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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Parameterization of Curves and Surfaces
I would like to know if the following settings are correct:
a) Parameterization of Curve:
Given the curve $C = \{(x, y, z) \in \mathbb{R}^3 \,|\, x^2 + y^2 = e^3\}$ I want to find a parameterization ...
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Area-preserving continuous deformation of the graph of $r=a(1+\cos(2\theta))$ into a circle centered at the origin (graph included)
I want to know if the equation I provide actually does indeed model the transformation that I desire to model
Model the starting shape by
$$f\left(t\right)=\left(a\left(\cos\left(2t\right)+1\right)\...
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Find the parametric equation of the surface cut from a sphere by a cone
Find the parametric equation of the surface cut from a sphere $x^2+y^2+z^2=16$ by a cone $z=\sqrt{x^2+y^2}$.
Attempt: I want to use spherical coordinates. Solving the sphere and cone, the surface is ...
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Mathematical representation of projection plane
I've gotten the following assignment, given this specification of a screen I need to project a video onto, I want the video to appear flat.
My idea was to find a parametric equation describing this ...
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Proof that $y = |x|$ is not regular smooth
I've been studying vector calculus, and I came across the notion of regular smoothness.
In particular, the textbook I am using claims that $y = |x|$ has no regular smooth parametrization. No proof of ...
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Parametrized matrix has rank $2$, find the parameters
Given that the rank of $\mathbf A=\left(\begin{smallmatrix}1&2&1&1\\3&a&4&3\\5&8&6&b\end{smallmatrix}\right)$ is $2$, find $a$ and $b$.
I used matrix ...
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The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint F ds$.
The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint_C F ds$.
Attempt:
Firstly, to define the term inside the ...
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Motion of midpoint of elastic band connected by two gears
A mechanical device consists of two circular gears, one of radius 2 centered at (0, −2) and
the other of radius 1 centered at (0, 1). The gear of radius 2 rotates clockwise at unit angular
velocity (1 ...
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Distance of two parametrized lines
Consider the four points $A = (2, 4, 0), B = (3, 1, 1), C = (1, 1, 3), D = (0, 5, 1)$. Find the distance between the lines $(AB)$ and $(CD)$, i.e. the distance between the closest points on these two ...
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Distance between two planes parallel to two lines
Consider the four points A : (2, 4, 0), B : (3, 1, 1), C : (1, 1, 3), D : (0, 5, 1). Find the
distance between the lines (AB) and (CD), i.e. the distance between the closest points on
these two lines, ...
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Why can we parameterize differential equations?
When studying about ODE's I've stumbled upon some types that are solvable by using parameterization (as explained by my book which unfortunately isn't written in English), for example:
$$t = f(y, y')$$...
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Name for the paradox of conditioning on "equivalent" continuous random variables
I remember being shown the following example in class some time ago, but haven't been able to find any information about it on the internet.
The paradox
Let $(x, y)$ be a uniform random variable on ...
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Problem doing a line integral $\int_C P(x,y)dx+Q(x,y)dy$
Evaluate $$\int_C P(x,y)dx+Q(x,y)dy$$ where
$P(x,y) = y^2 $, $Q(x,y) = x$, and
$C$ is the part of the graph $x = y^3$ from $(-1,-1)$ to $(1,1)$.
I was trying the parametrization:
$$x = t $$
$$y = \...
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Parametrising a piecewise circular curve in 3d
The curve in question is a closed curve $C: = \overline{abcda}$ made of three circular arcs $C_1=ab, C_2 =bc$, $C_3 = cd$ and a straight line segment $ C_4 = da$. The arc $C_1$ lies on a plane $P_1$ ...
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Explanation for arc length in parametrized curve
let $\Delta s_i$ be a piece of arc length hence:
$$\Delta s_i = \int_{i-1}^i \sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}dt$$
Why is that the length of $\Delta s_i$
in 2d I know that the as $\Delta x$ -> ...
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Where is the hypothesis that $f_{n-1}(x, y)+f_n(x, y)$ is irreducible used in my solution?
I'm working through Shafarevich's Basic Algebraic Geometry, and one of the problems is: Prove that the curve given by the equation $f_{n-1}(x,y)+f_n(x,y)=0$ is rational if it is irreducible. Here $...
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Position of greatest speed and greatest speed.
In the beginning of a multivariable calculus class, I'm trying to calculate the following problem.
A particle moves in the plane along an ellipse curve so that at time t ≥ 0 it is at the point $ r(t) =...
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Preimage of a point in differential geometry of surfaces.
I'm studying how to work with surfaces in differential geometry. The definition of a regular surface is the following one:
A subset $S\subset\mathbb{R}^3$ is a regular surface if, for each $p\in S$, ...
