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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Linearly approximating coordinates on an N dimensional point from known positions in time.

Let's imagine an N dimensional point $P$. Let's assume that we know what $P$ looks like at time $t=0$ and $t=1$. Is there an elegant formula to find where $P$ is, assuming linear approximation, at any ...
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Local Parametrizations cover unit $n$-sphere

Say I have a set $\mathbb{S}_n$ of $2^n$ vectors $s\in \mathbb{R}^{n+1}$ which are functions of $n$ parameters $\{ \theta_1,\theta_2,\ldots,\theta_n\}$. How can I prove that the set of all ...
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Parameterization of a curve within a path integral?

I have a question about the following problem: Find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R}^{2}$, with the implied orientation. The curve $C$, which goes ...
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How to make a 2-d linear function using a third variable for the iterator? [closed]

Say, for example, you have the vector $\vec {PQ} = \langle8,4\rangle$. As we all learned in Algebra I, the "traditional" slope (y-units per x-unit) would be $\frac{4}{8}$, and the slope for ...
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Parametrizing the intersection of two ellipsoids to find arc length

Suppose I have two ellipsoids in $\Bbb{R^3}$, with one centered at the origin and aligned to the coordinate axis, and the other in general position. I can represent these by: $$ \frac{x^2}{a^2} + \...
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Finding at least $4$ parameterizations of the curve $y=x^2-6x+9$. I can only find two.

Finding at least $4$ parameterizations of the curve defined by $y=x^2-6x+9$ What I tried: One way: Let $x=t,$ then $y=(t-3)^2$. So coordinates of any point on the curves is $(t,(t-3)^2)$. Another ...
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Torsion As The Rate Of Change Of An Angle

Just as we define the signed curvature of a plane curve as the rate of change of the angle through which a constant vector must be rotated to bring it into coincidence with the tangent vector, is it ...
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How to parametrize the intersection of an ellipsoidal surface and a sphere?

Suppose you have an ellipsoid given by the set, $$\left\{ x \in\mathbb{R}^3 \mid x^TQx = 1 \right\}$$ where $Q = \mbox{diag}(a,b,c)$. Is there a way to parametrize the set $$\left\{ x \in \mathbb{R}^3 ...
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28 views

Parametric representation of the intersection of spheres

Goal: I am trying to find the curve of intersection of two spheres. $\begin{align*}x^2+y^2+z^2 &= 9 \\ (x-3)^2+y^2+(z-1)^2 &= 4 \end{align*}$ What I have done: One of the ways of achieving ...
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3D Parametric Equation changing over time

I can create a 3D Parametric Equation of a spiral but I'm having trouble getting the angle of "decent" to also change over time. $$x=u\sin(u)\cos(v)$$ $$y=u\cos(u)\cos(v)$$ $$z=-u\sin(v)$$ ...
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101 views

Problem with the parametrisation of this surface integral

I am facing troubles in understanding (read: "guessing") the correct way to parametrise this integral: $$\int_{\Sigma} \dfrac{1}{\sqrt{1 + x^2 + y^2}}\ \text{d}\sigma$$ Where $\Sigma = \{(x, ...
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Parametrizing any arbitrary Conic

Given the implicit equation of a Conic $C$, how to determine its parametric representation? I went thought this report and it has an algorithm: Fix a point $p$ on the conic. Consider the pencil of ...
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28 views

To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ with some additional constraints.

To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ whose unit tangents at start and stop are $\hat{v_0}$ and $\hat{v_1}$ and has the minimum length. Let us assume that the ...
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Parameterizing a curve using polar basis

Is it generally valid to parameterize a curve by: $\begin{pmatrix}r\\ t \end{pmatrix}_{\{\hat{r},\hat{\theta}\}},0\leq t\leq\frac{\pi}{2}$ when I want a curve that is a quarter of a circle with radius ...
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Parametric solution of a Diophantine equation of three variables

I came across this Diophantine equation $$4x^2+y^4=z^2$$ Primitive solutions of this equation can be found by \begin{align} \begin{split} x&=2ab(a^2+b^2)\\ y&=a^2-b^2\\ z&=a^4+6a^2b^2+b^4\\...
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Obtaining the Equation of a “Slanted Circle”

Consider the parametric curve $$A(a) = \langle -\sqrt{2} \sin(a),\cos(a),\sin(a) \rangle$$ for $0 \leq a \leq 2 \pi.$ We may obtain a unit parametrization of $A$ by $$B(a) = \frac{A(a)}{||A(a)||} = \...
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For the same curve,The critical point becomes a regular point under different parameter expressions.

