Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
0answers
35 views

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 ...
6
votes
2answers
97 views

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\...
0
votes
3answers
57 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 ...
1
vote
2answers
34 views

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 $0 ≤ \theta ≤ 2π$....
2
votes
3answers
88 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, ...
3
votes
0answers
230 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(...
0
votes
1answer
22 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, $$s(t) = 0.5 (t - \frac{ 1}{ t} ),$$ ...
0
votes
0answers
29 views

Converting parametrization of torus to proper implicit representation

Let $0 < b < a$. We define the torus, $\Bbb T^2$, by the following parametrization: $$\Gamma(s,t) = ((a+b\cos s)\cos t,(a+b\cos s)\sin t, b\sin s)$$ where $s,t \in [0,2\pi].$ Express $\Bbb T^2$ ...
2
votes
0answers
53 views

Explicit parametrization for a 3-ellipse? A 4-ellipse?

I searched around but was unable to find anything. For the usual $2$-ellipse we have the parametrization $x(t) = a\cos(t)$ and $y(t) = b\sin(t)$ for $t\in [0,2\pi]$. Is there anything similar for ...
1
vote
1answer
41 views

S is the part of the cylinder $x^2+y^2=2x$ parametrize $S$

S is the part of the cylinder $x^2+y^2=2x$ with $0 \leq z \leq \sqrt{x^2 + y^2}$ how would I parameterize this ? $x^2+y^2=2x$ can be made into $(x-1)^2 + y^2 = 1$
0
votes
1answer
23 views

Approximate rounding to nearest integer with a continuous function

I am trying to come up with a family of continuous functions, which will approximate a rounding to nearest integer function. I came up with the following solution: $f(x)=x-\beta*\frac{\sin(2 \pi x)}{...
0
votes
1answer
18 views

The 'curve must be traced from left to right' requirement for the parametric arc length formula

I read that, for the arc length formula for curve (f(t),g(t)) to work, the curve must be traced from left to right as t varies from the lower limit to the upper limit, or dx/dt>0 in the interval. 1....
4
votes
0answers
55 views

Help me see the connection between exponential functions and quadratic curves

We know that the degree 2 equations $x^2 + y^2 =1$ and $x^2 - y^2 =1$ can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it ...
1
vote
1answer
20 views

Alternative parameterizations of a statistical model

I'm currently learning about the basics of a statistical model and reparametrization through a textbook. In one of the examples, it says that a location scale normal model is usually labelled as (mu, ...
1
vote
2answers
67 views

Parametrize the surface $(x-z)^2+(y-z)^2+(y-x)^2=1$

So I know this equation is the equation for a cylinder. And I know that the equation $x^2+y^2=1$ can be parametrized to $x=\cos(t)$,$y=\sin(t)$, with $z=t$ making a helix. I want to know how would ...
0
votes
0answers
34 views

for each point on the curve, the line segment of the tangent line from tahe curve point to the $y-$ axis has length 1.

Show that the tractrix has the following property: for each point on the curve, the line segment of the tangent line from the curve point to the $y-$ axis has length 1. My attempt:- Let the given ...
0
votes
0answers
28 views

When does BCH formula yield a Markov transition rate matrix?

Given two $n\times n$ matrices of the following structure: $A_{ii} = -d_i, A_{in}=d_i$ with $d_i>0$, all other elements equal to zero. For which matrices B is $Z = \log(\exp(A)*\exp(B))$ a ...
0
votes
1answer
22 views

Explain why a hyperbola with center $(m,n)$ has the parametrized curve $r(t) = (m + a cosh, n + b sinh)$ “laying” on it

Struggling with how to approach a task given at university. The first part of the task asked to show that the equation $$4x^2-32x-9y^2+36y=8$$ resulted in a hyperbola, and to find the center, ...
0
votes
1answer
59 views

Why $g(t)=(\cos t, \sin t)$ and $h(t) (\cos 2t, \sin 2t)$ have the same image?

