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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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 ...
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How To Measure Work Done Against Friction On A Non-Linear Path?

I've been trying to solve how to find the Work done against friction as a ball rolls down a curve, but haven't been able to find anything matching what I want online. The solution I have came up with ...
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Parameterizing the parabola $9x^2 +y^2-6xy+4x-4y+1=0$

Find parametrization of curve given by equation: $$9x^2 +y^2-6xy+4x-4y+1=0$$ My attempt: Let's notice that \begin{split} 9x^2 +y^2-6xy+4x-4y+1=0 & \iff (3x)^2 -6xy + y^2 +4x -4y +1=0\\ & \...
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What is meant by Hermann Weyl's proof by homogeneity of the bijective property of the affine parameterization of the time continuum?

This is a specific question I have regarding my broader question: How would Hermann Weyl's development of the time "continuum" be handled in contemporary mathematical language? I've ...
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How would Hermann Weyl's development of the time "continuum" be handled in contemporary mathematical language?

The source of the quoted material is Space-Time-Matter by Hermann Weyl I started trying to summarized the following development of the mathematical treatment of parameterized time. Then I realized ...
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Why is this composition of curve parametrisations a diffeomorphism?

In our class, we wrote the following: Connected curves (1-dimensional manifolds) can be parametrised globally; $\overrightarrow{\gamma}: I \subseteq \mathbb{R} \to \Gamma \subseteq \mathbb{R}^3$, ...
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How to convert a Cartesian Equation to Parametric Equation?

I'm trying to 3D plot the following cartesian equation in Blender 3.1: $$ \left(x - x\left(\frac{z}a\right)\right)^2 + \left(y - y\left(\frac{z}a\right)\right)^2 = r^2$$ But in it's current Implicit ...
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What is the general formula (parametric expression) for a surface generated by rotating y=f(x) around y-axis?

Here is a simpler version of my question. Given curve $\displaystyle y=f(x)$ rotated around the y-axis. The generated surface can be described with parameters $\displaystyle x$ and $\displaystyle \...
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Why the wire is represented by $y$? why $\bar{x} = 0$?

Here is the problem in Stewart "Calculus, early transcendentals, 9th edition" My question is: Why the wire is represented by $y$ in the equation that expresses the density? why $\bar{x} = 0$...
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finding intersection of a parametrical representation

I've managed to answer the question and get S = 3, so by substituting S into L the intersection point is (11 6), but what I'm confused about is, when I substitute S into L' to double check my answer I ...
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Why is my parametrization for this curve incorrect for this line integral?

I'm trying to solve this line integral on Paul's notes: https://tutorial.math.lamar.edu/Solutions/CalcIII/LineIntegralsPtI/Prob2.aspx, but I don't know why my parametrization isn't working. For the ...
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Surface Parameterization and Coordinate Systems -- how do they mesh?

I am currently thinking about computing the surface area of surfaces in $\mathbb{R}^3$ through the lens of a traditional multivariable course, and I'm a bit confused about how parameterizations are ...
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What makes a function parametric? Why are they used so much?

I believe I lack a great deal in my understanding of parameterisation. The following vector-valued function $f(t) = (t\cos(t), t\sin(t))$ is said to be a parametric function but I don't understand why?...
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Proof $\iint_{x^2+y^2\le1}x \cdot f(\cos(x),y)dxdy=0$

I’m trying to proof : $$\iint_{x^2+y^2\le1}x \cdot f(\cos(x),y)dxdy=0$$ For a continuous function $f$ in $\mathbb{R}^2$ . My try was according to parametrization but I failed to get any help through ...
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Why are the red parts of this expression present $\int_a^b \sqrt{g_{cd}\frac{dx^c}{\color{red}{d\lambda}}\frac{dx^d}{\color{red}{d\lambda}}}d\lambda$?

I am now trying to teach myself about geodesics, and a passage of my notes reads: In this chapter geodesics have been introduced as generalizations of straight lines through $$\frac{Du^a}{ds}=0$$ ...
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Parameterization of the Costa-Hoffman-Meeks surface

I developed a library for the evaluation of some special functions such as the Weierstrass elliptic functions. I like to add some pictures to illustrate it, and this allows to check that the library ...
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Parametric Solution to system of two quadratic equations

Is there a way I can find a parametric (rational) solution to the following system: $$\begin{cases}& x^2-y^2 = 2\\ &8x^2-z^2 = 7 \end{cases}.$$ I know a solution $(x,y) = (\frac{p^2+2q^2}{2pq},...
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How to find the path parametrization when calculating the circulation of a vector field?

