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

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Consider the parametric curve given by: $x=3\cos(t)$, $y=t^{3/2}$.

The question asks to find the equation of the tangent to this curve at the point $t=\pi/4$. I've determined $$\frac{dy}{dx} =(\frac{dy}{dt})/(\frac{dx}{dt}) = -0.222$$ Have I got the right idea? ...
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53 views

Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then ...
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1answer
40 views

Parametric parabola

I was given my Math C assignment today and the moment I looked at question 1 I knew I had no idea what to do. This is the graph I was given: I was asked to provide an equation for the curve however ...
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1answer
75 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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1answer
50 views

Looking for a family of astroids

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the ...
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0answers
104 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
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2answers
41 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...
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1answer
140 views

Parametric equation of clock hands

I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to ...
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1answer
97 views

Does every algebraic variety admit a local parametrization at every non-singular point?

I am reading a text in which the first sentence of the proof of a theorem is: Let $X(t)=X(t_{1},\ldots,t_{n})$ be a local parametrization of the algebraic variety $X$... I guess that every ...
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0answers
55 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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1answer
53 views

Conditions that guarantee a composite Bezier curve in the cartesian plane represents a function?

Context I am allowing users of my application to control a curve connecting $(0,0)$ and $(1,1)$. There are a finite number of knots that are evenly spaced horizontally. The user can specify the ...
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2answers
211 views

Finding the parametric form of a standard equation

I need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did this ...
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37 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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1answer
31 views

equation of tangent line and parametric equation of tangent line (are they equivalent)

I am not sure about the steps to finding parametric equations of tangent lines and was wondering if these statements are equivalent Is there a difference if I am asked to find: equations of ...
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1answer
44 views

Is there an algorithm for parametrization of equations?

In this and this Math.SE questions askers wanted to parametrize their equations. It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate ...
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1answer
58 views

Parametric equation with image of a function.

Find all values of $a$ for which the image of the function $$y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}$$ contains $[2, 3]$. Now, I've already transformed it to ...
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1answer
16 views

Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
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2answers
45 views

Eliminating the parameter?

How would you eliminate the parameter where the x coordinate is in terms of t, but the t is squared. x= 3t - $t^2$ y= t + 1 I know to solve for y as a function of x, but I'm not sure how to do so ...
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3answers
48 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
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2answers
131 views

How to parametrize the curve $y^2 = x^2(x+3)$

I am emberassed to ask this, but I couldn't find a way. I want to write the curve $y^2 = x^2(x+3)$ as $$y=f(t) \quad \quad x=g(t) \quad \quad t \in \mathbb R$$ I guess I have to do something like ...
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1answer
60 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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1answer
53 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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2answers
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What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
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1answer
119 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
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1answer
664 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
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1answer
30 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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0answers
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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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2answers
414 views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two ...
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2answers
86 views

Parametrization by arclength

I could not re-parametrize the curve r[s_] := {-(5 + 2*Cos[2*s])*Sin[3*s], (5 + 2*Cos[2*s])*Cos[3*s], 2*Sin[2*s]} neither by hand nor with Mathematica. Is ...
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0answers
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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
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Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...
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1answer
39 views

Evaluate the countour integral $\int _\Gamma z dz$

Can someone please help me setup a) $\int _\Gamma z dz$ b) $\int _\Gamma \bar z dz$ and given the admissible parametrization of $\Gamma$ $\Gamma_1 : z_1: 2 + i(t - 1) ; 1 \leq t \leq 2$ and ...
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1answer
43 views

a question about how to parametrize a surface in $R^3$

Given a surface $$x^4/a^4+y^4/b^4+z^4/c^4=1$$,how can I parametrize the surface using X(u,v). I tried to use $x=a\sqrt{cos(\theta)sin(\phi)}$,$y=b\sqrt{cos(\theta)sin(\phi)}$,and ...
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1answer
32 views

Parametrization of this surface

Consider the following surface: $$A=\{(x,y,z,t)\in\mathbb{R^4}:0\leq x,y,z,t\leq1,\quad x+y+z+t=1\}$$ I need to parametrize it to be able to calculate its volume. Of course, I thought on seeing it ...
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1answer
50 views

Finding points on a tangent line of a parametric equation that are parallel to another parametric equation

So I got the tangent line of the first equation to be 12t/(3t^2+4) and I changed the second parametric equation to the cartesian form and got y= -(12/7x+5) with 12/7 as my slope. I equated ...
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1answer
30 views

Parametrics Arc Length

Can anyone explain the difference between the arc length and total distance? I'm using the textbook here and they seem to be the same formula. Please help me figure this out.
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2answers
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Parametric differentiation

The parametric equations of a curve are $$\begin{cases}x(t)=e^{-t}\cos t\\y(t)=e^{-t}\sin t\end{cases}$$ Show that $$\frac{dy}{dx}= \tan\left(t-\frac{\pi}{4}\right)$$ I did the differentiation ...
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1answer
67 views

Show that $Y^2-X^3\mid f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R}[X,Y]$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
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1answer
126 views

Find where a curve crosses itself?

I have the curve $x=t^2,y=t^3-4t$. I made a $t_2$ such that $t_2>t_1$ and from $x$ found that $0=t_1^2 -t_2^2$, from here I solved for values by basic guess and test and then subbed them into the y ...
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2answers
332 views

Find points on curve $r=2\sin\theta$ where the tangent line is parallel to the ray $\theta = \pi/4$

I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$ I was ...
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2answers
102 views

Velocity of a curve given by parametric equations

In standard Cartesian equations, $\frac{dy}{dx}$ is the velocity function because it's the derivative of position. $$\frac{dx}{dt} = \sin^{-1}\left(\frac{t}{1 + t}\right) ...
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1answer
96 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
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1answer
82 views

Comet Problem Using Parametric Equations

A comet has an elliptical orbit that is $144$ billion miles across the $x$-axis and $48$ billion miles across on the $72$ years to complete one revolution. If the center of the coordinate system is ...
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1answer
21 views

Curve parametrization verification: Plane and paraboloid

I wanto to find a parametric equation of the intersection between the plane $x+y-z=2$ and the paraboloid $z=(1-y)^2$ I proposed $y=t$, wich implies $z=(1-t)^2.$ So: $$\begin{align} ...
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3answers
76 views

Parametric representation of a plane cut of a sphere at y=5

The sphere is given by $x^2+y^2+z^2=36$ Parametric Form: $$x=6\sin t\cos u$$ $$y=6\sin t\sin u$$ $$z=6\cos t$$ If the sphere is 'cut' at $z=5$ this problem is trivial. ...
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3answers
71 views

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} ...
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1answer
45 views

Easy question on the limits of an integral

So I would like to ask how exactly do we determine what limits to take when integrating both Cartesian and parametric equations. So let's say we have a graph of $y=x^2$. If we wanted to take the area ...
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1answer
26 views

Finding parameters for a quadrature formula

To compute the integral $\int_0^1f(x) dx$ numerical I want to use the following quadrature formula: $$Q(f)=\omega_0f(x_0)+\omega_1f(1)$$ The question is how one should choose $\omega_0,\omega_1 ...
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1answer
74 views

Reverse Parametric Equations

I have just learned about parametric equations. I have gotten the concept of turning the parametric equations to regular/ordinary equations, but am having trouble doing the reverse in this problem: ...
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1answer
25 views

How can I show that the set of images of a parameterized curve is smooth?

I know that in order for a set S to be smooth, it needs to be connected and that for each point $\vec{a} \in S$, there needs to be a neighborhood $N$ such that $S \cap N$ is a class $C^1$ function. ...