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Questions tagged [parametrization]

For questions on parametrization, the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

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50 views

Parameterization of a part of a cylinder

I have to find the parameterization of the surface that is part of the cylinder $$y^{2}=2-x$$ for $x\geq0$, bounded by the cylinders: $$y^{2}=z\quad\text{and}\quad y=z^{3}$$ for $0\leq y\leq1$. I ...
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2answers
45 views

Intersection of a sphere and a plane.

I have a curve: $\gamma =:[ x^2+y^2+z^2 = 1, x+y+z=1] $ How I can parametrize it? Or write it out in explicit form? I need this for compute the integral $\iint_{S}dS$, where $S$ is a surface of ...
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0answers
61 views

Parametrisation of a general quadratic surface?

Consider a general quadratic surface in implicit form: \begin{equation} ax^2 + by^2 + cz^2 + 2exy + 2fyz + 2gxz + 2lx + 2my + 2nz + d = 0 \end{equation} I can parametrise this in the form $f(x, y, z(...
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1answer
21 views

Calculation of Line-integral: parametrization problem

Calculate $$\int_Cxy^2\ \text{d}y-yx^2\ \text{d}x$$$C=\{(x,y)\in\mathbb R^2:x^2+(y-1)^2=1\}$ using Green's theorem: $\int_{C}P\ \text{d}x+Q\ \text{d}y=\int_D(\frac{\partial Q}{\partial y}-\frac{\...
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1answer
42 views

Relationship between tangent normal of parametric curve and range of parameter

It is intuitive that by the right-hand rule one can choose direction of motion along a parametric curve by selecting one of the unit normal vectors. However, the relationship to the parameter is less ...
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2answers
46 views

Parametrisation of this triangle

I have been doing an exercise problem, and there is something unclear about it. Namely, how was the triangle in the following example transformed to a circle $C'$, which is parametrised by $(x(t), y(t)...
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1answer
31 views

Volume / Surface of a Paraboloid through Integration

Given is a Paraboloid delimited as following: $$z_1 = a(x^2 + y^2),\ z_2 = h $$ That's my try for the Volume computation: First I find the radius of the circle resulting from the intersection ...
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1answer
143 views

Parametrization of parabolic hyperboloid.

Please, help me! How I can parametrize this surface: I have a parabolic hyperboloid. $ H := (z = xy) $ intersected by a cylinder whose base is a unit circle centered at the point $(0, 0, 1)$ I tried ...
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1answer
60 views

Parametric equation of a rotated circle in 4 dimensions

I'm attending a differential geometry course, and I'm stuck at one part of a question that we've been asked. The following rotated circle is given in 4 dimensions: $$x_1x_3 + x_2x_4 = \frac{1}{2}$$ ...
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3answers
130 views

Parametrizing a set

I'm trying to parameterize the following set $$ \left\lbrace\, r\left(\cos\left(\alpha\right),\sin\left(\alpha\right)\right) \;\middle\rvert \quad 0\leq\alpha\leq2\pi, \quad 0<r\leq 1,\quad \tfrac{...
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1answer
34 views

Surface area of paraboloide inside sphere

Find the surface area of the part of the paraboloide $z=\frac{x^2+y^2}{2}$ inside the sphere $x^2+y^2+z^2=3$ Setting $2z = x^2+y^2$ I obtain that the points of intersection between the paraboloide ...
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0answers
39 views

Parametrization of graph (arc length)

I've got a curve in $\mathbb R^2 : (x(s) , y(s)) $ ($s \in (0,L)$). For the tangent, it holds: $t(s) = \left( \matrix{x'(s) \\ y'(s)} \right) $. Since we have arc length, it holds also $||t|| = 1$. ...
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3answers
91 views

Parametric Curve Derivative Formula Misunderstanding

I was reading this site on determine the derivative and tangents to a parametric curve. However, he begins by saying: "suppose that we were able to eliminate the parameter from the parametric form ...
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2answers
63 views

How do I compute $\int_{0}^{2\pi} \frac{x(t)x'(t) + y(t)y'(t)}{x^2(t) + y^2(t)}dt$?

I'm a little rusty on this stuff, so bear with me. Let $F(t) = (x(t), y(t))$ be a closed continuously differentiable curve in $\mathbb{R}^2 \setminus \{(0,0) \}$. How do I compute the following ...
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1answer
47 views

Prove that $\varphi(s):[0,l]\rightarrow {\mathbb{R}^{3}}$ is a regular parametrization of the curve $C$

Given the curve $C$ simple, smooth and $\gamma:[a,b]\rightarrow {\mathbb{R}^{3}}$ regular parametrization of the given curve. For each $t\in[a,b]$ let $g(t)$ the arclength curve between two points $\...
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1answer
92 views

What does the question mean by 'find all the possible values' of $\int_{\gamma}\frac{1}{z}$

I'm working on some past exam papers and I wanted to see if I'm thinking of the following in the correct way; Given a contour integral $\int_{\gamma}\frac{1}{z}dz$ I want to find all of it's possible ...
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2answers
54 views

