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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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complex line integration $\int_{\gamma}zdz$

enter image description here I am confused with $y$ parameter and the one in $z=x+iy$. I know how to perform line integrals. But I am not sure how to compute parts i and ii I know the third part... ...
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Arc length parametrisation of a logarithmic spiral

$$\gamma(t)=(ae^{bt}cos(t),ae^{bt}sin(t),0)$$ (a) Find the arc-length parametrisation. So I think that the parametrisation is $\gamma(s)= \bigg((\frac{bs}{\sqrt{b^2+1}})cos(\ln\bigg(\frac{bs}{a\sqrt(...
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Parameterize union of two shapes in 3space

I am supposed to parameterize the union of the two shapes $x^2 + y^2 = 1, z = y$. I do not even know how to get the union of the two shapes. When I graph the two shapes the intersection does not ...
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Kinematics and the parametrization of a curve

An object moves along a path given by the equation $$ y(x)=2x^{2}-3x-11$$ with a constant speed of 5m/s. Find the velocity at x=2. My approach: We know that the speed of the object is given by $$ 25=...
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Find a parametric equation of the parallelogram two of whose lines are connecting $(1,-2,1)$ with $(1,4,3)$ and $(1,-2,1)$ with $(2,3,-1)$

I first try to find a parametrization of the plane passing through the points $A(1,-2,1),B(1,4,3)$ and $C(2,3,-1)$. We can take $\overrightarrow{AB}$ and $\overrightarrow{AC}$ as two linearly ...
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58 views

Hugoniot Locus given by parametric curves

I need to prove that the Hugoniot Locus of a point $\hat{u}$ of the equation $$u_t + f (u) _x = 0,\qquad f\in C^2$$ is the set of $n$ curves $$\begin{cases}\tilde{u}_p(\xi, \hat{u})=\hat{u}+\xi r_p(\...
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Finding cartesian equation of curve with parametric equations

A curve has parametric equations $x=a \sin(⁡t)+b \cos(⁡t)$ $y=a \cos⁡(t)-b \sin⁡(t)$ How do I eliminate t to find the Cartesian equation here? I've tried different weird approaches, i.e. squaring ...
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Triangle parameterisation

I get how to answer the qs below, the problem is actually finding path $2$ $ \left( 2, 0, 0 \right) $ to $ \left( 0, 1, 0 \right) $ I get $(2-t)i+tj$ yet the answer for path 2 is... $$ (2-t)i+(t/2)...
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parametrical representations of polygons

Could you please explain, how one gets this Parametric representation of a solid trapezoid ? I mean the procedure and not the answer. I have some linear geometry (as polygons), and I need to represent ...
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Does conjugate prior for natural exponential family needs jacobian to transform natural parameter back to original parameter?

From bayesian theory, we have that if $f(x|\eta) \propto \exp(\eta \cdot T(x)- A(\eta))$ - a natural exponential family, then the prior conjugate of $\eta$ is $\pi^*(\eta | \mu, \lambda) \propto \exp(\...
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2D curve with two parameters to single parameter

I have been thinking about the following problem. I have a curve in 2D space (x,y), described by the following equation: $$ax^2+bxy+cy^2+d=0$$ where $a,b,c,d$ are known. It is obvious that it is a ...
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1answer
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Parametric and vectorial functions

I need some help trying to do the following: Given $$L: (x,y)=(\sin t , 1+3\sin t)~, \quad 0<t< \pi $$ Find $a, b, c$ and $d$ so that $$L_2: (x , y) = (-2,-5)+u(a;b)~, \quad c<u<d$$ ...
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Parametrizing a segment

I was solving some algebra problems but I don't know how to do one of them. The task says "Determine three parametric equations of a segment HL, with H=(-3;5) and L=(2;-7) using a direction vector of ...
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Parametrize the line in the $R3$ which passes through the points

The two points are: $p = \left[ {\begin{array}{cc} -2 \\ 1 \\ 3 \end{array} } \right] q= \left[ {\begin{array}{cc} 5 \\ -3 \\ -1 \end{array} } \right]$ The following formula ...
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1answer
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Question about loops(closed paths) in $\mathbb{S}^{1}$

Let $$\alpha \left(s \right) =\left( \cos{2\pi s},\sin{2\pi s}\right)$$ and $$\beta \left(s \right) =\ \left( \alpha \land\left( \alpha \land \overline{\alpha} \right)\right)\left( s \right)$$ with $...
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How to parameterize this surface: $x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+x_{2}^{2}$ s.t. $0<x_{1}^{2}+x_{2}^{2}<R^{2}$?