I know this might sound silly, but I don't know why $g(t)=(\cos t, \sin t)$ and $h(t)= (\cos 2t, \sin 2t)$ have the same image, that is the unit circle. I understand that if we eliminate the parameter ...
0
votes
0answers
65 views

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{...
0
votes
1answer
42 views

Parametric equation of a rose with $5$ petals, orthogonal to $x+y+z=3$, with radius $4$, center $(1,1,1)$, and a petal in the direction of $(0,0,3)$

I am kind of stuck in this exercise. Write the parametric equation of a rose-shaped curve with 5 petals, radius 4, centered on $(1,1,1)$, orthogonal to the plane $x+y+z=3$, such ...
1
vote
1answer
42 views

Incorrect radius of ellipsoid

I've seen the correct way of finding $r(\theta,\phi)$ for the purposes of integrating for area, but that left me wondering why we can't just use: $$ x = a \cdot \cos{\theta} \cdot \sin{\phi} \\ y = b ...
0
votes
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/...
0
votes
1answer
32 views

A Projectile Motion Problem

I need more help, but this time it if for a problem involving projectile motion. The question is " A shot leaves the thrower’s hand 6.5 ft above the ground at a 45° angle at 44 ft / sec. Where is it ...
0
votes
0answers
19 views

evaluating integral over graph of semicircles with parameterization

Evaluate the path integral of $f(x, y) = y$ over the graph of the semicircle $y = \sqrt{1-x^2}, -1 \leq x \leq 1$ solution: $\vec{C}(t) = (cos(t), sin(t))$ $\vec{C}'(t) = (-sin(t), cos(t))$ $||\...
3
votes
0answers
52 views

How do I parametrise the hyperbola $xy = y^2 -1$?

This equation is actually the solution to the intersection of the two surfaces $z = x^2 - y^2$, and $z = x^2 + xy - 1$. I am to parametrise the solution curve, which is noted in the title. A first ...
0
votes
0answers
26 views

General formula for parameterization of a hypocycloid with 4 rotations?

We are given $x^{2/3} + y^{2/3} = 1$ which has the parameterization of $x = \cos^3(t)$ and $y = \sin^3(t)$ for $0 \leq t \leq \pi /2$ my question is, would there be a general formula to get the ...
0
votes
0answers
36 views

Curve of intersection of the cone is $z = \sqrt{x^2 + y^2}$ and the ellipsoid is $3x^2 + y^2 + z^2 = 2y$. Find parameterization

Curve of intersection of the cone is $z = \sqrt{x^2 + y^2}$ and the ellipsoid is $3x^2 + y^2 + z^2 = 2y$. Find parameterization Solution: $z^2 = x^2 + y^2$ $3x^2 + y^2 + z^2 = 2y \implies 4x^2 + 2y^...
0
votes
0answers
61 views

Rational solutions of $k^4(x^2+4)-4xk^2=z^2$, where $k$ is given rational.

How to generate some parametric family of rational solutions to $k^4(x^2+4)-4xk^2=z^2$, where $k$ is given rational.
0
votes
1answer
48 views

Integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer.

Can we generate some parametric family of integer solutions of $kx^2=y^3-1$, where $k$ is given positive integer. I don't even know if there are finite or infinite number of solutions. For $k=7$, ...
0
votes
2answers
55 views

Rational solutions of $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.

How to generate some parametric family of rational solutions to $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.
2
votes
2answers
86 views

Find the unit normal to an ellipse given by an equation

The equation of the ellipse is given as being: $$x^2 -xy + y^2 = 7$$We're instructed to find a unit normal to the curve at a general point $P(x,y)$, and also at point $(-1,2)$ in particular. My ...
0
votes
2answers
44 views

Consider the vector field $F=-c \frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$.

$$\mathbf{F}={-c}\frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$$ (vector field was rewritten here to make it easier to see) Consider the vector field above and using $c=1$, find by direct calculation the ...
2
votes
2answers
48 views

Differential geometry, show a curve parameterizes a circle

So I have this exercise from differential geometry and I am struggling with how to show that this is indeed a valid parameterization. Let $n(\phi) = (\cos(\phi), \sin(\phi))$. Show that the curve $$...
0
votes
1answer
22 views

Finding parametrization of the curve of intersection

Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces. By equating them together, I get $y^2 +xy -1 =0$. Letting $x=t$, I ...
0
votes
1answer
16 views

Continuous paths joining positive x-axis to negative x-axis, through upper half plane

I need a way to parametrise all continuous paths from the positive to the negative x-axis, which go through the upper half plane (in $2$ dimensions). I do not care about the speed of the ...
1
vote
2answers
73 views

Draw $r ≤ 3 + 2\sin (\theta)$

Currently I'm stuck at this fairly easy task. All I have to do is sketch the region $r \le 3+2\sin \theta$. My guess would be that the circle has the origin $(0,3)$ with $r = 2$, as I use the formula $...
0
votes
0answers
81 views

First Order Nonlinear PDE - Discrete Ray Method

In the application of the discrete ray method to the mixed differential-difference equation below , as $n \to \infty $: $G'_n(x) = e^{-\frac{x^{2}}{2\,n\,(n+1)}}\,G_{n-1}(x)$, $G_{0}(x)=1$ and ...
0
votes
1answer
58 views

Let C be the graph of hypocycloid $x^{2/3} + y^{2/3} = 1$ oriented clockwise. Parametrize the curve and find its arc length.

