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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.

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Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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1answer
280 views

Find surface area by calculating surface integrals

Fix a radius $r > 0$ and two angles $ϕ_1$ and $ϕ_2$, with $−π/2 < ϕ_1 < ϕ_2 < π/2$ Find the surface area of the portion of the sphere of radius r with latitudes between $ϕ_1$ and $ϕ_2$. ...
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Parametrisation of cardioid $r=1-\cos \theta$

I know cardioid $r=1-\cos \theta$ can be parametrized to $$\gamma:[0,2\pi]\rightarrow \mathbb{R^2}, \space \gamma(\theta)=((1-\cos \theta)\cos \theta, (1-\cos \theta)\sin \theta)$$ But how is this ...
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355 views

Circular Arc Prametrization not Using Radius

In an optimization problem I have to parametrize a circular arc. Thus far, I have reduced a more general problem to the figure below: The figure shows a symmetrical circular arc, with chord length L,...
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Any parametrization $h\colon S^1\to \Gamma$ possible?

Consider the following model, and, in particular, the second picture given in the link, namely the blue line which represents a trajectory which I call $\Gamma$. Is it possible to give any ...
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351 views

How to parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute a tangent plane

How do I parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute the tangent plane at $(1, \frac{1}{3}, 0)$ using the resulting parametrization? I know that the tangent plane should be $$\nabla(x^...
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88 views

Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
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532 views

How to prove that a curve is an injective regular parameterization?

I have the above assignment: Prove that the curve $σ:\Bbb R → \Bbb R^2$ given by $$σ(t)=\left(\frac{t}{1+t^4}, \frac{t}{1+t^2}\right),$$ is an injective regular parameterization, but not a ...
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277 views

Parametrization of $a^2+b^2+c^2=2d^2$

Is a complete parametrization of primitive solutions to the equation $$a^2+b^2+c^2=2d^2\qquad a,b,c,d \in \mathbb{Z}$$ known? A reference would be great. Solutions to $a^2+b^2=c^2$ give solutions ...
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56 views

Is there a way to parametrise general quadrics?

A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && ...
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How to reparametrize with respect to arc length?

In part (a), I calculated that s(t) = $\int_0^t \! (\frac{t^2}{2} + 2t) \, \mathrm{d}t$ = $t^2/2 + 2t$ I'm unsure how to solve part (b). My attempt is: $t^2/2 + 2t$ = s $\iff$ t = -2 $\pm$ $\sqrt{...
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1answer
42 views

Parametrizing a surface with a single parameter.

Let $S=\{(x,y,z)\in \Bbb R^3: x^2+y^2=z^2 \wedge 0\leq z\leq 4\}$. I'd like to create a function $\vec \Sigma:A\subseteq \Bbb R\to \Bbb R^3/\vec \Sigma(t)=(x(t),y(t),z(t))$ and $\text{Im}(\vec\Sigma)=...
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1answer
221 views

Parametric line segment in 3-space

If one wants to parametrize a straight line segment in $\mathbb{R}^3$, which goes from $(1,0,0)$ to $(0,1,\pi/2)$, would this approach be correct? First, we come up with the $xy$-plane equation, ...
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1answer
55 views

Parametrization of intersection of curves

If one wants to parametrize the curve of intersection in $\mathbb{R}^3$ of the surfaces $y=x^2$ and $z=x^3$, would it be correct to parametrize this curve as $\bf{g}$$(t)=(t, t^2,t^3)$? The reason I'...
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2answers
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I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do.

I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do: doing an analogy with how we represent functions from $\Bbb R$ to $\Bbb R$ or from $\Bbb R^2 \to \Bbb R$...
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1answer
25 views

One parametric family that interpolates continuously between identity and natural logarithm on (0,1]

I am looking for a family of continuous functions $f_p$, $(0,1]\to\mathbb{R}$, and $p\in [1,\infty)$ that fulfill $$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$ for $x\in (0,1]$. I ...
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1answer
56 views

Reverse-engineering a parametrization

Let's say you have a polynomial depending on complex parameters $A,B$: $$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$ One parametrization of zero points of this polynomial is given by $$ x(t) = -(t + t^{-1})...
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1answer
50 views

Analytic formula for parameterizing the below family of curves

I'm trying to find an analytic formula for a curve that can look like any of the curves below depending on one or more parameters. My initial thought was to use exponentials, something that might ...
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1answer
60 views

Which is this curve?

What curve is given from $$\gamma (\theta)=\left (\sin (\theta +c)\cos \theta , \sin (\theta +c)\sin \theta \right ), \text{ where } c \text{ is a constant } $$ ? How can we find it?
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1answer
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Parametrization of the curve

A cycloid is a flat curve that is traced by point of the rim of a circle while the circle rolls without slippage on the line. Show that if the line is the axis $x$ and the circle has radius $a>0$, ...
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Surface area with double integral - how to parameterize?

Problem: Find the surface area of the part of the cylinder $x^2+z^2 = a^2$ that is inside the cylinder $x^2+y^2 = 2ay\;$ and also in the positive octant $( x \ge 0, y\ge 0, z\ge 0$). Assume $a > 0$....
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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 ...