The following equation represents a surface in $\mathbb{R}^{4}$, that is a 3-dimensional manifold: $$ x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+x_{2}^{2}\qquad\text{s.t.}\qquad 0<x_{1}^{2}+x_{2}^{2}<R^{2} $$...
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Using the Jacobian matrix to calculate a parametrized area?

$ x = \cos(\theta)r\\ y = \sin(\theta)r\\ z = r $ I have done this parametrization, and now I want to integrate to get the area of a cone. According to my textbook, I'm supposed to multiply with the ...
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Parameterisation of $B_r(x_0) \cap S$, $x_0 \in S$, $S$ a hypersurface

So I have a $(d-1)$ dimensional, $C^1$ hypersurface $S$ and $x_0 \in S$. The claim is: For $r>0$ small enough, the set $B_r(x_0) \cap S$ is parameterised by a single map $\phi: U \subset \...
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If parallel transport from $p$ to $q$ does not depend on curve joining $p$ to $q$, then the curvature of $M$ is identically zero.

I'm currently working through Do Carmo's book Riemannian Geometry and came across the following question: Let $M$ be a Riemannian manifold with the following property: given any two points $p, q \in ...
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Family of functions with curvature parameter

I am looking for a family of functions to model some data. I found this question, and the functions I need are quite similar: I need to define a family (one parameter) of monotonic curves Just as in ...
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55 views

Parameterization of a parallelogram in $\mathbb{R}^3$

I have a parallelogram in $\mathbb{R}^3$ with the vertices $(0,0,0),(1,1,-1), (1,1,1), (2,2,0)$. How would I find the parameterization of this? Thanks!
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Intersection of a parametric equation and a plane

Use $cos(t)$ and $sin(t)$, with positive coefficients, to parametrize the intersection of the surfaces $x^2+y^2=36$ and $z=6x^3$. I have found $<6cos(t), 6sin(t)>$, but I haven't pined down $z$....
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Polar Coordinates for Two Parallel Lines which has to follow an Arc

What I have What I want to obtain Hi, I have these two lines (What I have ), which one is described by two points with the next polar coordinates: LINE 1: Point 1: ( R1*cos(alfa1) ,R1*sin(alfa1) ,...
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1answer
36 views

Polar and parametric curves

I was solving a calculus problem on polar coordinates and I came across with some doubts, I don't know how to solve it. It says: "Given the curve $C: (x+1)^2+y^2=1$ parametrize the arc of a curve that ...
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4answers
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Parametrization of a line segment using angle as parameter

I know this is probably elementary level for most people here, but I've been stuck on this problem for no less than 4 hours and I am completely clueless as to how to figure this out. Is it possible ...
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1answer
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Restricting a function on the sphere to a one-parameter family of arcs

Let's consider the following constrained optimization problem: Given the quadratic form $x^TAx$ defined using the matrix $A$ below, find the maximum and minimum values subject to the constraint $\|...
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1answer
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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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1answer
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Plotting Complex Parametric Curves

I have to plot out by hand the following curve $z(t) = 3+ie^{it}$ for $0\leq t \leq \pi$ I know that circles in $\mathbb{C}$ can be parameterized as $z(t) = c + re^{it}$ where the circle has radius $...
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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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How to give a parameterisation of $u(x_0,y)=y$ in the form $\Gamma(s)$?

I am working to solve a PDE with the given initial condition $$u(x_0,y)=y$$ where $x_0$ is a constant. In order to solve the PDE I need to parameterise the curve in the form $\Gamma(s)$. I have ...
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1answer
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Parameterization of the initial condition of nonlinear PDE

(NOTE: I am not asking for the solution to this PDE) I have the PDE, $$u_x u_y + ln(x^2)=0,$$ with the condition that, $$u(x_0,y)=y,$$ where $x_0$ is a constant. I am to find the explicit solution (...
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36 views

Dupin Cyclide: Cartesian coordinates to parametric coordinates

I have been given points in Cartesian coordinates that lie on Dupin's cyclide. I am simply trying to extract the corresponding parametric coordinates. Given two parameters $u,v \in [0,2\pi]$, the ...
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Given a line from -i to i and a semi circle from -i to i, evaluate the integral over both and explain why they are different

