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

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12 views

How to clip Bézier curves using Casteljau's algorithm?

I am attempting to approximate intersections of Bézier curves using iterative clipping. This common method is described here and here. It basically works like this: Find bounding lines outside one ...
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0answers
31 views

Finding the mean value of y

I don't understand how to obtain the limits for the $t$-values considering that they gave us the $x$-values in the first part of the equation. I've considered substituting the $x$-values into the ...
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3answers
22 views

Check if two vector equations of parametric surfaces are equivalent

Give the vector equation of the plane through these lines: $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and ...
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1answer
11 views

Check me- Speed(Arc Length) of Parametric Equations

A ferris wheel has height = 100 ft and completes 1 revolution in 3 minutes at a constant speed. Compute the speed of a rider in the ferris wheel. Ferris wheel = circle modeled by x= cos t ; y = sin t ...
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17 views

Which property can be used to derive a differential equation for a reparametrization

With $0\le t\le1$, two space curves given by: $$c_1(t)=(1,t,0)\quad\quad c_2(t)=(0,t,2t(1-t))$$ One of them, say $c_1$, must be reparametrized by $r(t)$ in order to minimize the area between the ...
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7 views

what is a non parametric iterative algorithm?

I am trying to uderstand mean shift algorithm.Mean Shift is a non parametric iterative algorith.What does it mean?
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2answers
26 views

Rearranging equation $t = \frac{T}{2\pi} (\psi - \epsilon \sin \psi)$ in terms of $\psi$

I was playing around with the maths for orbits and trying to make a parametric equation that, well.. worked. I found a worksheet with parametrics with another variable ($\psi$), but I wanted to be ...
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1answer
20 views

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), ...
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9 views

Using Parametric Equations To Simulate Motion [on hold]

In the diagram, a large number of subatomic particles are simultaneously emitted from a radioactive substance located at the origin. The particles all pass through a vertical slit with the same ...
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2answers
52 views

Is there a general way to parameterize all implicit functions?

We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and ...
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2answers
78 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
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1answer
18 views

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
20 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
33 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
17 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
24 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
166 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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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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0answers
8 views

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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0answers
48 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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2answers
124 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
138 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
47 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
27 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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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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2answers
43 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
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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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
25 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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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
21 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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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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1answer
47 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 ...
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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
22 views

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
46 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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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 ...
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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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1answer
23 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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1answer
27 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
27 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
53 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
56 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
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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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 $t$?