So I have this vector field $$V(x,y)=(xy,x+y)$$ that I am calculating its circulation with 2 methods (Using a parametrization and using Green's theorem), The domain we're working on it is $${(x,y)\in ...
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Stability of trust region algorithm as a function of the number of free parameters

I use a trust region regression algorithm for a Least Square Minimization. I used the first two terms of the Taylor series as a model for the confidence interval, based on the usual methodology of ...
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Rational Parametrization of a curve

currently, I'm learning how to parametrize basic curves, but I don't understand the Rational Parameterization of a curve. For example: $\frac{(x-x_0)^2}{a} + \frac{(y-y_0)^2}{b} = 1$ or $y-y_o = a\...
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Calculating linear speed of a dough hook stand mixer.

I'm trying to calculate the effective linear speed of the dough hook arm of a stand mixer, such as a KitchenAid mixer. If I can figure out the arc length I think I can get the speed easily enough. I ...
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Find the point where $\tau(t)$ is minimal.

Question. $\vec{r}(t)=\left[\frac{4}{9}(1+t)^{3/2},\frac{4}{9}(1-t)^{3/2},\frac{1}{3}t\right]$ is a parametrization of a curve $C$ ($t\in\left[-1,1\right]$) find the point where $\tau(t)$ is minimal. ...
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Parameters in survival function not changing output.

I am looking at this paper, and they are modeling "cohesiveness" in a group by doing the following: First, they define a "common action" of all individuals in a group by using the ...
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1 answer
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Surface area under a parametrized curve

I have a curve in the form $\gamma[t_P,t_Q]$ with the endpoints $P=\gamma(t_P)$ and $Q=\gamma(t_Q)$. Also the curve can be written as $r=r(\phi), \phi_P\leq\phi\leq\phi_Q$. The task is to prove these ...
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Given a vector valued function, draw curve, find the arc length, and find the curvature

I'm doing some practice problems and am stumped: Given: $r(t) = \cos(t^3)\,i + \sin(t^3)\,j + t^3\,k$. Draw the curve for $t \geq 0$, using any projections or additions that help explain the curve. ...
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Parameterization of the circle [closed]

Is $x = r\cos θ$, $y = -r\sin θ$ a good parameterization of the circle $x^2+y^2=r^2$? I know that parameterization $x = r\cosθ$, $y = r\sinθ$ is common but I'm wondering if this is okay too.
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How to Find Arc Length Parametrization of Combined Functions?

The arc length parameterization for the circle is easily written as $$x(t), y(t) = r\cos(t), r\sin(t)$$ But how can we do this if our geometry is composed of multiple arcs of circles with different ...
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Curve $\Gamma$ is parametrized by $\Gamma(u)=(u+\sqrt3\sin u,2\cos u,\sqrt 3u-\sin u)$. Find circular helix & isometry $T$ with $T_{\gamma}=\Gamma$.

Question The curve $\Gamma$ is parametrized by $$\Gamma(u)=(u+\sqrt3\sin u,2\cos u, \sqrt 3u-\sin u)$$ Find a circular helix and an isometry $T$ such that $T_{\gamma}=\Gamma$. I have tried to ...
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1 answer
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Is it possible to parametrize a chart by arc-length?

Let $M$ be smooth connected 1-dimentional manifold. I think that there is an atlas of $M$ such that every chart of it is parametrized by arc-length I know that there is an atlas of $M$ such that every ...
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Showing that $\forall h \in k[x_1,\dots,x_n]: \exists N\in \mathbb{N}, F: g^Nh = F(t_1,\dots,t_m,g_1x_1,\dots,g_nx_n)$

This is the a-part of problem 11 from the book Ideals, Varieties and Algorithms (4th ed., pp. 142). Let $k$ be a field and define the rational parametrization $x_i = \frac{f_i(t_1,\dots,t_m)}{g_i(t_1,\...
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1 vote
1 answer
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Showing that equations $f_1,\dots,f_n\in k[t]$ fill up the variety $\mathbb{V}(\left<x_1 - f_1(t),\dots,x_n - f_n(t)\right>\cap k[x_1,\dots,x_n])$

This is the a-part of the exercise 10 (pp. 141) in Ideals, Varieties and Algorithms. Let $k = \mathbb{C}$ and define a curve $\gamma(t) = (f_1(t),\dots,f_n(t)$, $f_1,\dots,f_n\in k[t]$. I want to show ...
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How did they come to this parameterization of an ellipse?