Let gamma 1 be a straight line from -i to i and let gamma 2 be the semi-circle of radius 1 in the right half plane from -i to i. Evaluate $$\int_{\gamma_1}f(z)dz$$ and $$\int_{\gamma_2}f(z)dz$$ ...
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Finding Maximum Value of CST Parameterization over an interval

I have a CST parameterization for a shape over an interval (0,1), so I have y as a function of x like so $$y = C(x)*s(x)$$ where $$C(x) = x^{n1}*(1-x)^{n2}$$ and $$S(x) = \sum_{i = 0}^{n} A_i(x)^i(1-x)...
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Meridians and Parallels on a Unit Sphere

Let $S$ be the unit sphere in $\Bbb R^3$ with centre $(0, 0, 0)$ $\sigma(u, v) = (\cos v/\cosh u,\sin v/\cosh u,\tanh u)$ is a parametrization of $S$ minus the north and south poles. Show that ...
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1answer
29 views

Reason for Z-axis orientation in torus knots

My understanding is that when |p|≠ 1 ≠|q| and they are coprime for (p,q) torus knots, those knots are chiral and while rotation and translation in three dimensions cannot map a chiral knot to its ...
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How to show that two parametrizations are representing the same line in plane?

(Two parameterizations): $$\vec r_1(t) = (t^3, t + 1), t ∈ [0, 1] $$ $$\vec r_2(t) = (t^6, t^2 + 1), t ∈ [0, 1]$$ How can I show that these two parameterizations represent the same line in plane? ...
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Isn't a smooth map having rank 2 injective by the corollary of inverse function theorem?

In the book of E. Bloch, at page 171, it is stated that Corollary: Let $U\subseteq \mathbb{R}^2$ be an open subset, and let $f: U \to \mathbb{R}^3$ be a smooth map. If for $p \in U$, $Df(p)$ ...
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Why does a a torus knot parameterization invert the sign of the z values?

The torus knot page on Wikipedia contains the following (p,q)-torus knot parametrization:\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned} where\begin{...
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The image $M=\gamma(S)$ is a Mobius Strip, what are the missing points of the parameterization of the interior and what is another parameterization?

Let $$S=\{(r,\theta) | -\frac12 <r<\frac12 , 0\leq\theta<2\pi \}$$ and $$S'=\{(r,\theta) | -\frac12 <r<\frac12 , 0<\theta<2\pi \}$$ so $S'$ is the interior of $S$. Define $\gamma ...
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2answers
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What are the conditions for a curve/surface to be parametrized?

I wanted to know what's on the title because I see a lot of people parametrizing curves/surfaces or saying that they can be parametrized somehow, but I never really saw a proof that it can really be ...
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1answer
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Reparametrization by Arclength Definition

In my textbook, reparametrization by arclength is described as follows: Take a regular vector function $\alpha (t)$ whose arclength is $s(t)$. $s(t)$ is an increasing differentiable function, so it ...
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Proving the dimension of a manifold is well defined

Show that if $M \subseteq \Bbb R^n$ is a $k$-dimensional manifold, then there is no $x \in M$ and $U_x$ a neighborhood of $x$ such that $M \cap U_x$ has a good parametrization from $V_x \subseteq \Bbb ...
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How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\...
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3answers
56 views

Parameterizing the boundary curve of the surface defined by $x+y+z \geq 1$ and $x^2 +y^2+z^2=1$

I am unsure how to parameterize the boundary curve of the surface defined by $x+y+z \geq 1$ and $x^2 +y^2+z^2=1$, where $x,y$ and $z$ are real numbers. The boundary curve should be the circle ...
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Area of Parametric Curves

Compute, in terms of $A, B, h$, and $k$, the area enclosed by the curve defined by the parametric equations: $x(θ)=Acosθ+h$ $y(θ)=Bsinθ+k \quad \quad \quad \quad \quad \quad $ for $0 ≤ \theta ≤ 2π$....
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3answers
80 views

Area Inside A Loop Formed By Parametric Equations

We are given: $x=49-t^2$ $y=t^3-16t$ The curve apparently makes a loop which lies along the x-axis. I need help finding total area inside the loop. I don't know where to even start. If it helps, ...
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221 views

Do Carmo 3.4. exercise 8: Vector Field on a Surface

I'm having trouble trying to start this. Here is the problem statement: Show that if $w : S \to \mathbb{R^3}$ is a differentiable vector field on a regular surface $S \subset \mathbb{R}^3$, and $w(...
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1answer
21 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, $$s(t) = 0.5 (t - \frac{ 1}{ t} ),$$ ...