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

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1answer
55 views

Parametric problem: do these 2 comets collide. Am I solving this correctly?

$\text{comet1} = x_1(t), y_1(t)$ $\text{comet2} = x_2(t), y_2(t)$ set $x_1(t) = x_2(t)$ and solve for $t$. Since $t$ had a square, I had 2 possible values for $t$ ($t_1$ and $t_2$). substitute ...
4
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2answers
479 views

Finding surface area of a cone

I will describe the problem then show what I tried to solve it. I need to find the area of the cone defined as follows: $$z^2=a^2(x^2+y^2)$$ $$0\leq z\leq bx+c$$ where $a,b,c>0$ and $b<a$. ...
4
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1answer
88 views

Parametric plots: Determine if 2 comets collide at a given time. Am I solving it correctly?

There are $2$ comets comet 1 $(x(t), y(t))$, comet 2 $(x_1(t), y_1(t))$ I need to determine if these two comets collide. From reading my steps below, is this the proper way to solve this? $1.$ set ...
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0answers
17 views

Can one characterize which surfaces are capable of being described by a closed-form parameterization?

Speaking intuitively, I can visualize a lot of surfaces in my mind; but it seems that some of the ones I can imagine are not capable of being described by the 'usual suspects', i.e., elementary ...
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2answers
50 views

The parametric form of a line

For the parametric representation of a line L with the following points, is my answer correct: P1 = <2,2,0>, P2 = <0,-2,-4>, P3 = <3,4,2> Is this correct: X = P1 + s.P1P2 + t.P1P3 = ...
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0answers
43 views

$\frac{dy}{dx}$ of a parametric curve

Given $x = sin^2(t)$, $y = cos^2(t)$, I need to find $\frac{dy}{dx}$ in every non-singular point of the curve. So $\frac{dy}{dt} = -2sin(t)cos(t)$ and $\frac{dx}{dt} = sin(2t)$. To find the ...
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1answer
41 views

Parametric motion question

What exactly happens when both $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}x}{\mathrm{d}t}$ equal zero? I know that if $\frac{\mathrm{d}y}{\mathrm{d}t} =0$ then its a vertical tangent with ...
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2answers
2k views

Finding the coordinates of a point of intersection from a pair of parametric equations.

A curve is given by: $$x = 2t + 3 $$ $$y = t^3 - 4t$$ The point $A$ has parameter $t = -1$. Line l is a tangent to the curve at $A$. Line l cuts the curve at point $B$. Find the value of $t$ at ...
2
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2answers
94 views

Parametric equation for plane with $\langle 0,1,1\rangle + s.\langle 1,0,-1\rangle$ and $\langle 0,0,-3\rangle + t.\langle 2,1,2\rangle$

In $3$-dimensional space, two lines $l_1$ and $l_2$ are given parametrically as follows: $$ X = \langle 0,1,1\rangle + s.\langle 1,0,-1\rangle \text{ and } Y=\langle 0,0,-3\rangle + t.\langle ...
2
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2answers
124 views

Implicit form of a parametric surface

Let $\Sigma$ be the surface in $\mathbb{R}^3$ parametrized by $$ (u,v) \mapsto \Big(\;p_X(u,v),\; p_Y(u,v),\; p_Z(u,v)\;\Big), $$ where $p_X, p_Y, p_Z$ are polynomials. Is there a standard way to ...
2
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1answer
57 views

How to interpret this task?

I have a task given to me in my homework I can not figure out what asks of me. The task is worded like this: A curve in a plane is given by $$ x(t) = 3(t - \sin(t)) $$ $$ y(t) = 3(1 - \cos(t)) $$ ...
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1answer
135 views

Proper methods of solving parametric equations

I'm learning parametric equations in this section. Although I understand why the following works, I'm having difficulty understanding why the method employed for solving it is the correct one. I'm ...
1
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1answer
127 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
3
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2answers
102 views

Having trouble solving question involving parametric equations

I have been given the following: $$y = a \cdot \cos^3t$$ $$x = a \cdot \sin^3t$$ $$0 \leqslant t \leqslant {\frac\pi2}$$ I am supposed to show that the mean value of $y$ over the interval ...
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2answers
337 views

Eliminating parameters to obtain surface equation

Given the following vector equation, how do I eliminate the parameters $u,v$ to get an equation of a surface in rectangular coordinates? $$\vec{r}(u,v)=3u\cos(v)\hat{\imath} + 4u\sin(v)\hat{\jmath} ...
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1answer
57 views

Line integrals of vector fields

Consider the vector field:$$\vec G = \left(\frac y{x^2+y^2}, \frac {-x}{x^2+y^2}\right)$$ compute $\int_\Gamma \vec G$ where $\Gamma$ is the proportion of a parabola $y=a(x-1)^2$ from (1,0) to (2,a). ...
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1answer
59 views

Find parametrics equations of a line

Consider the line in $R^2$ that is given by the equation $d_1x_1 + d_2x_2 = c$ for numbers $d_1, d_2$ and $c$ in $R$ where $d_1$ and $d_2$ are not both zero. Find parametric equations of the ...
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1answer
548 views

Representing A Plane Curve By A Vector Valued Function

I am given the function $x^2+y^2=25$, and I am suppose to write this as a vector valued function. I have always been awful at these sort of problems, even with parametric equations, which requires ...
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0answers
106 views

Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
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0answers
74 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
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1answer
66 views

3D Surface parametrization basics

I'm studying 3D rendering: I have a surface and the points on the surface are given by some function f such that $p = f (u, v)$ Since I'm a newbie this is unclear to me: how can a function of u and v ...
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2answers
77 views