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Finding x-axis-intercept of a parametric equation and slope
let $x = t^2$ and $y = t^3-3t$
now the equation for the slope is:
$$\frac{dy}{dx} = \frac{3t^2-3}{2t} = 0$$
now at the point (0,0) is the first intercept with the x-axis with slope $\infty$ what is ...
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Parametric solution of quartic diophantine equation in three variables
How can I handle the quartic diophantine equation in three variables $x$, $y$ and $z$ $$x^4-x^2=y^2-z^2$$ in general, i.e, does exists a (three-variable) parametric solution?
What I've tried is ...
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Second derivate of parametric equations (Intuitive) [closed]
let:
$$x=f(t)$$
$$y=g(t)$$
hence:
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$
How can I derive the formula for the second derivative of a parametric equation?
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The curve $y^2=x^3+Ax+B$ can be parameterized if and only if $x^3+Ax+B$ has a repeated root.
I'm working through Shafarevich's Basic Algebraic Geometry, and one of the problems asks the reader to prove the problem in the title. I found the "if" direction fairly straightforward, but ...
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Parameterizing the equation of a line in 2-D
I'm attempting to read Emil Artin's little book on the Gamma function with a borderline adequate background at best.
In order to make it past the first page I need to understand what a convex function ...
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What is the parametric path of an equalateral spherical triangle in the first octant on the unit sphere with the edges equadistance from each plane?
I want to clarify a bit. The vertices need to be the same distance a path that follows from (1,0,0), (0,1,0), and (0,0,1) and converges at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}...
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Inverse of parametrization for a smooth surface
Let $M = \{(s\sin(t), s\cos(t), s^2+t)|s,t\in \mathbb R)\} \subseteq \mathbb R^3$. An obvious parametrization for $M$ would be $r:\mathbb R^2 \to \mathbb R^3, r(t,s) = (s\sin(t), s\cos(t), s^2+t)$. ...
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finding a region bouded by 4 parametric curves
suppose i have the following curves:
$C_1=\{(t,t+2)|t\in\mathbb{R}\}$
$C_2=\{(t+2,t)|t\in\mathbb{R}\}$
$C_3=\{(t^2+t+1,t^2-t+1)|t\in\mathbb{R}\}$
$C_4=\{(t^2+t+2,t^2-t+2)|t\in\mathbb{R}\}$
define ...
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Line integral of an exact differential and how to integrate $\sqrt{1-x^2} \cdot (1-2x^2)$?
I want to calculate the following path integral:
$$\int_\mathcal{C} ye^{-(x^2+y^2)} (1-2x^2) \:\mathrm{d}x + xe^{-(x^2+y^2)} (1-2y^2) \:\mathrm{d}y ,$$
where $\mathcal{C}: x^2+y^2=1$, with $x,y\geq0$,...
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Parametrizing arbitrary rectangular spirals
Parametrizing a square spiral can be done by modifying the equations found in this OEIS entry
$$
k(n) = \frac{\pi}{2}\left \lfloor \sqrt{4n-3} \right \rfloor
$$
$$
x(n) = \sum_{k=1}^{n} \sin(k(n))
$$
$...
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Finding every solution of $a^2+b^2+c^2=3$ in $\mathbb{Q}(i)$
Specifically, $a,b,c\in\mathbb{Q}(i)$ are complex numbers with rational parts whose squares sum to $a^2+b^2+c^2=3$. There's an answer to this question over $\mathbb{Q}$ already here but I couldn't ...
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Why are the singular values equal to the first partial derivatives.
I am studying computer science so please go easy on me.
I am also too bad at math to extract the mathematical essence that is needed to answer this question so I'm just gonna explain the whole setup.
...
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Parametrization of circle in clockwise
How to parametrize a circle in a clockwise direction? The problem was "$C$ is the portion of the circle $x^2 + y^2 = 1$ from $(0,1)$ to $(1,0)$ traced clockwise."
What I did was $c(t) = (\...
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The inverse of an arc length parametrization of $\partial U$ is Lipschitz
Let $U$ be a simply connected open bounded subset of $\mathbb{C}\cong\mathbb{R}^{2}$ with smooth boundary (thus the boundary $\partial U$ is diffeomorphic to a circle). Let $L\geq0$ denote the total ...
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Flux through tetrahedron and parametrization of triangular surfaces
I'm struggling with a math problem in my Math for Physics class. We just introduced Gauss's law and now I'm supposed to calculate the flux through a tetrahedron:
Let
$$\vec{A}: \mathbb{R}^3 \space \...
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Parametrization of a Cycloid Example Solution
In this example solution, for $\overrightarrow{OC}$, why is it $12\sqrt{2}\lt\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\gt$ ? Isn't the formula for the general equation supposed to be $\lt a\theta, a \gt$ ...
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Change of Variables for Parametrized Manifolds
Suppose $M=\varphi(A)$ a parametrized $k$-manifold in $\mathbb R^n$, given a diffeomorphism $g$ in $\mathbb R^n$, $N=g(M)$ is still a parametrized $k$-manifold. Now for a scalar function $f:N\to\...
