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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.

Parametrization is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. "To parameterize" by itself means "to express in terms of parameters".

Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrisation consists thus of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.

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Parameterizing same curve in different directions

$$\textbf{r}_1(t)= \sin t \textbf{i}+ \cos t \textbf{j}$$ and $$\textbf{r}_2(t)= \cos t \textbf{i}+ \sin t \textbf{j}$$ represent the same curve but traversed in different directions. $\textbf{r}_1$ clockwise and $\textbf{r}_2$ anticlockwise. It is…
Eiraus
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Parametrization for the ellipsoids

Can someone help to describe some possible parametrizations for the ellipsoid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1?$$ I am thinking polar coordinates, but there may be the concept of steographic projecting (not sure how to…
mary
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Why is two variables enough to parametrize a surface?

For a surface S, how do we know that two variables is always enough to parametrize the surface? I am thinking that it has something to do with the number of directions you can move in. For parametric curves, you could only move in two directions:…
Akash
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Parametric Equation of $y^2-2x^2=x^3-2y^3$

Context The context of the problem is the derivative doesn't exist at the point $(0,0)$ if you look at the curve as is in the $xy$ plane. It an intersection point. So by parametrizing the curve we can determine a suitable $t$ to determine…
Mando
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Confusing Parametrization

Problem: Determine whether the following statement is true: The line $y=3x$ can be parameterized as $$x=\cos (2t)$$ $$y=3\cos(2t)$$ $$-\infty
user532874
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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 1D curve embedded in a 2D space. So I would think…
Radek Svoboda
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Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $

Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $ in the plane $ x+y+z=1 $ with positive orientation . $$ $$ I have thought the parametriation: \begin{align} x(t)=1+ 3 \cos (t) \hat j +3 \sin (t) \hat k \\ y(t)=1+3…
MAS
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Polar coordinates for $x^n + y^n = 1$, $n \in \mathbb{R}_{>0}$

The equation $x^n + y^n = 1$ has a familiar parametrized version for $n=2$, which is a circle $(\cos(\theta), \sin(\theta))$. For even n, the greater the value, the closer it comes to a square (infinite norm). Do you know an algebraic manner to…
Bruno Peixoto
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Rational paramaterization of curve.

I'm trying to find a rational parameterization of $$x^2+4xy+4y+2=0$$ We can see that there are no integer points on this curve, but how to prove it? We can get this form: $(x+1)(x+4y-1)=-3$ and we can analyze the conditions (for example, if $ab=-3$…
PabloZ392
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Proof of parametrization

Suppose that I have an implicit curve $$F(x, y)=0 $$ and a candidate $$ \overline{r} : I \rightarrow \mathbb{R}^2$$ for its parametrization. $$ \overline{r}(t) =x(t)\hat{i}+y(t) \hat{j}, \ \ \ t\in I $$ How could I prove rigorously that the…
mathslover
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Theory behind conservation of number of free parameters?

It is an often-used rule in mathematics and statistics that if one representation of an object has $n$ "free" parameters, then another representation of that same object requires at least that number of parameters. E.g. I was just now reading along…
Him
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How to parametrize this surface $x^2 +2y^2+3z +4 = 0$?

How do I parametrize this paraboloid S: $x^2 +2y^2+3z +4 =0$? I first isolated the z-component to make it a function of $f(x,y)$, howevere that gets me $z= (-x^2-2y^2-4)/3$ which leaves me stuck on how to parametrize this into $\vec{r}(r,\theta)$
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Parameterized ellipse

The general equation of an ellipse with center in the Cartesian axes origin is $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ This ellipse can be parameterized by returning it to a circumference in some way: replacing $$\dfrac{x^2}{a^2} =…
user3204810
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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 that $f(x,y)$ is the equation of an ellipsoid. I…
StudentMaths
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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 sense they are equal. I'll give a trivial…
Michael Stachowsky
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