The topic is about line integrals, the question: Calculate $\int_\gamma \vec x \cdot d\vec s$ for $\vec v = (0,xy^2)$ and $\gamma$ die ellipse with the equation $x^2+(y/2)^2=1$ once to go counter ...
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$z_0+r*e^{2\pi imt}$ meaning of $m$

I know that $f:[0,1] \to \mathbb{C}, f(t)=z_0+r*e^{2\pi imt}, r>0, m\in\mathbb{Z},z_0\in\mathbb{C}$ parameterizes the circle around $z_0$ with radius $r$. But how does $m$ affect the ...
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3 votes
1 answer
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Finding an envelope for a moving circular sector

Preamble: I want to find the curve which bounds a moving circular sector, i.e. an envelope for the following family of plane curves. Suppose that we are given a "perspective" point $T$ and ...
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Finding a curve lying on a given surface

I have a bohemian dome given by the parametric equation $$x= a \cos u$$ $$y = b \cos v + a\sin u -1$$ $$z=c\sin v$$ , where $$a= 0.5$$ $$b = 1.5 $$ $$c= 1$$ , and $u,v \in [0,2\pi)$ . I know that I ...
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4 votes
1 answer
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Finding points on a parametric curve where curvature changes

I am a engineer working on Wankel motors, where a simil-Reuleaux triangle rotates eccentrically in a 8-shaped form: (from Wikipedia) Fascinated by this mechanism, I was studying the meaningful ...
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When are two space curves with same image equivalent?

Let $I_1, I_2$ are two non-degenerating intervals of $\mathbb{R}$ and, let $\gamma_j : I_j \to \mathbb{R}^n,\quad j=1,2$ be two parametrized regular $\mathcal{C}^r$-curves with same trace (image in $\...
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Isometric parameterization of curves

In section 1.3 of "A Geometric Approach to Differential Forms", we want to integrate the function $f(x,y) = y^2$ on the top half of the unit circle, For the 2 parameterizations of the top ...
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1 answer
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I am having trouble solving this problem. I don't know how to parametrize this curve and sketch the direction of the curve.

I'm pretty sure how to find arc length using the formula. But I'm stuck on 1a. I solved for x for the second equation and got $ \ x \ = \ \frac{(2-z)}{\sqrt{3}} \ \ . $ I then substituted this into ...
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3 answers
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Parametrizing a circle in a counterclockwise direction

How do I parametrize a circle in a clockwise direction? For instance, if the circle is in a counterclockwise direction, the parametrization would be $$c(t) = (r \cos t,r \sin t).$$ I've seen a lot of ...
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1 vote
1 answer
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Find the surface area of cylinder between the intersection curves and cone, eliptic paraboloid.

The parametric equation of the eliptic paraboloid: $${r}(u,v) = \left(u\cos(v); u\sin(v); \dfrac{1}{4}u^2-\dfrac{21}{4}\right)\qquad 0 \leq v \leq 2\pi,\,\,0 \leq u \leq +\infty$$ The parametric ...
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How to calculate surface of area with given equations $x^2 + y^2 \leq a^2$ and $az = xy$

For $a>0$ I'm trying to calculate the surface of area with given equations $x^2 + y^2 \leq a$ and $az = xy$. I think it should be done with $$\iint_D |\vec{r_u} \times \vec{r_v}| \,du\,dv $$, which ...
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1 vote
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How do I find the parabola parameter equation?

Show that $x(t)=\cos^4t, y(t)=\sin^4t$ is a parametrization of the parabola $(x-y-1)^2=4y$. I think that to solve this problem, we need to know how to find the parametric equation of the parabola. So ...
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1 vote
1 answer
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Show that $x(t)=2r\cos^2(t)$, $y(t)=2r\sin t\cos t$ is a regular parametrization of the real circle of radius $r$, centre $(r,0)$.

Show that $x(t)=2r\cos^2(t)$ and $y(t)=2r\sin t\cos t$ is a regular parametrization of the real circle of radius $r$, centre $(r,0)$. I set up the equation of the circle as follows. $(x-r)^2+y^2=r^2$ ...
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1 vote
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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 (...
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1 vote
1 answer
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Angle between two curves on curved space

Today I did an exercise where I found the angle between two curves $\mathbf{x}(\lambda_1)$ and $\mathbf{x}(\lambda_2)$ on the surface of a unit sphere with line element $ds^2 = d \theta^2 + sin^2 \...
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How to parameterize intersection of surfaces

S1 is the circular cylinder of radius 2 with the y-axis as its central axis. S2 is the surface described by $y = x^2 - z^2$. Curve C is the intersection of these two surfaces. I want to calculate the ...
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Parametrized surface is vaguely tetrahedral?

My son has been looking at space curves parametrized as $$ x\mapsto (\cos(ax), \sin(bx), \sin(cx)) $$ for various integer triples. When $(a,b,c)$ involve few, but distinct, primes, the image seems to ...
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1 answer
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What are some nice continuous mapping from the closed ball $\overline{B}^n$ to the $n$-sphere $S^n$

While proving that the quotient space $\overline{B}^n/S^{n-1}$ is homeomorphic to $S^n$, I needed to construct a continuous function $p:\overline{B}^n\to S^n$. I figured that by fixing two points $R = ...
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$V(y^2-x^3)$ not isomorphic to $\mathbb A^1_K$

We can find everywhere that $\phi :\mathbb A^1_K \to V(y^2-x^3)$ sending $t$ to $(t^2,t^3)$ defines a bijective map but not a morphism of affine varieties. To me it's not clear why it's even ...
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