneous rate of change ...
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0answers
50 views

How do I find the area bounded by a $y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$

I am having difficulties with this problem:$y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$. Graphng two of the functions I get the following: Graphing both functions shows that they intersect at $...
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1answer
25 views

Show that the following map is surjective

Situation: Consider the curves $ C_1 = x^2 + y^2 +x^2y^2 = 1 $ and $C_2 = w^2 = 1 - z^4 .$ Define a map $\lambda$ from $C_1$ to $C_2$ with $\lambda(x,y) = (x,(1+x^2)y)$. This map should be ...
0
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1answer
36 views

If there is anything more general than NURBS

NURBS is a generalization of B-splines. I have also heard of T-Splines which potentially sounded like generalizations of NURBS, or more general than them at least. Wondering if there are anything more ...
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2answers
93 views

Understanding Bezier Curves vs. Circular/Elliptical/Other Arcs

From what I've read, you cannot construct an elliptical or circular arc with a single bezier curve (though I read maybe you can if the arc is less than 1/4 of a circle or something small like that, or ...