Determine the arc length of the following parametric curve

Determine the arc length of the parametric curve given by the following parametric equation: $φ(t)= (\sqrt{t}, t+1, t)$ $t\in[10,20]$ In order to do this I simply tried it to solve it by the ...
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1answer
41 views

Parameterization of a path to the summit of $z=1-x^2 -y^2$

I'm trying to parameterize a path to the summit and a path along the level surface $x^2+y^2=1$. Let's say i start from the point $A=(-1,0,0)$ and i want to end up at the point $B=(1,0,0)$ by taking ...
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1answer
29 views

Minimizer of parametrized convex function

So let us consider the following minimization problem parametrized in $z$ \begin{align*} \min_{x \in \mathbb{R}} L_z(x) = \frac{1}{2} \left(x - z\right)^2 + |x| \end{align*} The goal is to then find ...
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1answer
48 views

Trouble getting to Mercator Projection 'parametrisation' of the Sphere

I am trying to show that the parametrisation $$\varphi(u,v) = (u \cos (v), sech (u), \tanh (u))$$ of the unit sphere $S^2$ (the Mercator's Projection) can be given by the change of coordinates $\log \...
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0answers
47 views

What is the Relationship Between a Sand Pendulum and Infinite Series?

I'm working on a pendulum that drops a stream of sand as it swings, similar to the one in this video. How could this pendulum be modeled or represented by infinite series? My idea so far is to use ...
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3answers
91 views

Find the parametrization of the curve resulting from intersection of a function and a curve

I have the following function $f(x,y) = 2-x^2-4y^2$ and the surface $2x+4y+z-1 = 0.$ How do i go about finding the parametrization of the curve resulting from intersection of these surfaces? I see ...
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1answer
46 views

Evaluate Surface integral: $\vec F=y{\hat i}+x{\hat j}+zy{\hat k}$, S is part of $z=x^2+y^2$ above $z=1$. Assume S has an upwards orientation.

I want to evaluate ${\int}{\int_S}{\vec F}\cdot d{\vec S}$ with the given information in the title but cannot for the life of me figure this one out. I have looked at Stoke's theorem, and also the ...
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1answer
51 views

Parametric formula for the tangent line of the curve $x(t) = \cos(t) $, $y(t) = 1 + \sin(t)$, at $(x,y)=(\frac{\sqrt3}2,\frac32)$

Suppose you have the following parametric equations for t $\in [0,2\pi]$: $x(t) = \cos(t) $, $y(t) = 1 + \sin(t)$ Give the parametric formula for the tangent line of this curve at $(\frac{\sqrt{3}}{...
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1answer
40 views

Parametrization for $(x^2+y^2)^3=x^2y$

Greetings I want to find the surface given by the curve $\Gamma:(x^2+y^2)^3=x^2y\,$ The graph looks pretty nice( see the picture bellow) and in my book it says that after the parametrization the Area ...
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1answer
331 views

Find the area of the cap cut from the sphere by the cone implicitly and explicitly

Find the area of the cap cut from the sphere $x^2 + y^2 + z^2 = 2$ by the cone $z = \sqrt{x^2 + y^2}$. Doing this implicitly is straightforward, but I'm wondering what I'm doing wrong when I try to do ...
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1answer
192 views

Intersection between a cylinder and a plane

I have to find the intersection between these two surfaces in $\mathbb{R}^{3}$: this cylinder $$x^{2}+y^{2}-8x-8y+28=0$$ and this plane: $$x-y=0$$ and then find a parametric curve $\gamma(t)=(x(t),y(t)...
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4answers
88 views

Curve dense inside the unit circle

For $\alpha$ a real, irrational number, I have been to prove that any point $(x,y)$ such that $x^2 + y^2 \leq 2$ can be written as $$(x,y) = (\cos( u) + \cos( \alpha u), \ \sin( u) + \sin(\alpha u)) $$...
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1answer
35 views

Questions on arc length parametrization

Based on this page, suppose we have the following parametric curve: $$v(t) = (cos(t), sin(t), t)$$ The arc-length parametrization would be: $$\tilde{v}(t) = (cos(\frac{t}{\sqrt{2}}),sin(\frac{t}{\...
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1answer
22 views

Provide parametisations for the triangle with vertices $1, 2i$ and $-1$

Sketch these paths and provide parametisations for them: The straight line from $z=0$ to $z=2$. My idea: From the logic that a line from $p$ to $q$ in $\mathbb C$ can be parametised as $...
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1answer
81 views

Can you give a piecewise smooth parametrization of the astroid

The astroid is given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$. It is not smooth as the derivative of the function at $0, \frac{\pi}{2}, \frac{3\pi}{2}$ and $2\pi$ is $0$. However, is it ...
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1answer
77 views

Question on equivalent parametric curves

I'm confused about the concept of equivalent parametric curves. Based on my understanding, two parametric curves, $\phi$ and $\psi$, are equivalent, if there is a strictly monotonically increasing ...
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1answer
19 views

Finding a path of a particle given that it moves parallel to $-\nabla f$

I am told that a particle starting off at $(-1,1)$ moves parallel to $-\nabla f$ where $f(x,y)=x^2-3y^2$. So $-\nabla f=(-2x,6y)$. The question wants the path of the particle and I considered ...
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3answers
266 views

If I have an oblique cylinder can I trim it in to a rectilinear cylinder?

