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

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5
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89 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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279 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.
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41 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
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43 views

What's the next “recursion” here?

Plotting a single 3d helix is x = cos(t); y = sin(t); z = t; From this equation: x = [R + a cos(\omega t)] cos t y = [R + a cos(\omega t)] sin t z = h t + a sin(omega t) Comes the awesome ...
2
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41 views

How can i translate a parametric equation to cartesian

The parametric equations are : $$ x=16\sin^3(t)$$ and $$y=13\cos(t)-5\cos(2t)-2\cos(3t)-\cos(4t) $$ with $t$ from $-\pi$ to $+\pi$ so I'm new to this kind of equations and i really don't know how to ...
2
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0answers
64 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
2
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0answers
42 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
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34 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
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46 views

Line integral - new parametric equation

We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$. Let now $\hat{r} = r \circ \phi$ ...
2
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129 views

Given the equation of a cylinder $ x^2+z^2=1,$ find the parametric and locus form of the curve of intersection with plane

Given the equation of a cylinder $x^2+z^2=1,$ describe the curve of intersection between the cylinder & the planes z=x & y=x in the parametric form & the form F(x,y,z)=0. I am so lost ...
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24 views

Parametrisation of surface

Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$. I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure ...
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54 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
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60 views

Find the length of the parametric curve (Difficult)

Find the length of the parametric curve $$x = t$$ $$y = f(t)$$ $$f(t) = \int_0^t {s \over (s^2-1)} \ \mathrm{d}s$$ $$0\leq t \leq 1/2$$ First I create the $x'$and $y'$ Then put it into the ...
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60 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object A starts at the origin (0,0) and moves straight up the y axis with a speed v. Object B starts at point (1,0), always moves towards object A and has a speed of 2v. ...
2
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0answers
138 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...
2
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100 views

Launch angle required to hit coordinate (x,y) with air resistance

Finding Angle of Elevation to hit X, Y and Wikipedia Angle required to hit coordinate work, but don't calculate air resistance. Is there a way to find the launch angle of a projectile required to hit ...
2
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0answers
39 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
2
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583 views

Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space. When say quadratic ...
2
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0answers
381 views

Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
2
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0answers
1k views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
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2k views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
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29 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of a sinus modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ ...
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0answers
46 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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27 views

Collinearity of three points on a curve.

In the realm of elliptic curves, the collinearity of three points is of a fundamental importance because this condition allows us to define on the curve a law of Abelian group, the study of which is ...
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31 views

How do I detect collisions in a parametric equation$?$ (Lissajous curve)$?$

Let's say you have a simple parametric equation where $x= \sin(3t)$ and $y=\cos(7t).$ This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure. You can ...
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21 views

Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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0answers
17 views

Vectorial function representation

I'm trying to solve the following excersice: Given the following equations, represent by a vector function: $y=x^2+1$ from $(3,10)$ to $(-1,2)$ $y=4-x$ with $x\in [-2,3]$ ${x^2\over{25}} + ...
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0answers
40 views

Mathematic's difficulties to understand the parametrisation of an electrostatic potential field

I start to learn electrostatic and I have some problem with finding the upper and lower boundaries of my parametric variable that I used to represent the graph of the potential surfaces of two ...
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0answers
40 views

Intersection between a parametric equation and a linear equation

2Consider the parametric functions $f_1, f_2$ with $$f_1(x) = 3(60-x)\cdot \sin(3x)$$ and $$ f_2(x) = 3(60-x)\cdot \cos(3x).$$ Suppose you have a linear function: $$f_3 (x) = 1.5 x$$ How does one ...
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52 views

parsimonious definition of a zig-zag function between two bounds

Suppose I have some strictly increasing function $f:[0,b]\to[0,b]$ with $0<b<1$, $f(x)<x$ and $f'(b)=\frac{1-f(b)}{1-b}$ (i.e. tangent to the secant line to $(1,1)$). Now imagine a 'tunnel' ...
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0answers
54 views

Calculate parametric bounds of a circle in a 2D quadrilateral

Given a 2D quadrilateral defined by the points $(p0, p1, p2, p3)$ and a circle centered at $c$ with a radius of $r$, I want to find a quad in the parametric space of the outer quad that tightly bounds ...
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0answers
35 views

Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
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26 views

How to write explicity a curve on $S^n$?

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$. I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the ...
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0answers
29 views

Parametrization of arbitrary objects to display on an x-y-scope

I am trying to find an approach for general parametrization of an arbitrary geometric object or closed curve. Though I am not sure if I am on the right path with that. Basically I have an geometric ...
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0answers
36 views

Lissajous Curve

$$ \gamma(t)= (x(t),y(t))=(sin(2t),sin(3(t)) $$ Justify that we can reduce the domain of study to [0, $\pi/2$], by specifying the necessary symmetries to obtain the whole curve. I'm not really too ...
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0answers
146 views

Trying to find the volume of a 3D torus shape that I made

After playing around with 3D parametric equations on my calculator (modifying the equations of a standard torus), I came across a shape that I like. The equations are: $$x=(2+\sin t)\cos u$$ ...
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0answers
23 views

Find the line tangent to the parametric curve $\left\langle t^3-1,t^4+1,t \right\rangle$

Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only. I am given a parametric curve with the ...
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0answers
20 views

figuring out parametric equation of a moving dot of specific velocity along acurve

I currently need to model a dot moving along an arbitrary curve given it's velocity, initial point, and $y=f(x)$ form of equation. I vaguely remember from my high school teaching that it will possibly ...
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0answers
69 views

Polynomial parametrization of a quadric with two given points

Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function $$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$ where and A, b and c are unknowns. ...
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37 views

Interpretation of parametrization

Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function. Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$ ...
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42 views

Find all real numbers a such that equation 3^(x2+2ax+4a−3)−2=|(a−2)/(x+2)| Has exactly two different roots x1,x2 those belong to [−4;0]

Find all real numbers $a$ such that equation $${3^{(x^2+2ax+4a-3)}}-2=|{a-2 \over x+2}|$$ Has exactly two different roots $x_1,x_2 $ those belong to $[-4;0]$ Tried plenty different things to solve: ...
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47 views

Challenging Path Integral

Let $c$ be the curve of intersection of the plane defined by $x+y+z = a$ and the cylinder $x^{2} + y^{2} = a^{2}$ ($a > 0$). Evaluate the path integral: $\displaystyle\oint_{c} \sqrt{a^2 + xy} ...
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51 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
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32 views

Parametric curve arc length! Am I doing it right?

$x=3t^2+2$ and $y=2t^3-1$ on $[1,3]$ The formula for parametric arc length is $\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$ $x'(t) = 6t,y'(t)=6t^2$ Under the radical, I ...
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0answers
40 views

Parameterization of a closed curve on a sphere

I'm looking for a parameterization of a closed curve C on a sphere. assume the projections of C on y-z, x-z, x-y plane are f(x), g(y), h(z), respectively, and ${\oint}f(x)dx={\oint}g(y)dy=0$, and ...
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0answers
32 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
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982 views

Helix around helix parametric equation?

I know the parametric equation for a $3D$ helix is: $x = R \cos t$ $y = R \sin t$ $z = h t$ Can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix ...
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0answers
43 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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0answers
23 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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37 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, ...