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Questions tagged [parametric]

For questions about parametric equations, their application, equivalence to other equation types and definition.

In mathematics, a parametric equation of a curve is a representation of this curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. This contrasts with implicit equations that define a curve as the zero set of some equation in the coordinates.

The parametric forms of curves are well-suited for drawing on a computer, while their corresponding implicit forms are useful for analytic manipulations (intersections, etc.)

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Converting between explicit function and parametric function

Given an explicit function $y = f(x)$, how to convert it to the respective parametric functions $x = f_1(t)\; y = f_2(t)$? Given parametric functions $x = f_1(t)\; y = f_2(t)$, how to obtain the respective implicit function $f(x,y) = 0$?
Gavin
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Parametric Curve Tangent Equations

Let a curve be given in the parametrized form by: $r(t) = (2\cos t, 2\sin t), 0 \leq t \leq 2\pi$ Find the equations of the tangents to the curve at each of its points $(X_0, Y_0)$. Having gone through some text, it never really directly approaches…
mwtmurphy
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Find the coordinates of the point where the normal cuts the curve again problem

Find the equation of the normal to the curve $x=2\cos\theta$, $y=3\sin\theta$ at the point where $\theta=\frac{1}{4}\pi$. Find the coordinates of the point where this normal cuts the curve again. Okay so I found the equation of the normal to the…
Rogerc1979
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Calculate u in terms of time such that a particle maintains a constant speed following a parametric equation

I have a parametric equation given by: $x=\cos(6u)$ $y=\sin(4u)$ And I understand that the speed of a particle at any given t is: $\sqrt{\left(\dfrac{dx}{du}\right)^2 + \left(\dfrac {dy}{du} \right)^2} $ Which for my parametric equation would…
TenEighty
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Understanding Parameters

My textbook (New Tertiary Mathematics, Volume 1 Part 1, Pure Mathematics: The Core, by C Plumpton & P S W Macilwaine) introduces Parameters in the following manner: "The coordinates of a point on a curve can usually be expressed in terms of a third…
Au101
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Pythagorean Quadruple Parametric Equation in 3 variables

I am looking for a pythagorean quadruple generator in 3 variables. I know this one with 4 variables. $$a=2mp+2nq \\ b=2np-2mq \\ c=p^2+q^2-(n^2+m^2) \\ d=p^2+q^2+n^2+m^2 $$ Anyway to do this?
Seth Kitchen
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Parameterizing an ellipse

Given the ellipse $(x-1)^2 + \frac{y^2}{4}= 1$, parametrize the curve in polar coordinates. I've forgotten something very basic here. Can someone help get me started?
Alec
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How do I successfully combine these two paramaterized equations?

I'm working on a set of equations that would tell a hypothetical robot soccer player whether or not to pass a ball to a teammate. After a lot of algebra, I arrived at these equations for the partial boundary of a…
HDE 226868
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why can't we eliminate the parameter of straight lines in higher dimensions

Why we can't remove the parameter and find the Cartesian equation of straight lines in higher dimensions ?
Mohamed Osama
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Parameterizating a function generator

I'm trying to program a morph animation between a quarter of a circle (an arc) and a straight line, while keeping the length constant. In other words, I need to program a "function generator" $f(t), t=0..1$ so that $f(t)=y(x)$ for the given…
liorda
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How do I change parametric in t to cartesian when I can't re-arrage for t

I'm stuck looking at this parametric equation which I have to put in cartesian form $x=t^2+ \frac1t$, $y=t^2-\frac 1t$ Something to do with difference of two squares? I can't see how to eliminate $t$ $xy=t^4-\frac{1}{t^2}$ doesn't help. I…
Saxobob
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Find the values of parameter $a$ so that....

Determine all the values of real parameter $a$ so that the equation:$$(x-a)[log_4(x-5)-1]=0$$ admits a maximum number of real solutions. Thank you!
wonderingdev
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Is it possible that $f_1,f_2,\dots,f_n$ are not all differentiable, but $\alpha:I\to \Bbb{R^n}$ is differentiable?

Consider the parametrized curve $\alpha:I\to \Bbb{R^n}$. These notes say that $f_1,f_2,\dots f_n$ being differentiable $\implies$ $\alpha$ is differentiable. I wonder why the converse is not true. Is it possible that not all of $f_1,f_2,\dots f_n$…
algebraically_speaking
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Find rectangular equation from parametric equation???

Find rectangular equation from parametric $ x = t^{2} + t $ $ y = t^{2} - t $ I tried finding the equation but I am stuck here: $ x - t^{2} = t $ $ y = t^{2} - t $ $ y = t^{2} - ({x - t^{2}}) $ $ y = t^2 - x + t^2 $ $y = 2t^2 - x $ Is there even…
SwagmcMuffin
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Tangent Line of a Parametric Curve

Deduce the equation of the tangent line to the curve defined by the equations x=cosh(t), y=sinh(t), and z=ct I have somewhat of a good grip on the definition of a tangent line, but the lack of a given point is throwing me off compared to other…
James Snyder
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