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Questions tagged [curves]

For questions about or involving curves.

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1answer
21 views

Why does $\int_{C_M}{e^{iz}-1\over z}dz=0$ where $C_M$ goes through $0$?

I read an example evalutating $\int_{-\infty}^\infty {\sin x \over x}dx$ and the author defines a closed curve $C_M$ which is chaining $\Gamma_M:Me^{it}, t\in[0,\pi]$ with $[-M,M]$. He writes: ...
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1answer
27 views

What's the meaning of a derivative of a parametric curve?

A parametric curve $C$ can be defined as follows $$ C(p) = \{x(p), y(p) \}, \; p \in [0, 1] $$ where $p$ is the parameter. We can define the unnormalised tangent to the point of the curve ...
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1answer
30 views

Parametrize the surface

For the given curve $y^2-x^2=4, z=0, y>0$, parametrize the rectilinear surface that intersects the given curve and contains the point $(0,1,2)$. I found some solutions using vectors and stuff, but,...
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0answers
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Computing the intersection number of a line and a curve (Fulton Algebraic Curves 3.21.)

The problem is stated as follows: Let $F$ be an affine plane curve. Let $L$ be a line that is not a component of $F$. Suppose $L = \{(a+tb, c +td)\, | \, t \in k \}$. Define $G(T) = F(a+Tb, c+Td)$. ...
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1answer
21 views

Modification of the definition of basis function in open clamped B-Spline

An open B-spline is B-spline in which the end knots satisfy $t_0=t_1\cdots=t_d$ and $t_{m-d}=t_{m-d+1}=\cdots=t_m$. A minor modification of the definition of the basis function $$N_{i,0}(t)=\begin{...
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1answer
36 views

Is it possible for a finite integral closure of a DVR to not be a PID?

Suppose that we have a point $A$(local ring, DVR) of an abstract curve over $k=\bar{k}$ given by a field $k(X)$. Let $k(Y)$ be a finite extension of $k(X)$ and denote by $B$ the integral closure of $A$...
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3answers
59 views

Constrained parameters in least square curve fitting

I have some data points that need to be fit to the curve defined by $$y(x)=\frac{k}{(x+a)^2} - b$$ I have considered that it can be done by the least squares method. However, the analytical solution ...
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0answers
28 views

Intersection multiplicity and contact order of plane curves

A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then $$ I_p(X,...
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0answers
34 views

Transfer points on curve onto straight line: preserving distance

Please could I ask for advice. I'm a biologist so apologies if this is trivial. I have calculated the distances between points on two curved lines (an inner and outer part of my bacterial cell) and ...
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0answers
19 views

Best method of spline interpolation for any 2D curve

So I want to make a mobile app where you can draw lines and then interpolate them with a spline. I used this bezier spline method: https://gist.github.com/anonymous/06f4104d93f6cef6f341 But it draws ...
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0answers
27 views

Orthogonal monomials on a curve

Let $\Gamma \subset \mathbb{C}$ be a smooth curve such that monomials are orthogonal on it, i.e. with $n,m \in \mathbb{N} \cup\{0\}$ $$\int_{\Gamma} z^n \overline{z^m} |dz| = 0, \qquad \qquad \forall ...
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1answer
30 views

Why is the limit along the y-axis 0?

For a math course, my course book computes the limit of the function $f(x,y) = \dfrac{xy^2}{x^2+y^4}$ at $(0,0)$ along the $y$-axis (along $\mathbf{r}(t) = (0,t)$). It finds $\lim _{t \rightarrow 0}f(\...
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0answers
15 views

All the intersections between two plane curves

In commercial CAD software, one can find ALL the intersections of two plane curves very easily. As it is shown in this case, one of the curve is a B-spline and the other is a polynomial. Both of the ...
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1answer
58 views

How the curvature of a curve affects its behavior relative to a circle

I'm trying to prove Osserman's third lemma in his proof of the Four Vertex Theorem: Lemma 3. Let a smooth oriented unit speed curve $\gamma$ have the same unit tangent vector $\vec{t}$ at a point $...
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1answer
25 views

Question about the definition of involute of a plane curve

I'm studying about involute of a plane curve from here and there's a small point that is really bothering me and I can not understand it. Assuming curve is parameterized by arc length, the involute ...
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1answer
18 views

area under curve must be 1, function intersects with y-axis above 0

what is the best way to find a function that looks like a normal distribution, when the curve intersects the y-axis above 0 (say 0.3) and the area must be 1 (100%)? https://i.stack.imgur.com/5cJSy....
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2answers
22 views

How does replacing x with x-a shift a curved line to the right?

