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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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Parametrizing a triangle in $\mathbb{R}^3$?

I have the points $a = (1,2,3),b = (1,1,1), c = (3,1,2).$ I know this is the region S such that $S = a + \lambda (b - a) + \mu (c-a)$ for some $\lambda,\mu \in \mathbb{R}$. How would I find the restrictions on $\lambda, \mu$?
Twenty-six colours
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Equation of a plane $Ax+By+Cz=D$

I have question abous parametrization and plane equation.How can I write the equation of a plane in the form $Ax+By+Cz=D$ if it is given as $$x=2s+3t$$ $$y=s-2t$$ $$z=3s+t$$.
kejinatsu
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Parameterization of the Intersection of Two Parallel Cones

I need to parameterize the curve of intersection of two congruent, parallel cones, where $\xi$ is the distance between the axes of symmetry, $\zeta$ is the difference in $z$ position of the vertices, and $\alpha$ is the slope of the cones. I wrote…
pseudoeuclidean
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Mapping a unit square to parametrized area by "combining" two functions

I'm a layman in mathematics so I apologize if I'm not using the correct mathematical notations. I would like to map a unit square in $\mathbb{R}^2$ with corners in $(0,0)$ and $(1,1)$ to an area in $\mathbb{R}^2$ that is bounded by two parametrized…
QBrute
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Does this exhibit a parameterization of $S^1 \times S^1$?

Guillemin and Pollack asks: Explicitly exhibit enough parameterizations to cover $S^1 \times S^1 \subset R^4$. My solution is given by $8$ parameterizations (closely following their example of parameterizing a circle): $$f_1(x,z) = (x, \sqrt{1 -…
Dair
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Parametrize the "alpha curve"

I wanted to check if my solution would be correct for the question: Q. Parametrize the "alpha curve" $y^2$=$x^3$+$x^2$ A. y=tx $(tx)^2$=$x^3$+$x^2$ $t^2$$x^2$=$x^3$+$x^2$ $t^2$$x^2$-$x^2$=$x^3$ $x^2$($t^2$-1)=$x^3$ x=$t^2$-1 Substitute x=$t^2$-1…
immk
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Parameterization of objects centered on a specific line

Today while avoiding studying things I will actually be evaluated on I was pondering different things I could try to parameterize. Having no formal exposure to parameterization these types of tasks are fun to see if I can figure out. General…
Prince M
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Explain that two parametrizations represent the same curve

Task: I have the two given curve, in different intervals for t: $x_1(t)=t$ , $y_1(t)=t$ $x_2(t)=t^5$ , $y_2(t)=t^5$ for $t ∈[-1,0]$ and $x_1(t)=2t$ , $y_1(t)=2t$ $x_2(t)=16t^4$ , $y_2(t)=16t^4$ for $t ∈[0,1/2]$ I can think of many different…
netwon1227
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Can you use a parameterization in spherical coordinates to find the area of a surface?

In class, we are learning to find the surface area using the formula $\iint_D \|r_u \times r_v\|dA$ (where $u$ and $v$ are the variables the parameterization uses). The first question was about a sphere with radius R. Instead of writing the…
AndrewEllis
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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 $\lt a\theta, a \gt$ where $a$ is the radius? How do we figure out…
Valued Insight
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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 form I cannot use it as input. I need the input to…
Harry McKenzie
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If a curve is not smooth at a given interval, how can you find the length of that curve at that interval?

I'm wondering if you could somehow calculate the length of the curve even if it is not smooth at a given interval.
ameagari
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What does parametrize all such solutions implies

When I am ask to parametrize all solutions can I just give the answer or do I have to proove that my parametrization is good?
R-B
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Does smoothness depend on parameterization?

Consider an algebraic curve: $$ \vec{r(t)} = t \vec{i} + t^2 \vec{j}$$ It has a tangent vector: $$ \frac{d \vec{r}}{dt} = \vec{i} + 2 t \vec{j} \tag{1}$$ Now, consider parameterizing the first equation by sub $ t \to t^3$, then: $$ \vec{r(t^3)} =…
tryst with freedom
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Parametrization of a paraboloid

I need to parameterize the curve in the first octant to apply Stokes' theorem... $$Z=9-x^2-y^2$$ Using the paramaters... $$C_1 = (3\cos \theta,3\sin \theta,0); 0 \leq \theta \leq \pi/2$$ $$C_2 = (x,0,9-x^2); 0 \leq x \leq 3$$ $$C_3 = (0,y,9+y^2); -3…
Glohelt
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