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

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1answer
29 views

Is this parametric equation describe a circle?

Let $w=\varepsilon\beta(t)-i\sqrt{\beta(t)^2-1}$, where $\beta(t)=\cosh t$ and $\varepsilon >0$. the parametric function is defined as $x+iy=\frac{2w}{|w|^2+1}$ and $z=\frac{|w|^2-1}{|w|^2+1}$. ...
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2answers
35 views

Parametric curve: $x=\frac{a}{2}(t+\frac{1}{t})$, $y=\frac{b}{2}(t-\frac{1}{t})$?

What kind of shape is the parametric curve described by: $$x=\frac{a}{2}(t+\frac{1}{t})$$ $$y=\frac{b}{2}(t-\frac{1}{t})$$ $a,b \in\mathbb{R^+}$ ?
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2answers
51 views

On my grapher, $(\cos t, \sin (t+1))$ generates a geometric figure. What is that figure?

On Wolfram Alpha I am getting a graph like this: http://www.wolframalpha.com/input/?i=%28sin+t%2Ccos%28t%2B1%29%29 Is this an ellipse? I really don't know how to find it.
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0answers
17 views

Existence of vertical or horiaontal tangent vector of a circle with different representation of the same curve

I am recently working with this question but not sure if I am on the right track, which is: $$t\mapsto (\frac12 cos(t^2), \frac12 sin(t^2)),0\le t \le \sqrt{\pi} $$ which clearly is a semi-circle ...
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2answers
37 views

Is there a general equation for an n-ellipse?

I'm sorry if this question is too trivial, but even a more thorough search on Google brought me no answers so far. So please, is there a general equation for n-ellipses? Given N points on the ...
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2answers
95 views

Parameterization of an ellipse

If an object (like a planet) orbits around a more massive object (like the sun) the orbit will be an ellipse with the massive object at one of the two foci of the ellipse. The parameterization $$x(t) ...
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1answer
16 views

Parametrics, when $t$ is not in between $0<t<1$

I understand how to parameterize a line segment when the $t$ value lies in between $0$ and $1$, however I was wondering how to create a parametric equation for the line segment between say $(1.5,2)$ ...
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3answers
76 views

Trigonometric equation with parameter

Find $p$ for which $\cos^2(x) - \cos(x) + p + 1 = 0$ has EXACTLY two solutions for $0 \le x \le 2\pi$ I tried to substitute $t = \cos(x)$ and then I got two solutions, but I don't know what to do ...
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1answer
21 views

Area of parametric surface (theory)

In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ ...
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2answers
33 views

Finding an equation relating $x$ and $y$ with their respective parametric equations and using its differential?

How can I find the equation relating $x$ and $y$ directly without an additional parameter $t$, which both are related to initially. For example, $\frac{\mathrm{d}y}{\mathrm{d}t} = 2t+1$, ...
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1answer
21 views

Parametric line segment in 3-space

If one wants to parametrize a straight line segment in $\mathbb{R}^3$, which goes from $(1,0,0)$ to $(0,1,\pi/2)$, would this approach be correct? First, we come up with the $xy$-plane equation, ...
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3answers
44 views

Turning points of parametric curve

Find the slope of the curve at $t=\frac{1}{4}\pi$. $$\begin{cases}x=\sin t+\cos t \\ y=\frac{1}{2}\sin(2t)\end{cases}$$ $$\frac{dy}{dx}=\frac{\cos(2t)}{-\sin t+\cos t}$$ ...
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1answer
29 views

Turn the direction of movement of a parametric curve

Given is the parametric curve $K$ that satisfies $$\begin{cases}x=3\sin t \\ y=2\cos\left(t-\frac{1}{4}\pi\right)\end{cases}$$ How can you change the parametric equations if you want to turn the ...
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3answers
21 views

Parametric differentiation

A question from Active maths. At 60 years old this is my interest not my homework!! Let $$\begin{cases}y=e^t\cos t\\ x=e^t\sin t,\end{cases}$$ and prove that $dy/dx=\tan(\pi/4 -t)$. I ...
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1answer
39 views

Convert advanced parametric equation to regular/cartesian

can anybody help me to convert following parametric equation in a form Y =Y(X): $$ x = cos(t) \sqrt{(2 - cos^2(3t))} \\ y = sin(t) \sqrt{(2 - cos^2(3t))} $$ I've tried also with Wolfram Alpha and it ...
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1answer
16 views