My professor just took up notes regarding curved lines and surfaces. In the curved lines section, he said a curved line on the 2D plane is an equation with x and <...
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0answers
43 views

Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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0answers
29 views

Help me to find a statistical rule/law which most closely matches a fixed set of numbers

I am hoping someone can help me with this problem! I have a distribution of 2000 numbers but I only know the first 10. The first 10 numbers are: 2025, 1000, 335, 300, 187.5, 135, 99.5, 20, 17.5, and ...
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0answers
48 views

How to select numbers from large set in such a way that they appear random or to have a neat pattern

So I am wondering about how to generate a subset $8^{10} \subset 8^{32}$ IDs. That is, there is a space of $8^{32}$ possible IDs, but I only want to generate $8^{10}$ of them right now. This shouldn't ...
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1answer
45 views

How to evaluate integral of (x - y)(dx + dy) with Green's Theorem?

I want to evaluate the integral $\int(x - y)(dx + dy)$ along curve C where C is the semicircular part of $x^2 + y^2 = 4$ above $y = x$ from $(-\sqrt2, -\sqrt2)$ to $(\sqrt2, \sqrt2)$ using Green's ...
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0answers
32 views

What is the definition of ''geometrically irreducible closed curve''?

In the algebraice geometry, one says about "geometrically irreducible closed curve" over field $k$. For example, the theorem 5.4.5 (pp. 147) of ''Heights in Diophantine Geometry'' of E. Bombieri wrote ...
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1answer
13 views

Determine the period of a curve

A curve is expressed with the two equations: $x=3+\cos(t)$ $y=4\sin(t)$ How can I find the period of this curve? I was thinking to get the period of $y = 4\sin(\arccos(t-3))$ but I'm not sure of ...
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4answers
77 views

Finding the area between two curves.

Context: High School question. Find the surface area between the curve of the function $y=6-3x^{2}$ and the function $y=3x$ in the interval $[0,2]$ My approach: -We must find the points of ...
2
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1answer
43 views

Proving product of torsions of Bertrand curves is constant and positive, and also why can`t a curve be a bertrand mate of itself?

A year ago I asked [this] (Proving a few properties of Bertrand curves) same question (without the "why can't a curve be a Bertrand mate of itself" part) - see that post if you want to know the ...
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0answers
17 views

Simple closed curves have the same trace if and only if they are equivalent

I'm very stuck on Exercise 1.35 (4) in Kristopher Tapp's Differential Geometry of Curves and Surfaces which reads: Show that two parametrized simple closed curves have the same trace if and only if ...
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0answers
46 views

Unraveling a tangle [duplicate]

Let $f: [0,1] \to X$ be a continuous function from the segment $[0,1]$ to a Hausdorff space $X$ such that $f(0) \ne f(1)$. Can we claim that there is always an injective continuous function $g: [0,1] ...
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0answers
38 views

How do I find the smallest circle enclosing a compact region?

The Four-vertex theorem states that any simple closed plane curve has at least four vertices, i.e $\kappa'(t) = 0$ where $\kappa$ is the signed curvature function. The proof given by Osserman here ...
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1answer
370 views

Weighting a cubic hermite spline

I am trying to figure out a function behind the software's curve drawing algorithm. Originally, each node comes with 3 parameters : time, value, and tangent. I have found that it fits cubic Hermite ...
2
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1answer
30 views

If a curve has unit speed, is the magnitude of its tangent and normal vectors equal to $1$?

If a curve has unit speed, is the magnitude of its tangent and normal vectors equal to $1$? I am having trouble seeing this. if r is the curve, then the tangent is $r'$. Also, normal vector is $r''/|...
2
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1answer
54 views

Help with approximation of the length of a curve in $\mathbb{R}^n$

I would like to have some hints on this exercise since I struggle to begin. Let $c : [a,b] \rightarrow \mathbb{R}^n$ be a curve such that $$ l(c) := \sup_{a<t_0<...<t_k<b} \left \{ \...
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1answer
58 views

Is a Weierstrass curve a topological manifold? [duplicate]

Obviously a Weierstrass curve is not a smooth manifold, but it seems like a Weierstrass curve should be a topological manifold (which I now see is a suspicion supported by this post), since it is a ...
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3answers
29 views

Trigonometric Curves, Finding Range

Sketch the graph of $y=2\sin x + 1$ for intervals $0° \leq x \leq 360°$. Hence state the range of values of $x$ in this interval which satisfies the inequality $2\sin x + 1 \geq 0$. The graph ...
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1answer
37 views

Secant variety of a curve

$\underline {Background}$: Let,$X$ be a curve in $\mathbb{P}^{2}$ which is not a line.We denote secant variety of $X$ to be $\sigma_2(X)$ $\underline {Question}$: To prove $\sigma_2(X)=\...
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1answer
88 views

