Questions tagged [curves]

For questions about or involving curves.

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Definition of curves with two variable

I have a question regarding a diagram that I have made. It reflects measurement results. For different duty cycles, a gain factor was measured. The time reflects the turn-on time of transistors. I ...
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34 views

derivative of composition curve

a quick question: Let $(M,g), (N,h) $ be pseudo-Riemannian manifolds, $\gamma:I \rightarrow M$ a curve. $\gamma^{'}(t_0):= d \gamma \dfrac{\partial}{\partial t} |_{t_0}$ Let $F:M \rightarrow M$ be ...
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1answer
36 views

Understanding of differential of a curve as a vector field

I have trouble understanding the definition for a geodesic curve: Let $\gamma:I \rightarrow M$ be a curve, $M$ pseudo.Riemannian manifold. Then $\gamma$ is geodesic, if $\nabla_{\gamma^{'}} \gamma^{'...
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Why (possibly) are all of these integrals either $\frac \pi2$ or $\pi$?

I have encountered with a rather interesting vector field while studying Green's Theorem: $$\vec{F}(x,y)=-\frac{y}{x^2+y^2}\hat{i}+\frac{x}{x^2+y^2}\hat{j}$$ This field turned out to be quite special ...
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2answers
72 views

How to solve a differential equation when the right-hand side of the equation is a non-parametric curve?

I just asked another question (see here: How to calculate the shape of a curve given y coordinates and slope?) and was advised by the user who answered my question to ask a new question. I would ...
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1answer
37 views

How to calculate the shape of a curve given y coordinates and slope?

I apologise in advance if my description of the problem does not use the correct terminology but I'm still learning! Let me know if something is ambiguous or not clear and I'll try to rephrase it. ...
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34 views

How to draw astroid given by Parametric equation $a(t)=(\cos^3(t),\sin^3(t) )$

I wanted to draw Asteroid with parametric equation $a(t)=(\cos^3(t),\sin^3(t) )$ I know that $x^{2/3}+y^{2/3}=1$ is equation . Also $x^2+y^2=1$ is equation of circle. By using online app I had ...
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49 views

Determine value for which a curve is geodesic

Let $(N,g_N)$ be Riemannian manifold, $I \subset \mathbb{R}$ open with coordinate $r$ and $M:= I \times N$ with metric $g=dr^2+f^2 g_N$. Now let $\gamma:J \rightarrow N$ be a geodesic curve. I need ...
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1answer
22 views

Using Green's theorem to compute integral on curve

Prove: $$\ \int_C (\sin x - y^2)dx +(x-y \tan^{-1}(y^2))dy = 2.4 $$ where $\ C $ is the curve from $\ (1,2) $ to $\ (-1,2) $ on $\ y = x^2 + 1 $ Using green's theorem $$\ \int \int_D (Q_x - P_y)dx ...
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1answer
27 views

Find the points of the trajectory in which the velocity is 9

I have a parametric curve given by $M:[0,2]\rightarrow\mathbb{R}; x(t) = 3t^2, y(t) = t^3,z(t)=6t$. I need to find the points of its trajectory in which the velocity is 9. I also need to represent ...
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Weird Notation in Differential Geometry (Variational Vector Field)

I can't seem to understand what my professor wants to say with his notation. So I hope that some of you can help me. We're looking at some variational curve $\gamma_t : I \to \mathbb{R}^2, \ t \in (-\...
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1answer
27 views

Regular homotopy between a circle and an oval

How do I build a regular homotopy between an oval $$\frac{x^4}{a^4}+\frac{y^4}{b^4}=1$$ and a circle $$\ (x-x_0)^2+(y-y_0)^2=R^2$$? I know I need to find parametrizations for both curves and there ...
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1answer
24 views

How do you render the union of a simple closed curve and its interior?

In this question there is confusion about the answer because it appears (to me, anyway) that the origin was supposed to be rendered as not on or inside a simple closed curve; but the question is ...
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2answers
52 views

Extremum of ln(x)/x = C (analytical)

I want to find the maximum value of y of the following equation: $A\cdot \sqrt{x^2 +y^2} = e^{\frac{\sqrt{x^2 + y^2}-x}{B}}~(1)$ I tried to use polar coordinates with $x = r \cos(\phi)$ and $y=r \...
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Vertical curve in a Riemannian warped product is a geodesic

Let $(N,g_N)$ be Riemannian manifold, $I \subset \mathbb{R}$ open with coordinate $r$ and $M:= I \times N$ with metric $g=dr^2+f^2 g_N$. Let $c_p: I \rightarrow M, (t \mapsto (t,p))$ for a fixed $p$ ...
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There are no non-constant morphisms from $\mathbb{A}^1$ to a cubic curve

Consider the cubic curve $X$ defined, over an arbitrary field $K$, by the equation $y^2 =x(x−1)(x−\lambda)$ in $\mathbb{A}^2$, where $\lambda \neq 0,1$. Show that there are no non-constant morphisms $...
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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
31 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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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
25 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
39 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
104 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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33 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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35 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
31 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
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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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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
59 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
28 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
25 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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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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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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49 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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34 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
14 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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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 ...
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1answer
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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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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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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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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
387 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 ...
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1answer
32 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''/|...
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1answer
59 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
59 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
31 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
89 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 ...