Numerically Invert the Wakeby Percent Point Function

I am looking at the possibility of using the Wakeby Distribution to attempt to model color components in image rows and columns (it is a very silly idea for "compressing" images that I want to see ...
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1answer
38 views

Arclength of parametric curve

Find the arclength of the curve defined by $$x=\cos^2(t)$$$$y=\cos(t)$$ from $0$ to $4\pi$. I know using the formula that the arclength is given by ...
2
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2answers
56 views

parametric equations of a curve

The parametric equations of a curve are $$x=2\theta-\sin 2\theta$$ $$y=2-\cos 2\theta$$ The question asks that ''For the part of the curve where $0<\theta<2\pi$, find the coordinates of the ...
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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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2answers
24 views

parametric equations with cubed sin and cos

It has been a while since I have had calc 3, I know how to find the rectangular equation from parametric equations; however, I do not remember how to find the rectangular equation given these ...
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1answer
24 views

Parametric differentiation for equation of a tangent.

Given $y=t^3-\frac{5}{2}t^2$ and $x=\sqrt t$, for $t>0$, a) Use parametric differentiation to express $\frac{dy}{dx}$ in terms of $t$ in simplified form. b) Show that $\frac{d^2y}{dx^2}=at^2+bt$, ...
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1answer
33 views

Unparametrize x = 7 cos t, y = 4 tan t

"Express the given parametrization in the form $y = f(x)$ by eliminating the parameter. $x = 7 \cos t, y = 4\tan t$" $y=\pm4 \sqrt {\frac {49} {x^2} - 1}$ is correct?.
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1answer
46 views

Maximize velocity with parametric equations

Suppose we are asked to find the value of t at which an object is at its maximum velocity, if it travels on a path governed by: $x = 2 + 8cos(t)$ $y = sin(t)$ Here's what I understand: ...
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2answers
62 views

How to verify a plane is parallel to a line?

Plane equation is: $$7x-5y+2z-9=0$$ Line parametric form: $x=t, y=-4+3t, z=9-4t$
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2answers
57 views

Derivative with parametric set of equations

I am struggling for hours on a fairly simple problem. I have the following set of parametric equations: $$i = I (q_1^2 + q_1 - q_2^2 - q_2)$$ $$v = V (2(q_1 - 1) + \log q_1 - 2(q_2 - 1) - \log q_2)$$ ...
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2answers
24 views

Parametric equations - stating values.

A curve is defined by the parametric equations: $x=\cos2t, y=\sin2t, 0<t<π.$ a) Use parametric equations to find $\frac{dy}{dx}$. Hence find the equation of the tangent when $t=\frac{π}{8}$. ...
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1answer
15 views

General vector curve for the surface of a sphere

I want to prove a theorem about a vector curve ${\bf c}(t): \mathbb{R} \to \mathbb{R}^3$ (for $t \in [a, b]$) which lies on the surface of a sphere in $\mathbb{R}^3$. It is in my understanding that ...
0
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1answer
59 views

Find a vector function represented by the curve of intersection?

I'm struggling with the following problem: Given $\, z = \sqrt{x^2 + y^2}\,$ and $\, z = y+1\,$ find the vector function represented by the curve of intersection of the surfaces using the ...
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1answer
45 views

solving for initial velocity using the position vector

I am having trouble wrapping my head around this problem. The big picture is that i have to calculate the initial velocity v= needed for a soccer ball to cross a goal line. this is a homework ...
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1answer
26 views

Constructing parametric equation for $x=3z\cos(\ln z)$

I was trying to transform this $$x=3z\cos(\ln z)$$ in parametric form: $$x=x(t)$$ $$z=z(t).$$ To this end I made a substitution $\ln{z}=t$ and I got: $$x=3{e^t}\cos(t)$$ $$z=e^t$$ $$t\in ...
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1answer
45 views

two-dimensional bounded area defined parametrically

How do I define this without using piecewise function? I think it has something to do with Bilinear Surface but not sure how to get started. $x_1=-1, x_2=1, x_3=0, x_4=1$ $y_1=0, y_2=1, y_3=1, ...
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2answers
246 views

Parameterization of a curve for complex integral

I have problem with parameterization of a curve in order to evaluate a complex integral. Most docs that I've tried to read didn't explain the topic very well, especially, in case where the curve in ...
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1answer
61 views

Finding the parametric equation for a longbow curve about a circle

In the figure the circle of radius $a$ is stationary, and for every $\theta$, the point $P$ is the midpoint of the segment $QR$. The curve traced out by $P$ for $0<\theta<\pi$ is called the ...
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1answer
36 views

How to solve parametrized limits?

