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.

675 questions
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How do I parametrize a cone sitting on the $xy$ plane

I know how to parametrize a cone surface that has its vertex on the origin. However, how should one parametrize a cone that is sitting on the $xy$ plane, that is to say the cone has its base on $xy$ ...
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Parametrization of $x^2-y^2=1600$

While trying to compute the line integral along a path K on a function, I need to parametrize my path K in terms of a single variable, let's say this single variable will be $t$. My path is defined by ...
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Finding points on a plane or line in $\mathbb{R}^3$

I'm just getting familiar with analytical geometry and basically I have two probably very simple questions. Question 1: Let's assume I have some plane, i.e: $\pi: x - 2y + 4z - 8 = 0$ What should I ...
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Parameterize union of two shapes in 3space

I am supposed to parameterize the union of the two shapes $x^2 + y^2 = 1, z = y$. I do not even know how to get the union of the two shapes. When I graph the two shapes the intersection does not ...
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Find a parametric equation of the parallelogram two of whose lines are connecting $(1,-2,1)$ with $(1,4,3)$ and $(1,-2,1)$ with $(2,3,-1)$

I first try to find a parametrization of the plane passing through the points $A(1,-2,1),B(1,4,3)$ and $C(2,3,-1)$. We can take $\overrightarrow{AB}$ and $\overrightarrow{AC}$ as two linearly ...
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Finding cartesian equation of curve with parametric equations

A curve has parametric equations $x=a \sin(⁡t)+b \cos(⁡t)$ $y=a \cos⁡(t)-b \sin⁡(t)$ How do I eliminate t to find the Cartesian equation here? I've tried different weird approaches, i.e. squaring ...
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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 ...
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Meridians and Parallels on a Unit Sphere

Let $S$ be the unit sphere in $\Bbb R^3$ with centre $(0, 0, 0)$ $\sigma(u, v) = (\cos v/\cosh u,\sin v/\cosh u,\tanh u)$ is a parametrization of $S$ minus the north and south poles. Show that ...
In the book of E. Bloch, at page 171, it is stated that Corollary: Let $U\subseteq \mathbb{R}^2$ be an open subset, and let $f: U \to \mathbb{R}^3$ be a smooth map. If for $p \in U$, $Df(p)$ ...
(Two parameterizations): $$\vec r_1(t) = (t^3, t + 1), t ∈ [0, 1]$$ $$\vec r_2(t) = (t^6, t^2 + 1), t ∈ [0, 1]$$ How can I show that these two parameterizations represent the same line in plane? ...