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

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1answer
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Parametric equations of a cycloid

Given a parametric equation of a cycloid ($t \in R$): $$ x(t)=r(t-\sin(t)); \\ y(t)=r(1-\cos(t)). $$ A vector $v=(x'(t),y'(t))$ if is not equals to zero then is a tangent vector to the curve at ...
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2answers
18 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
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1answer
27 views

To prove a hyperbola being orthogonal via parametric equations

Hi could anyone help me with this problem. Prove that the hyperbolae x^2-y^2 =a and xy=b are orthogonal to each other at each point they intersect.Here a and b are non zero parameters i first do a ...
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1answer
16 views

parametric equations multivariate calculus

could anyone help me to solve this problem Given a parametrization of the tangent line to the curve,(x(t),y(t)) at t=a is: ...
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1answer
23 views

Finding equation of directrix when the parametric equation of parabola is given.

If the parametric equation of the parabola is $( x = t^2 + 1 , y = 2t + 1 )$, then find the equation of the directrix. This was the question in my last test in which I got stuck and wasted much of my ...
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2answers
22 views

Equations of a projective variety from parametric ones

How does one find equations of a variety given parametric equations (i.e. a regular map) in projective space? For example, I got stuck in finding the equations of the curve in $\Bbb{P}^2$ described by ...
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2answers
157 views

Do these two parametric equations represent the same curve?

Could anyone help me with this $x = 1 + \cos t$, $y = −2 + \sin t$, $π ≤ t ≤ 2π$; $x = t$, $y = −2 −\sqrt{2t − t^2}$, $0 ≤ t ≤ 2$ For the following parametric equations, how do I determine whether ...
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2answers
122 views

How do I find equation of this curve?

I need to find equation of the curve as shown below, for which, I need to find equation for upper part. lower part is half circle. upper part is a constant distance from circle with line passing ...
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2answers
36 views

Finding the equation for a (inverted) cycloid given two points

If I have two points on a Cartesian plane, and I know that they are connected by a cycloid, then how do I find the equation for that cycloid? For background information, I have been playing around ...
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0answers
43 views

Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
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1answer
29 views

Geometric explanation of a contour's image

$$\gamma(t)= t^2 + i\, t^4 , \quad t\in [ -1, 1]$$ What is the geometric explanation of the image of the above contour? Intuitively , I think it's ellipsoid-like, but I don't know how to put it in a ...
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Comparing normal distributions using a two sample Kolmogorov-Smirnov test

I have used a two sample Kolmogorov-Smirnov test to compare the distributions of two sets of data. I know that the K-S test is a non parametric test, however the distributions of data I'm comparing ...
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2answers
136 views

Find the length of the curve $x^{2k}+y^{2k} =1$

I want to find an expression for length and find the limit $k\rightarrow \infty$ The answer is obviously 8, if we look at the graphs.
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2answers
45 views

Parametric Curve Representation of a Square from a Circle

Given the parametric equation of a unit circle $$ \vec r(\theta) = \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix}, \quad 0 \leq \theta \leq 2\pi $$ It seems that there is some function $$ f ...
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1answer
26 views

Area of the surface generated by revolving curve around y-axis

So I did something wrong in my solution because I'm not seeming to get the right answer. $$\int_c^d 2\pi (4 \sqrt{9-y}\sqrt{1-\frac{4}{9-y}})~\mathrm{d}y$$ combine square roots and move out ...
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0answers
22 views

Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
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1answer
433 views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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1answer
183 views

Motion on a parametric surface

Please excuse what will surely turn into a long rambling question, full of incorrect terminology. I'm trying to figure out the mathematics of moving on a parametric surface - that is, for some ...
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2answers
42 views

Area inside curve given by parametric equation

I have this parametric equation: $$ \text{C}: \qquad \vec r(t)=\left(\cos^3(t), \sin^3(t)\right), \qquad t \in [0, 2\pi] $$ How to find the area inside of $\text{C}$? I have this formula, but I ...
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0answers
60 views

Surface parametrization and calculating its area

I have to find the parametric equation of the surface of the sphere inside the cylinder and above the $z=0$ plane, as shown in this picture. $$ \text{Sphere: }x^2 + y^2 + z^2 = 1\\ \text{Cylinder: ...
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1answer
243 views

Is this valid parametric equation to create control points for a helix in 3D space?

