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

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1answer
51 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
0
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1answer
56 views

Cartesian Equation and Parametric Equation Help

I just need some help with a maths question that I am trying to get done for a maths tutorial homework sheet. The question is... Let L be the line through D = (6,5,4) and E = (1,0,6), and let P be ...
0
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1answer
38 views

Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
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0answers
22 views

How to know which variables to parametize in a large matrix?

(dont want anyone to solve the problem, just don't understand one thing) So I have a homework problem where I got a 3x6 matrix, and I have to parametrize the equations and solve for each variable in ...
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2answers
39 views

How to apply chain rule to:$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$?

How do I apply chain rule to the following: $$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$$ Where $$\Big( \frac{dy}{dx} \Big) = \Bigg(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \Bigg)$$ I don't see the ...
1
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1answer
108 views

Intersection of graphs of parametric equations with trig functions

I've had this problem for a while, but I have not been able to solve it. Any help would be really appreciated, thanks in advance! Problem: Find the sum of all possible values of the constant $k$ such ...
0
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1answer
50 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
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0answers
126 views

Finding the equation for ellipse velocity

I am trying to figure out how to do a homework problem for my math class. The homework for the night is, given a set of parametric equations, has three parts, the first of which is to find the speed ...
0
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0answers
21 views

Comparison of Parametric and Polar Equations

Having been introduced to parametric equations, I cannot help but question the similarities between parametrized functions and polar functions. A parametric circle is defined by the following: ...
-1
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1answer
556 views

Eliminate the parameter to find a Cartesian equation of the curve

I've done every problem on this subject except I can't get this one. (every other problem had x = something and y = something. $$y = (t+1)^{1/2},\quad y = (t-1)^{1/2}$$
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1answer
64 views

Evaluation of $\int_{-\pi}^{\pi} \cos(ax) \sin^n(bx) dx$

As it is a kinda famous integral I thought I would find something on MSE but I didn't so here I am. If there is, link it in the comments and I will delete the question. How do I evaluate ...
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2answers
113 views

Not sure how to differentiate implicitly using parametric equations…

I am not sure (not taught before explicitly) how to apply implicit differentiation on parametric equations when I am solving the question posted below. Question Two positive numbers $x$ and $y$ ...
0
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2answers
39 views

Determine max/min speeds of trochoid

Find the minimum and maximum speeds of the point of a trochoid and the locations where each occurs. I know a trochoid has equations $ (x)t = at - b \sin{t} $ ; $ y(t) = a- b \cos{t} $ for trochoid ...
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4answers
41 views

Find $\frac{dy}{dx}$ for $x=2\theta+sin2\theta$ and $y=1-cos2\theta$

The parametric equations of a curve are $$x=2\theta+\sin2\theta,\:y=1-\cos2\theta.$$ Show that $\frac{dy}{dx}=\tan\theta$. I can use the chain rule to get ...
1
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3answers
61 views

Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
0
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1answer
214 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
41 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 ...
1
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3answers
46 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 ...
0
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1answer
30 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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2answers
40 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 ...
1
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1answer
28 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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2answers
131 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
125 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 ...
0
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2answers
69 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
26 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 ...
2
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1answer
60 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 ...
0
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1answer
26 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
140 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 ...
0
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2answers
42 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 ...
4
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2answers
219 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 ...
0
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1answer
40 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
19 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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2answers
166 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 ...
6
votes
2answers
149 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.
1
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3answers
498 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 ...
1
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1answer
46 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 ...
1
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0answers
34 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, ...
1
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2answers
76 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 ...
0
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2answers
342 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
83 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: ...
0
votes
1answer
30 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
35 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
27 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 ...
0
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2answers
940 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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vote
1answer
468 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
votes
1answer
46 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
23 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 ...
0
votes
2answers
76 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.
2
votes
1answer
47 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 ...
0
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1answer
36 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 ...