A Jordan Curve Symmetric About the Origin Does Not Pass Through Origin

I've been thinking about this statement for a while, and I think it's true, but I'm not sure of how to prove it. The statement is A Jordan curve $J$ that is symmetric about the origin $p$ does not ...
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1answer
30 views

Tangent line of non-simple curves

I have a small confusion about curves. Given a non-simple parametric curve $\alpha : I \mapsto \mathbb{R}^{2}$. How does the tangent vector $T(s)$ and the tangent line $\{\alpha(s) + t T(s) : t \in \...
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1answer
32 views

Get points in the plane of an Euler spiral given by curvature

I am currently working on a visualization of a road network. I am using OpenDrive as my standard for road description. I now have a problem with visualizing the curved parts of roads. These are given ...
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0answers
25 views

Doubt about curves with the same curvatures

I am having some trouble with this theorem: In Klingenberg's "A course in differential geometry" on page 13 it claims: Let $c_1, c_2: I \to \mathbb{R}^n$ be two curves ($C^{\infty}$) ($n-1$)-...
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0answers
18 views

Does a curve with a box-counting dimension greater than 1 have to have infinite length?

If I have a curve that occupies a finite space (e.g. the unit square) and it has a box-counting dimension > 1, can it still have a finite length? If not, is there a proof of this?
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1answer
58 views

Why does a pointed surface minus a countable set of points contain a curve?

Let $S$ be a surface over $\mathbb{C}$ and let $s_1,\ldots, s_n$ be closed points of $S$. We consider this data as fixed. It is not hard to see that there is a curve passing through $s_1,\ldots,s_n$....
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1answer
130 views

A map is injective if it is nonzero at the generic point?

Proposition IV 2.1 in Hartshorne's states that if $f: X\to Y$ is a finite separable morphism of curves. Then $f^{*}\Omega_Y\to \Omega_X$ is injective. And he proves this by saying that it will be ...
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1answer
16 views

Directional Derivative in the direction in which $z$ is growing

Find the directional derivative of $f(x, y, z) = xy + 2xz - y^2 + z^2$, at the point $P = (1, -2, 1)$, passing through the curve $x = t, y = t -3, z = t^2$, in the direction in which $z$ is growing. ...
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1answer
36 views

Finding and minimizing the length of a string wrapped around a cylinder.

A string of length $l$ is wrapped around a cylinder of diameter $d$ and height $h$. The string does $n$ turns and starts at one end of the cylinder, ending at the top. The pitch of the resulting ...
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0answers
34 views

Curve length on a surface

Consider a surface with the equation $z=x^3 − 3xy^2$. So I have parametrization $p(u,v) = \begin{bmatrix} u \\ v \\ u^3-3uv^2\end{bmatrix}~.$ I have found that the first fundamental form of this ...
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1answer
36 views

Computing $(\frac{\alpha''(s)}{|\alpha''(s)|})'$

Let $\alpha:I\rightarrow\mathbb{R}^3$ be a curve parameterized by arc length $s$ with curvature $k(s)\ne0$ for all $s\in I$. I am attempting to compute $(\frac{\alpha''(s)}{|\alpha''(s)|})'$. This ...
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1answer
22 views

Arclength-Parametrized Space Curve Inequality

Give any arclength parametrized space curve $\alpha(s)$ (where space curve just means its codomain is $\mathbb R^3$), I want to show the following inequality: $$\lVert \alpha(s) \rVert ≥ \lvert \...
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0answers
5 views

Generating Linked Curves in 3-space.

I am working on a differential geometry project. I have created a program which reads in two closed curves in 3-space and uses a numerical integration to calculate the linking number. I would like to ...
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0answers
39 views

Finding Parametric Equation of Curve with some conditions

Let $S$ be a Sphere (in 3d space ,i.e. $\mathbb{R^3}$) and $\gamma : \mathbb{R} \to S$ be a curve that is parameterized by length. For all $t$ , we have $|\gamma''(t)| = k<1$ and $k$ is a constant. ...
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0answers
45 views

How to prove that Frenet frame is independent of the choices of parameters?

When I am reading ''A course in differential geometry'' of Klingenberg, I cannot be sure the Frenet frame defined in this book is independent of the choice of parameter of a curve. As a result, the ...
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0answers
25 views

Topology of uniform convergence

Let M be a connected smooth manifold and $x_0 \in M$. Let $P(x_0,M)$ be the space of all continuous curves $\gamma:[0,1] \rightarrow M$ starting at $x_0$. I wonder what the topology of uniform ...