Is this a valid way to compute new points that are on a helix and if not what is it wrong? The Cartesian coordinates of each new helix control point could be described by the following ...
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1answer
337 views

Shortest distance between a 3D parametric surface and a point

Right now I'm working on a library for finding the distances between objects in Lua. I've had some trouble finding the distance between a point and a bounded plane. I'm using these parametric ...
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1answer
753 views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
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0answers
27 views

Area of the portion of the cylinder $x^2+y^2 = 9$ for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$

Problem: Find the area of the portion of the cylinder $x^2+y^2 = 9$, for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$ I first solved this by parametrizing the surface. $x = 3\cos(u)$ , ...
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1answer
24 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
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1answer
22 views

At what extent I can use trigonometric functions and properties with parametric curves?

I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the ...
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2answers
33 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...
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2answers
20 views

Explanation of how to go from polar to parametric equations.

I was wondering how you can make a polar equation parametric, and I just don't get it. My book says that for $r = f(\theta)$, $x = f(t) \cos t$ and $y = f(t) \sin t$, but this makes absolutely no ...
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3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
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1answer
25 views

parameterize the following functions [closed]

Help determining the parameterized solution of the following functions $$a) { \left( x-2 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 }=4\quad if\quad 1\le y\le 3$$ $$b) \frac { { \left( x+3 \right) ...
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2answers
22 views

Parametric equation for intersection of curve

Here's the three part question: A) Find parametric equations for curve which is the intersection of the cylinder $x^2 + z^2 = 1$ and the plane y = -x. B) Show that the curve lies on the surface $x^2 ...
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0answers
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Parametric equation of a circle with given radius and starting point

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v = 0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ ...
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1answer
41 views

Parametric equation of a circle given starting point.

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v=0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ and ...
2
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1answer
42 views

Parametric & Trigonometry

$$x=7\sin(t)+\sin(7t)$$ $$y=7\cos(t)+\cos(7t)$$ How would I solve this one out? I have to simplify the two enough to graph it. I squaring the two and adding them together, but I hit a roadblock: ...
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0answers
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Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
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2answers
45 views

Parametric Equations

$x=3\sin^3t$ $y=3\cos^3t$ How would I even begin to work out this one? I'm supposed to graph it, but I have no clue what how to even start it.
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0answers
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Partial derivative of straigh-line parametrized integral

I would like to evaluate the following $$ F(\mathbf{r}_1,\mathbf{r}_2) = \int_0^1 ds~f(\mathbf{r}_1 + (\mathbf{r}_2 - \mathbf{r}_1)s) $$ where $\mathbf{r}_{1/2} = (x_{1/2} , y_{1/2})$, i. e. a ...
2
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1answer
38 views

Improper parametric arc length

The first thought I had to solve this problem was using the integral, $$ \int_1^\infty \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\: \:dt $$ Once you solve for the derivatives ...
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0answers
13 views

How can I find the projection of a sphere onto a direction vector subject to constraints?

Given a direction defined by a unit vector, I can find the projection of a point on that direction by using the dot product. The projection interval of a set of points is the max and min values of the ...
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1answer
26 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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2answers
30 views

Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
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0answers
26 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
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1answer
34 views

Help Needed Changing Parameter

Given that $r(t)=(4(\sin(t)−t\cos(t)),4(\sin(t)+t\sin(t)),(3/2)t^2)$ is a vector-value position function. Find the arc length function $s$. I need to change the parameter before deriving to calculate ...
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0answers
50 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
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1answer
55 views

How to find the parametric equation of $x^y=y^x$ without Lambert W function?

This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos ...
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2answers
2k views

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where ...
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2answers
51 views

Multivariable calculus - scalar field

I don't know how to solve this problem. Determine if $\mathbf{F}$ is or not the gradient of a scalar field. If it is find the corresponding potential function f. $\mathbf{F}(x,y,z)= 3y^4 ...
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2answers
88 views

Find Vector and Parametric Equation

I'm having some trouble finding answers to these problems. When i try to find help online, all i find are (x,y,z) problems and I'm simply looking for a PreCalculus (x,y) problem solving technique: ...
0
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1answer
27 views

How would I parametrise a straight line?

If I want to parameterise a straight line and I have the equation, eg $y=2x+1$ and I also have two co-ordinates it passes through, would it ok to use the co-ordinates to parameterise in terms of $t$?
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2answers
34 views

“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...