When I say "the same curve",I mean they have the same image in the $R^2$ plane. When I say "a critical point of $\mathbb{r}(t)$",I mean "$\mathbb{r}^{'}(t_0)=\mathbb{0}$"....
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Gradient is perpendicular to level set and implicit function theorem

My lecture notes on the gradient state the following: For $f : U \to \mathbb{R}$ differentiable consider the level set $N_w = \{v \in U : f(v)=w\}$ where $U \in \mathbb{R}^n$. Suppose that $c : I \to ...
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Parametrization along the coordinate axis.

I'm looking for a parametrization along the coordinate axis that goes from $(x,y)=(x_0,0)$ to $(0,0)$ and then from $(0,0)$ to $(0,x_0)$, and I know that my x-component will have to be $x_0-t$ but I ...
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Given the parameterization of a curve,show that it's a circle with its center placed at the origin

Show that the curve defined by $$\gamma(s):=\left(a\cos\left(\frac{s}{a}\right),a\sin\left(\frac{s}{a}\right)\right)$$ Is on a circle with the radius $a$ and the center $(0,0)$,also show that $\gamma(...
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Find the value of $\left\Vert \frac{d^3\gamma(s)}{ds^{3}} \times\frac{dN}{ds} \right\Vert$ for the given curve

Given a curve defined by : $$\gamma(t):=\left(4\cos\left(3t\right),4\sin\left(3t\right),3\cos\left(2t\right)\right)$$ Assuming the curve is parameterized by its arc length and is differentiable three ...
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Parameterize $\gamma(t)=\left(\int_{0}^{t}\sin\left(\frac{ks^{2}}{2}\right)ds,\int_{0}^{t}\cos\left(\frac{ks^{2}}{2}\right)ds\right)$

Parameterize the following curve by its arc length: $$\gamma(t)=\left(\int_{0}^{t}\sin\left(\frac{ks^{2}}{2}\right)ds,\int_{0}^{t}\cos\left(\frac{ks^{2}}{2}\right)ds\right)$$ First of all we need to ...
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How to use Spherical Coordinates to define an area.

Let $G=\{ (x,y,z) : x^2+y^2+z^2 \leq 4 \text{ and } x^2+y^2 \geq 1\}. $ Use spherical coordinates to describe the area $G$. I imagine that this describes the area in between the sphere with radius 4 ...
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Parametrization the curve of intersection of sphere $x^2+y^2+z^2=5$ and cylinder $x^2+\left(y-\frac{1}{2}\right)^2=\left(\frac{1}{2}\right)^2$.

I've seen similar posts here but none of the answers helped me. I am trying to parametrize a curve of intersection of a (top half $z>0$) sphere $x^2+y^2+z^2=5$ and cylinder $x^2+\left(y-\frac{1}{2}\...
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52 views

Dimension of polynomially parameterized set of points

I'm seeking a general method for determining the dimension of a set of points that have polynomial parameterizations. Any information about what this type of problem would be called would be helpful, ...
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How to parametize an intersection of sphere and a plane

I need to find some "nice" parametrization of intersection of sphere $x^2 + y^2 + z^2 = 1$ and a plane $Ax + By + Cz = 0$. I know that the curve we get is an ellipse, but have no idea how to ...
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A Question on the Rate of Change of the Arc-Length

Main Question Consider some curve $y(x)$, going from a point $(x_0,y_0)$ to a point $(x_1,y_1)$. Let $L$ be the length of the curve, and the function $F$ be the rate of change of the length of this ...
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Solution to Exercise from Section 7.4.1 in O'neil's Elementary Differential Geometry

Exercise: Show that a reparametrization $t → \alpha(f(t))$ of a nonconstant geodesic $\alpha$ is again a geodesic if and only if f has the form $f(t)=at+b$. If $\alpha$ is a geodesic then it ...
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Integral of $\cos(z)/z$ along two circles, using residues or parametrization

In a problem I was asked to calculate the complex integral $\int_C\frac{\cos(z)}{z}dz$ where $C$ is (1) the unit circle or (2) the circle $|z|=3$, both positively oriented. I wanted to approach this ...
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How to find tangent vector to curve (which is the intersection between two curves)

Consider the curve C obtained by intersecting the surfaces defined by $x^2+y^2+z^2=3$ and $x^2-y^2+z^2=1$ At the point $(1,1,1)$, what is the tangent vector to the curve? I tried parameterizing the ...
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66 views

Ambiguity in a ellipse integral

Suppose we have a ellipse $x^2 + 4y^2 = 1$, and we want to integrate a density $f(x,y) = 3 \cdot |xy| $ over the ellipse. I was trying to solve this problem in two different ways and i find 2 ...
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General formula for parametric equations of line of intersection of two planes

I've started to learn about the intersection of two planes. I was wondering if there was a general formula for the parametric equations of the line of intersection between any two planes. Here's my ...
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How to parametrize a circle not centered at the origin

I'm trying to parametrize the circle centered at $2i$ with radius $1$. I'm trying to parametrize it to find the integral of $\frac{1}{z}$ from that circle so I'm not sure if I can put this to make it $...
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$\nabla \times \bf{u} \neq 0$ but $\oint_{c} \bf{u} \cdot \textit{d}r \textit{=0}$?

