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

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1answer
83 views

Is a parameterization defined to be surjective and/or injective?

A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know ...
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1answer
182 views

Find the second derivative ${{{d^2}y} \over {d{x^2}}}$ in terms of t when $x = 3 - 2{t^2}$ and $y = {1 \over t}$

This is my attempt: $\eqalign{ & x = 3 - 2{t^2} \cr & y = {1 \over t} \cr & {{dx} \over {dt}} = - 4t \cr & {{dy} \over {dt}} = - {t^{ - 2}} = {{ - 1} \over {{t^2}}} ...
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1answer
76 views

Parametric to Implicit ( {x(t),y(t)} --> P(x,y) == 0 )

I have this parametric equations: $x(\theta) = r Cos(\theta) - \frac{v_{0}^2Cos(\theta)Sin(\theta)}{g}$ $y(\theta) = \frac{v_{0}^{2}Cos^2(\theta)}{2g} + r Sin(\theta)$ This is for $\theta \in ...
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1answer
106 views

position question from velocity and given point.

A particle moves along the $x$-axis so that at any time $t\geq 0$, its velocity is given by $v\left(t\right)=\sin\left(2t\right)$. If the position of the particle at time $t = \frac{\pi}{2}$ is $x = ...
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2answers
405 views

Explanation of the area under the curve given by a parametric equation

My textbook says the area under a graph is given by: $\smallint ydx$ And it then goes on to say by the chain rule: $$\smallint ydx = \smallint y{{dx} \over {dt}}dt$$ Could someone explain to me how ...
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1answer
1k views

Find the cartesian equation of: $y = \sin (2t)$ and $x = \cos (t)$

$\eqalign{ & y = \sin (2t) \cr & x = \cos (t) \cr} $ therefore: $\eqalign{ & \sin (t) = {y \over {2\cos t}} \cr as: & \cos (t) = x \cr & {\rm{ sin(t) = }}{y \over ...
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1answer
79 views

Parametric Equation along a line segment

I am having some trouble understanding how to determine the parametric equation of a line segment between A(1,1) and B(-1,1). I did some research and came across the following relation: ...
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4answers
390 views

Some general question about parametric equations

My textbook doesn't explain this very well, what I want to know is the purpose of parametric equations, what is a parameter? what is the advantage of these equations over a function y=f(x), what do ...
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0answers
87 views

Vector Tangent to Curve of Intersection

I am having problems solving this. Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$. I'm able to do this kind of thing using ...
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3answers
49 views

How to take parametric equations (x, y) to create a derivative formula?

I always thought that if I take the derivative of the y and x equation and divide y' by x', then that would be the derivative in formula form. Is this correct?
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1answer
29 views

How to check the visibility of these three points?

For question d part i, I have calculated the distances from $Q$ to $P_1$ and $P_2$ respectively and found $P_1$ to be closer with a distance of root $6$, with $P_2$ having a distance of root $24$. ...
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1answer
77 views

What is the normal form for this line?

I have calculated the parametric form of a line as: $L = P_1 + tP_1P_3 = <2,2,0> + t<1,2,2>$. If I am given a point $ K = <1,-1,-1>$, how would I show the normal form of plane $E$ ...
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1answer
1k views

Parameterization of the surface a torus

For a calculus question I have I need to parameterize the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the z-axis (with $0<a<b$). I've had a go at ...
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2answers
124 views

Circumference parametrization

Let $C=\{(x,y)\in \Bbb R^2: (x-x_0)^2+(y-y_0)^2=r^2\}$ and let $\varphi :[0,2\pi]\to \mathbb{R}^2$, $\theta \mapsto (x_o+r\cos \theta, y_0+r\sin \theta)$, with $r>0$. I'm trying to prove that ...
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1answer
56 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 ...
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2answers
492 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
97 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
51 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 ...
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2answers
96 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
127 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 ...
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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
136 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 ...
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1answer
128 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 ...
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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
350 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
588 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
78 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\\ ? ...
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1answer
107 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?
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1answer
328 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
381 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: ...
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2answers
1k 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) ...
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0answers
665 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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1answer
169 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
316 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
53 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
304 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 ...
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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
122 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?