There was an interesting conversation on twitter today about the net of an oblique cylinder. I misunderstood the question and produced a net of a right cylinder sliced by a plane with the equation $ax+...
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2answers
38 views

How to make the angular parameterization of $x^2+y^2=1$ unique?

If there exists a relation $x^2+y^2=1$ between two real variables $x$ and $y$, one can always make the parameterization $x=\pm\cos\theta$ and $y=\pm\sin\theta$. Can we make the parameterization unique ...
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1answer
16 views

Surface intersection equation

I am required to find the vector function of a curve which is formed when these two surfaces intersect. A hint would be more than enough.
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1answer
15 views

Showing torsion of a curve is constant

I tried to simply find the torsion first by computing $- \frac{{\mathbf N\times B'}}{||\mathbf{r'(t)}||} $ and I was hoping to obtain a constant, but the algebra was too messy and so gave up on that ...
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0answers
23 views

Equality between two parameterizations of the same function

If I have a function $y = f(x)$, and I parameterize it so that I get two new functions $x(t)$ and $y(t)$, I'm curious about when I can say that two distinct parameterizations are equal, and in what ...
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0answers
96 views

Rational parametric equations of a conic

By a linear transformation, a conic known by its implicit equation can be put in a reduced form such as that of a unit circle $x^2+y^2=1$. The latter can be represented by rational parametric ...
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1answer
150 views

Is Reparametrization trick the same as Normal distribution?

Reparametrization trick replaces $z\sim N(\mu, \sigma)$ with $z=\mu + \epsilon\sigma, \ \epsilon \sim N(0, 1)$ for backpropagation. Intuitevely, I can imagine that this is related to sampling from $N(...
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1answer
97 views

What is the parametrisation of the flat torus: $\mathbb{T}^2 = \mathbb{E}^2/\mathbb{Z}^2$?

Is the parametrization of the flat torus $$\mathbb{T}^2 = \mathbb{E}^2/\mathbb{Z}^2$$ just the set of charts $$(\pi,U_{ij})\quad \pi(x)=x\quad U_{ij}=(i,i+1)\times(j,j+1)\quad i,j\in\mathbb{Z}^2$$ ...
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1answer
90 views

Proving an integral using parametrization and Cauchy's Integral Formula

Prove that $$\int_0^{\infty} {e^{-t^{2}}}\cos(t^2) \, dt = \frac{1}{4} \sqrt{\pi} \sqrt{1+\sqrt{2}}$$ by integrating the function ${e^{-z^{2}}}$ in the counterclockwise direction around the boundary ...
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1answer
45 views

Extrema and Critical Point for Multi variable Functions

Find the extrema of $F(x,y) = x^2 + xy + x + y^2$ on the set $S=\lbrace (x,y): x^2+y^2\le9$$\rbrace$. a): Find all critical points on the interior of $S$ b). Parameterize the boundary of $S$ in ...
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3answers
80 views

Parametrizing $x^2(x^2+y^2)=4(x-y)^2$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$x^2(x^2+y^2)=4(x-y)^2$$ is converted into an explicit function of the parameter $t$ that can be analysed ...
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0answers
44 views

Contour integration along a broken line giving different answer when compared with non broken line

From my limited knowledge of contour integrals I believe I understand that if $$\int_{C_{1}+C_{2}}f(z)\:dz=\int_{C_{1}}f(z)\:dz+\int_{C_{2}}f(z)\:dt$$ where $$\int_{C}f(z)\:dz=\int_{C}f(\gamma(t))\...
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1answer
59 views

What is $\int_\gamma \frac{x\,dy-y\,dx}{x^2+y^2}?$

Let $\gamma$ be the oriented piecewise $\mathcal C^1$-arc consisting of the line segment from $(1,0)$ to $(0,1)$, followed by the line segment from $(0,1)$ to $(-1,0)$. What is $$\int_\gamma \frac{x\,...
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1answer
66 views

Parametrizing $(2x+y)^2(x+y)=x$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using ...
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2answers
258 views

How to calculate surface area of the intersection of an elliptic cylinder and plane?

Given the equations $x^2+2y^2 \leq 1$, and $x+y+z=1$, how do I find the surface area of their intersection? I approached this question by first parameterizing the equation for the elliptic cylinder. ...
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1answer
34 views

Moving Along An Arclength-Parameterized Path By Altering Rate of t's Change

This is a problem that originally stems from code I'm working on, but I've tried to de-computer it as much as possible. I've got a Bezier curve represented by 4 points: $P_0$ is the start point, $P_1$...
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2answers
29 views

Parameterizing a non-centered circle

So basically I am stuck with parameterizing a curve. Half of a unit circle is centered at $(1,0)$ in the first quadrant and traced clockwise from $(0,0)$ to $(2,0)$. I am not so sure how to ...