Consider the vector field $\vec{u}=(xy^2,x^2y,xyz^2)$ The curl of the vector field is $$\nabla \times\vec{u}=(xz^2,-yz^2,0)$$ Consider the line integral of $\vec{u}$ around the ellipse $C$ $x^2+4y^2=...
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Find the directional derivative of a function along the curve.

Question: Find the directional derivative of f=x^2·y·z^3 along the curve x = e^-u; y = 2 sin u + 1; z = u - cos u at the point P where u = 0 My working: At u=0, x=1, y=1, z=-1 so let u = (1,1,-1). ...
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Parameterize a circle (given x, y, z cooridnates on its circumference) in 3D

I have a circle defined by a set of $x, y$ and $z$ coordinates. The circle exists across $3$ planes. I would know how to parameterize a $2$-d circle (say in just the $x$-$y$ plane) into polar ...
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Why is the derivative of the binormal vector parallel to the normal vector?

In my notes, it says to consider an arc length parameterization r(s). Then we can show that B' points in the direction of N such that we can write B'=-𝜏N. I understand why ||B'||=𝜏 but am unsure how ...
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Parametrization of a quartic

I know how to find a parametrization of a cubic genus $0$ curve. But I don't know it for quartics and i couldn't find it anywhere. Can you give me an example of a parametrization of a quartic? What ...
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Correct Interpretation of Notation

I was reading a parametrization and they used a peculiar way to write their equations which I am unfamiliar with as to how to properly interpret it. K refers to Kelvins in this case and what I am ...
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54 views

Surface-Curl integral questions

Good morning, I have some questions about a surface integral with curl. The exercise is the following: Be $(\Sigma, \omega)$ an oriented surface with boundary where $$\Sigma = \{(x, y, z): x^2 + ...
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38 views

Helix around Helix around Circle

I'm trying to find the parametric equations for a helix around a helix around a circle (helix on helix on circle) That is: I would like to start with a circle, add a helix around it and a helix around ...
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Parametrizing a manifold

I have a conceptual question about parametrizing a 2-manifold in $\mathbf R^3$ and computing its area/volume. If there is anything that I didn't phrase correctly, I would be glad to change it. What ...
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25 views

How to write a matrix from $SU(2)$ in terms of one angle and one complex number $z$ , where $z$ is from sphere $S^{2}$

For given a matrix from $SU(2)$ , how can represent it in terms of two parameters: one angle and one complex number $z$ from the sphere $S^{2}$ ? Does this have any links with : $\mathrm{SU}(2)$ axis ...
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1answer
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Is it possible to have function with a variable number of parameters?

I was wondering if there is a type of parameterized function where the number of parameters changes over time? How would you describe the derivative/properties of the number of parameters over time ...
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Understanding a parametric question

The parametric form of a relation is given by: $$x = rsin\theta$$ $$y = rcos\theta$$ for $0≤ θ ≤ 2π$ Eliminate θ so as to find the relationship explicitly between $x$ and $y$. What is this question ...
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Find area enclosed by curves $x^3 + y^3 = x^2 + y^2$, $x$-axis and $y$-axis using Divergence Theorem

I know how to deal with the axis part, but I struggle solving the integral with the curve $$x^3 + y^3 = x^2 + y^2$$ I tried to parametrize the curve using polar coordinates. $$x = r(t)\cos(t)$$ $$y = ...
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Quadratics with Integer Intercepts and Extrema

I'm a math teacher and was creating an exam for my Algebra 1 students when I tried picking an equation that had integer Intercepts (both x- and y-), as well as extrema. I wanted to do so because I ...
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84 views

Deep doubt on a double surface integral

I don't understand how to proceed with an exercise. I will write down what I have done so far. The exercise is: Evaluate the following integral $$\iint_{\Sigma}\dfrac{1}{x^2+y^2}\ \text{d}\sigma $$ ...
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How many parameters does the set of all spheres, which satisfy the given condition, depend on?

How many parameters does the set of all spheres, which satisfy the given condition, depend on? (i) Spheres that pass through the given point. (ii) Spheres that touch the given line (iii) Spheres that ...
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26 views

The sum of all parameter values for which the graphs of the functions have exactly one common point is?

The sum of all parameter $a\in\mathbb{R}$ values for which the graphs of the functions $y=(a+2)x^2-ax-3$ and $y=ax-4$ have exactly one common point is? The answer in textbook is -1. This is what I ...

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