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Parametrizing Unitary matrices using Hermitian matrices. Covers the whole space?
An $n\times n$ unitary matrix can always be written in the form,
$$ U=e^{i\,H}\,,$$
where $H$ is a Hermitian matrix. If we use the generalized Gell-Mann matrices as the basis for Hermitian matrices, ...
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Prove that $\varphi\circ X_1$ is also a parametrization if $\varphi$ is a diffeomorphism and $X_1$ is a parametrization
Suppose that $X_1:U_1\subset R^2 \to S_1$ and $\varphi:S_1 \to S_2$ is a diffeomorphism, I want to prove that $\varphi \circ X_1:U_1 \to S_2$ is a parametrization of $S_2$
Here is my attempt:
Suppose ...
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Stokes' theorem for the vector field $\vec{F}(x,y,z)=(y,-x,xz)$ on the surface $z=9-x^2-y^2$
Verify Stokes' theorem for the vector field $\vec{F}(x,y,z)=(y,-x,xz)$ on the surface $z=9-x^2-y^2$ with $z\geq 0$.
So I've tried finding the parametrisation which is $$\vec{r}(\theta,r)=(r\cos\theta,...
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Is it appropriate to change the number/form of a functions arguments?
Short version: Suppose I have a function $$R^*=q(Z_i,\lambda p_mR_{-i},p_mp_t,p_m)$$, is it equivalent to change #/form of arguments and write this as $$f(Z_i,\lambda, p_m,R_{-i}, p_t)$$, or, if I ...
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Parametrization of a circle $\gamma(t)= (\cos(t), \sin(t))$ [duplicate]
In my multivariable calculus class, we are learning about parametrization.
In my class, my teacher said that the circle $$\gamma(t)= \left(\cos(t), \sin(t)\right)$$
with $t \in [0, \frac{\pi}{2}]$ is ...
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Find the area of the part of the plane $x+y+z=a$ limited by the cylinder $x^2+y^2=b^2$
Find the area of the part of the plane $x+y+z=a$ limited by the cylinder $x^2+y^2=b^2$
I'm having issues on how to parametrise the surface, I've tried with $\vec{r}(b,\theta)=(b\cos\theta,b\sin\theta,...
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Accounting for "Glued" Edges in Finding Boundary Curves of Parametrized Surfaces
Let $T: [0,2] \times [0,2\pi] \to \mathbb{R}^3$ be defined by $T(r,\theta) = (\cos\theta, \sin\theta, r)$, which parametrizes a cylinder with radius $1$ and height $2$.
To find the boundary curve of ...
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equation of circle and ellipse to parametrize an arc
an equation of an arc is defined by the equation of a circle:
$$(x-a)^2+(y-b)^2=R^2$$
so
$$x=x(y)=a\pm \sqrt{R^2-(y-b)^2}$$ I want to be safe from using the $\pm$ solutions, so I thought if we can ...
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Dodgy limit swap
Suppose one has
$$F(b)= \int_{0}^{\infty} f(x, b)e^{g(x, b)}dx. $$
Next, suppose that $F(0) = \infty$ and $g(x, 0) = 1.$
Further suppose that $ \lim_{b \rightarrow 0}bF(b) $ converges.
Finally, let $$ ...
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Why did the author used the intersection of the line $y = tx$ with the curve to find the parametrisation?
I want to parametrize the curve $y^2=x^2+x^3.$ When I was reading the textbook by Theodore Shifrin "DIFFERENTIAL GEOMETRY:A First Course in Curves and Surfaces". I found the parametrisation ...
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How do I find the surface integral over a paraboloid in spherical polar coordinates
The question I'm trying to solve asks us to find $\iint_{S}\vec{r}\cdot \vec{dS}$ over the surface described by the paraboloid $z = a^2 - x^2 - y^2$. They offer the parameterisation that $x=a\sin(\...
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Finding a Function $\phi$ Given a Parametric Equation
I am trying to find a function $\phi$ that satisfies the equation $\phi(\psi(t)) = t$, where $\psi(t)$ is given by the following parametric equation:
$\psi(t) = \left(\frac{t^2-1}{1+t^2},\frac{2t}{1+t^...
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Family of curves sharing the same length
Do you have an example of a family of curves $C$ that share the same length $L$?
By family, I mean a set of curves that can be expressed in a generic form - using one or multiple parameters.
Put ...
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How to shift the circular hole of Dupin cyclide while keeping the curve circular without deformation?
I have been interested in studying parametric functions. But when I dealt with Dupin cyclide, I found it difficult to shift the circular hole, for example, if I wanted to shift it towards the x-axis ...
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Determine p so that the line q does not have any point in common with the circle
I am preparing to take an entrance exam for a university in my country, which will happen soon. As part of my preparation, I have been practicing with some sample math tests provided by the university....
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