I am a bit confused at the moment. This exercise in particoular shattered my self confidence: $$\lim\limits_{x\to 0}\frac{\sinh(x) + 1 - (1 + 3x)^{\frac13}}{\ln\left(1 - 2x^α\right) + 2x^3}$$ with ...
2
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1answer
46 views

What's the equation of this parametric surface?

disclaimer: my math is sketchy at best AND english is not my first language, so... i might have some issues naming things - but i'll try my best to be clear :) given this parametric curve: see it ...
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2answers
42 views

finding t for parametric equation tangent line

Here is the problem I'm trying to solve: Find the tangent line at the point (0,2) $$x=2 \, \cot(t)$$ $$y=2 \, \sin^2(t)$$ $$\frac{dy}{dx} = -2 \, > \sin^3(t) \, \cos(t)$$ The tangent line ...
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1answer
13 views

Proof: Given a $C^1$ parametrization of $f=f(x,y):\mathbb R^2\to \mathbb R$, substitute variables $(x,y) = (s+t,s-t)$

I'm studying Mathematical Analysis and trying to solve example problems as I go. Specifically, this problem comes after an introduction to $C$ parameterizations of $r:I\subseteq\mathbb R \to \mathbb ...
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1answer
44 views

On the implicit function theorem and the gradient.

I was following some MIT notes and came across this proof I had a doubt about the existence of $r(t) = \langle x(t), y(t), z(t) \rangle$ a parametrization of a curve on the level surface. Then I ...
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2answers
37 views

Need help to understand a math task about algebraic and parametric equations

Can anybody please explain this for me?: Find the algebraic and parametric equations of the circle with centre (-2,3) that passes through (1,-1) How do I find the algebraic and parametric ...
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1answer
30 views

Parametric curve parametriced by length

Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that ...
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0answers
59 views

Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
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1answer
51 views

Identifying self-intersection points in one parametric graph.

My question for you is how to identify self-intersection points in a parametric curve of the form x = f(t), y = g(t). The specific problem asks for the t values of the intersection where $x = ...
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1answer
48 views

Parametric Equation explanation

Explain how the expression $tX + (1-t)Y$, $0\le t\le 1$, produces a segment that connects point $X (x_1, y_1)$ with point $Y (x_2,y_2)$. So I rearranged the problem such that $t(X - Y) + Y$ which I ...
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1answer
126 views

Paramaterization of paraboloid and plane.

Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have set each equation equal ...
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8answers
123 views

How can I parametrize $|x|+|y|=1$

I need parametrize $|x|+|y|=1$ but I don't know how to parametrize. I know that it is a rotated square, I would like understand so if you can explain to me like if I was still, thanks
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1answer
40 views

How do I represent a Mobius Band Triangle Parametrically

I am trying to describe a Mobius band in the shape of a triangle like this: parametrically in terms of its $x$, $y$, and $z$ functions. Is this even possible? I know a basic mobius strip can be ...
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2answers
52 views

Convert between parameteric ellipse equations

I have the parametric equation of an ellipse in this form: $$x(t)= a\cos(t)$$ $$y(t)=b\cos(t+\phi)$$ It's an ellipse centred about the origin, with a tilt angle. So three parameters. How can I ...
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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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2answers
76 views

How can I know whether the airplanes collide by using parametric equations

Recall that a line hes equation y=mx+c. Suppose one airplane moves along the line y=2x+3 while the other airplane moves along the line y=3x-2. By plotting a graph, even though the lines are intersect, ...
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1answer
325 views

Converting between explicit function and parametric function

Given an explicit function $y = f(x)$, how to convert it to the respective parametric functions $x = f_1(t)\; y = f_2(t)$? Given parametric functions $x = f_1(t)\; y = f_2(t)$, how to obtain the ...