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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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Finding the parameter values of parametrized curve

I'm kinda confused on how to find the values that a parameter takes given two points of parametrized curve, this is the problem I have: Parametrize $y=x^2$ in the interval of the two points $(-1,1)$ and $(1,1)$ find the values of the parameter…
Sergio Garcia Castro
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Equation of plane containing intersection of two other planes and perpendicular to a third plane?

Let's say you have two intersecting planes. I know that to find the line intersection of them is simply the cross product of their two gradients. Let's say we get the Line $L=(2,3,0)+t<1, -4 ,\ 2>$. Now we want to find a plane that contains the…
Eliot Behr
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r4 parametrization

I'd like to ask how to parametrize linear equations in $\mathbb{R^4}$? I have two equations: $x+2y-2z+w=-1$ $x+y+z-w=2$ I can't come up with how to eliminate two variables
batonas su braškėm
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linear equation of a plane from parametric equation

How to find a linear equation of a plane that passes through the point $(6,0,-2)$ and contains the line with parametric equations $x=4-2t$, $y=3+5t$, $z=7+4t$?
user42789
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True polar parametrization of epitrochoid

I need to cnc machine the internal walls for a wankel engine on a rotary table. For that I need the true polar parametrization of an epitrochoid with roll circle ratio 0.5 with respect to: 1. fixed circle radius R; 2. crank point distance d on the…
user702411
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Parameterization of a parallelogram in $\mathbb{R}^3$

I have a parallelogram in $\mathbb{R}^3$ with the vertices $(0,0,0),(1,1,-1), (1,1,1), (2,2,0)$. How would I find the parameterization of this? Thanks!
hoya2021
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Intersection of a parametric equation and a plane

Use $cos(t)$ and $sin(t)$, with positive coefficients, to parametrize the intersection of the surfaces $x^2+y^2=36$ and $z=6x^3$. I have found $<6cos(t), 6sin(t)>$, but I haven't pined down $z$. I have tried $6(\sqrt{36-t^2})^3$ and $6(cos(t))^3$
NeedHelp
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Continuous paths joining positive x-axis to negative x-axis, through upper half plane

I need a way to parametrise all continuous paths from the positive to the negative x-axis, which go through the upper half plane (in $2$ dimensions). I do not care about the speed of the parametrization, just as long as I can describe the set of…
pureundergrad
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Doubt about the procedure of parametrization

I can't understand how parametrics equations are found. For example, I realize that the parametrization of the curve given by the intersection of the plane $\ 2x+2y+z=2$ and $z=x^2+y^2$…
J.Doe
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Curvilinear abscissa

The curve arc length does not depend on the given parameterization of the this curve, because there are infinite ways to represent it according to the parameter t. For example $$\vec{f}(t) = [t,t^2] \; ; \; \vec{g}(t) = [t^3,t^6]$$ they are both…
user3204810
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Parametrisation of a curve(intersection of a circular cone and a plane)

I am trying to find a parametrisation of the intersection of the graphs of the functions: $f(x, y) = \sqrt{x^2+y^2}$ and $g(x, y) = 20 + x − y$. I used a graphing tool, which gave me the following result: I tried $x=cost, y=sint, z=20+cost-sint$ but…
Relax295
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Simple parametrisation

I have been trying to parametrise the curve $C$: Consider the cylinder $S$ given by the equation $x^2+y^2=1$ and the curve $C$ that is given by the intersection between the cylinder and the plane given by the equation $x + y = 0$ So, I can…
Relax295
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Finding a path of a particle given that it moves parallel to $-\nabla f$

I am told that a particle starting off at $(-1,1)$ moves parallel to $-\nabla f$ where $f(x,y)=x^2-3y^2$. So $-\nabla f=(-2x,6y)$. The question wants the path of the particle and I considered integrating the tangent vector to find the position but…
math111
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Help with parametrization of a surface

I'm having issues with the following problem. Let $a$ be a positive real number. A spacecurve $K_{a}$ in the $(x,z)$-plane is given by $$s(u)=(a\sin(u)\cos(u),0,\cos(u)),\ \ u\in[0,\frac{\pi}{2}]$$. Determine the parametrization of the surface $F$…
James
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Parametrizing a tangent

Find parametric equations of the straight line tangent to the following space curve at the point $P(−3,−9,0)$ on the curve. $$r(t)=(8t^2+63t+46)i+(2t^3-98t-9)j+\frac{70\sqrt3}{\pi}(1+2\cos(\frac{4\pi t}{21}))k$$ I found the tangent equation as…
mathnoob123
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