Let C be the graph of hypocycloid $x^{2/3} + y^{2/3} = 1$ oriented clockwise. Parametrize the curve and find its arc length. Attempt: let $x = \sin^{3}t, y = \cos^3({t})$ $r(t) = (\sin^{3}(t), \cos^...
0
votes
1answer
92 views

parametrization of a moving wheel

Attempt: Given, radius of wheel $0.5 m$ Total distance travelled by the point during first revolution = circumference of the wheel therefore, distance travelled by the point $2 \pi r$ where r is ...
2
votes
0answers
39 views

Difference between a parametrized surface and manifold

What is the difference between a parametrized surface and manifold? Is it true that if $M \subset \mathbb R^n$ is an $n$-dimensional parametrized surface it is also a (parametrized?) manifold? I am ...
1
vote
4answers
90 views

Parameterize $3x^2 + 4xy + 3y^2 = x+3y$

I am calculating the curve that is formed by the graph between the function $$f(x,y)=3x^2+4xy+y^2$$ and the plane $$z=x+3y$$ Then I'm left with the equation $3x^2+4xy+y^2=x+3y$ that I want to ...
0
votes
0answers
44 views

Reparametrization of a circle

Let $\,\mathbf{x}\longrightarrow\mathbb{R}^2\,$ and $\,\mathbf{y}:J\longrightarrow\mathbb{R}^2\;$ be two parametrizations of a circle, given by $$\mathbf{x}\left(\theta\right)=\left(\cos\theta,\,\sin\...
1
vote
2answers
113 views

parametrize the boundary of a region

I need to parametrize the boundary of this region : $D=\{y^2+z^2\le x^2+18,x^2+y^2\le 16\}$ So It's a one-sheet hyperboloid (radius=$\sqrt{18}$)+ cylinder with radius 4 I know how to parametrize ...
1
vote
2answers
48 views

Easier way to parametrize an ellipse.

I want to calculate the parametrization of the curve $$E=\{(x,y,z):z=x^2+y^2,x+y+z=1\}$$ To do so I did a change of variables taking as basis the vectors $\frac{1}{\sqrt{2}}(-1,1,0),\frac{1}{\sqrt{6}}(...
-2
votes
1answer
37 views

“Parameterizations are non-unique”, how can we prove it?

$$ \textbf{"Parameterizations are non-unique"} $$ I have seen this statement in several books and at Wikipedia. However, I have never seen a proof of the statement. How can we prove it?
0
votes
1answer
83 views

Parametrizing the curve of intersection between a elliptic cylinder and a sphere

How can I form the parameterization of a curve of intersection given a sphere $x^2+y^2+z^2 =1$ and an elliptic cylinder $2x^2 + z^2 = 1$ ?
0
votes
1answer
19 views

Breaking a path, in 2 integrals

Let $\overrightarrow V, \Gamma$, be a vectorial field and a path such that: $$\overrightarrow V=-(x^2+y^2)\overrightarrow i-(x^2-y^2)\overrightarrow j$$ and $$\Gamma=\{(x,y)\in\mathbb{...
4
votes
1answer
28 views

Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$

Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$ and $\Gamma$ is the intersection between: $x^2+y^2+z^2=r^2$ and $x-y+z=0$ My attempt: $\Gamma$ is some kind of ellipse in the ...
1
vote
1answer
18 views

Calculate $\int_{\Gamma} \omega$ when $\omega =z(z-y)dx+xzdy-xydz$, $\Gamma=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$

Calculate $\int_{\Gamma} \omega$ when $\omega =z(z-y)dx+xzdy-xydz$ $\Gamma=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$ $$x^2+y^2=(z-1)^2$$ $x\geq0, y\geq0,z\geq0$ $\Gamma_{1,2,3}$ are ...