Parametrizing this curve

How can I parametrize the trajectory so that it is a smooth path $h:[-1,1]\rightarrow \mathbb{C}$? I think that I should use $$h=\left\{\begin{array}{ccl}t+i |t|&:&-1\leq t \leq 0\\ ? ...
0
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1answer
105 views

parabola in homogeneous coordinates

So if I have the parabola Y = X^2, how do I go about representing this homogeneously? I know I can parameterize it as F(t) = (t, t^2), but then what? The reason I ask is because I have a 3*3 matrix ...
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4answers
3k views

Parametrization for intersection of sphere and plane

Given is the sphere $x^2 + y^2 + z^2 = 4$ and the plane $x + y = 2$ in $\mathbb R^3 $. How can I find a parametrization for the intersection of the two?
3
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1answer
321 views

How are the parametric equations describing the cupid curve derived? [duplicate]

No doubt as some people have already seen, today morning wolfram posted the best valentine ever. The graph depicting cupid with its arrow and floating hearts around it involves something like 6 pages ...
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1answer
1k views

Converting parametric equation to implicit form

So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$ $$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$ If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
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2answers
46 views

Parameterized curve describing trajectory of thrown object

We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve $$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$ with ...
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1answer
365 views

Domain of parametric equation

If there is a parametric equation $x=2\cos{2t}$ and $y=6\sin{t}$, $0\le t \le \frac{\pi}{2}$ the Cartesian equation is $y=3\sqrt{2-x}$. How do I find the domain of the Cartesian equation? I tried: ...
3
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2answers
980 views

Plotting parametric equations in gnuplot

I am trying to plot the following parametric equation in gnuplot: fx(t) = -35*cos(t) + 65*cos(-.35*t) ...
2
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0answers
657 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= ...
0
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1answer
167 views

equation for the region inside a circle

What equation or group of equations fill the entire or part of a region inside a circle without using inequalities? Update I don't know if this problem is already solved, I'm trying to find the ...
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3answers
313 views

Understanding cubic bezier curve

I do not have experience of Mathematics past a-level, so please excuse the incorrect terminology. I am trying to better understand the fundamentals of how a cubic bezier curve works. I am going by ...
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2answers
52 views

what's the algebra (if any) behind converting f(x) for a circle to a parametric equation

I'm sure there has to be some algebra behind it. My problem called to covert $$(x - 2)^2 + (y - 9)^2 = 4$$ if $x = 2 + 2cos(t)$ then $y = ? $ I know the answer is $9 + 2\sin(t)$ but I simply got ...
2
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2answers
297 views

parametric curves, parameter and integration

I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between ...
4
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3answers
2k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
2
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1answer
119 views

Parametric Equation of a Circle Using a Line

Consider the unit circle $$ x^2+y^2=1. $$ How can I parametrize it using the line $y=m(x+1)$, where $m$ is its slope?
0
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1answer
406 views

Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$ \iint\limits_S \, x^2yz\ \mathrm{d} S $$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
2
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1answer
231 views

Finding parametric equations

I am trying to understand volume and surface integrals. I do get the idea of the process (find a parametric equation of the volume/surface, integrate afterwards). But I just cannot make up parametric ...
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1answer
118 views

Find the Frenet frame

Consider the following space curve: $$ \gamma(x)=(e^x\cos(x), e^x\sin(x), e^x). $$ My main goal is to find the Frenet Frame T,N,B. So far I have found the arc-length using the following formula: $$ ...
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2answers
598 views

Converting $x=\frac{1}{2}\cos\theta\;;\;\; y=2\sin\theta $ to Cartesian form

How can we transform these parametric equations to Cartesian form? $x=\frac{1}{2} \cos\theta, \quad y=2\sin\theta \quad\text{ for}\;\;0 \leq \theta \leq \pi$
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2answers
112 views

Converting $x = \sin \frac{t}{2}, y = \cos \frac{t}{2}$ to Cartesian form

How can we transform these parametric equations to Cartesian form? $$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$$
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3answers
131 views

Help me to sketch this parametric curves

Is there any defined process to sketch parametric curves? Thanks in advance. $$x = \cos^2 t, \quad y = 1 - \sin t, \quad 0 \leq t \leq 2\pi.$$
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1answer
65 views

Find a function f(x) such that the parametric curve could be obtained by flipping the graph

Find a function f(x) such that the parametric curve could be obtained by flipping the graph of f across the line with slope 1 that goes through the origin. parametric curve with coordinates (t^14, ...
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3answers
324 views

How do we prove that two parametric equations are drawing the same thing?

For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) ...
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1answer
232 views

Arc Length Of Parametric Curve

I attached the problem as a file: Where did the trig functions go? I sifted through the different trig identities and formulas, but couldn't find anything that I could use. What should I do?
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2answers
2k views

Finding Where A Parametric Curve Intersects Itself

The problem I am working on is to find the where the curve intersects itself, using the parametric equations. These are: $x=t^2-t$ and $y=t^3-3t-1$ For the graph to intersect itself, there must be ...
2
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1answer
2k views

Finding Where A Parametric Curve Crosses Itself

The parametric functions I am dealing with are: $x=2\sin2t$ and $y=3\sin t$ I know for a parametric graph to cross itself, there must be two distinct $t$, $t_1$ and $t_2$, that when placed into the ...
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1answer
1k views

Finding Parametric Equations For A Rectangular Equation

I am trying to find a general way of finding parametric equations for a rectangular equation. The problem I am working on is $y=x^3$, and I have to find two examples of parametric equations. ...
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2answers
247 views

Parametric Equation Problem

The problem is, "to determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain." (a) $x=t;\quad ...
1
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1answer
144 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...