Tagged Questions

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

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0
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1answer
38 views

How to parametrize a curve using polar coordinates

How do I parametrize a curve $(x^2+y^2)^2=40^2(x^2-y^2)$ using polar coordinates? then what period is $\theta$ in?
0
votes
1answer
17 views

Equation for simple parametric curve

My math skills are rusty. I want to find the parametric equation for the 5 vertices curve below. It consists of an ellipse with a rotating axes. I get stuck after this: $$x = a \cos(t) \cos(\theta) - ...
0
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0answers
10 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
0
votes
1answer
11 views

curve with vanishing tangent vector assumption

I am just reviewing some assumptions in Parametric representations The book says we assume 3-d curve has non-vanishing tangent vector. Why do we need to assume this Simply if we take $R^3$ then ...
1
vote
0answers
25 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 ...
0
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0answers
11 views

Failure propability for two parametrs of a distribution

In a population with a random distribution we have calculate the confidence intervals for parametr $θ_1$ as $(a_1, b_1)$ with failure propability $p_1$ and for parametr $θ_2$ as $(a_2,b_2)$ with ...
0
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2answers
22 views

Show that if $v_1$ and $v_2$ are any two vectors in this plane, then for any two scalars, $c_1v_1 + c_2v_2$ is also a vector in the plane

Let $a,\,b$ and $c$ be constants (not all zero) and consider the equation $ax + by + cz = 0$, which has a graph that is a plane that passes through the origin in $\mathbb{R}^3$. Show that if $v_1$ and ...
0
votes
1answer
20 views

Is this parametric equation correct?

Am I able to just put the planes point into the equation and leave it at that? or am I wrong here? My parametric equation: (x,y,z) = (3,-2,1)+ t(2,1,-3) + s(1,-2,4)
-3
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0answers
31 views

Deriving a parametric equation for a (simplistic) bird? Disclaimer: This requires mathematical creativity. [on hold]

Derive a parametric equation of this rather adorable bird? Its a 2-dimensional picture as you may notice.
0
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0answers
15 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
0
votes
2answers
32 views

Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...
0
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2answers
20 views

How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
1
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2answers
20 views

Creating a parametric Equation when given the points of a collinear line?

$(-70, 3)$, $(88, 81)$, and $(246, 159)$ are three collinear points. Write parametric equations for $x$ and $y$. (In other words, write equations that produce points when $t$-values are assigned.) ...
1
vote
1answer
16 views

Parametrizing shapes, curves, lines in $\mathbb{C}$ plane

I've been struggling with parametrizing things in the complex plane. For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how ...
0
votes
1answer
24 views

Orthogonal parameterization

Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$ With complex parameters $a,b,c$ and $d$ Now find any ...
1
vote
1answer
18 views

Ray-sphere intersection: t-value of the intersection points

You have a sphere centered at [1,2,3] with radius 3, and a ray from [10,10,10] in the direction [-1,-1,-1]. Write the implicit equation for the sphere, the parametric equation for the ray, and compute ...
0
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0answers
20 views

Parametrization for intersection between sphere and elliptic cylinder

Given the sphere: $$x^2 + y^2 + z^2 = 12$$ and the ellyptical cylinder: $$(x-1)^2 / (7/3) + (y-2)^2 / 7 = 1$$ Give a parametrization for the intersection curve. I'm confused on how to do it. The ...
1
vote
1answer
8 views

How do I successfully combine these two paramaterized equations?

I'm working on a set of equations that would tell a hypothetical robot soccer player whether or not to pass a ball to a teammate. After a lot of algebra, I arrived at these equations for the partial ...
3
votes
1answer
18 views

Parametrization of the implicit curve $F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0$

I am trying to find a parametrization for the curve defined implicitly by $$ F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0, $$ where $a_1$, $a_2$, $a_3$, $a_4$ and $b$ are constants satisfying ...
1
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2answers
50 views

How to show that the curve $ (x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $ is an ellipse?

Show that the curve $$(x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $$ is an ellipse in the plane it lies on. $$x^2 + y^2 = (\sin t)^2 + (\cos t)^2 = 1$$ $$x^2 + (z/c)^2 = (\sin t)^2 + (\cos ...
0
votes
4answers
42 views

Using parametric differentiation for $\frac{\operatorname d \! y}{\operatorname d \!x}$?

Hi so I'm in my calculus class and the teacher gave us a problem to do. I'm not quite sure how to attack this question. He's given us a couple of steps but I don't understand. If someone can further ...
1
vote
1answer
21 views

Determine whether the points (5, -6, 10) and (3, 3, 8) are on the line x = 2 + t, y = 3 - 3t, z = 4 + 2t

I've gone about trying to solve this by assuming the x co-ordinate lies on the line, and then determining whether the other points lie on that line according to that ie. If x = 5, then 5 = 2 + t, ...
0
votes
2answers
34 views

Parametric form of square

What is the appropriate parametric equation of the boundary of a square? For example, the unit circle has a parametric equation $x(t)=\cos(t)$ and $y(t)=\sin(t)$.
1
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1answer
25 views

Parametrization of a Complex Path/Contour Integration

How would I parametrize the path which is a straight line from 1 to a complex point z? Does $\delta (t) = z^t$ make any sense?
0
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0answers
48 views

Converting this 3D plane parametric equation to non-parametric

I had these 3 planes to put into parametric equation: (5, 4, −8),(1, 6, −3) and (7, −2, 5) so I put it into this parametric equation: (x,y,z) = (5,4,-8) + t(-4,2,5) + s(2,-6,13) But i am having ...
4
votes
4answers
28 views

If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$

If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$. This is how I tried solving it: $$ \frac{dx}{dt} = 2t\sin3t + 3t^2\cos3t \\ \frac{dy}{dt} = 2t\cos3t - 3t^2\sin3t \\ ...
0
votes
1answer
21 views

Parameterizing $y = 2 -\sin \frac{\pi x}{2}$

I am trying to parametrize the part of the curve $$ y = 2 -\sin \frac{\pi x}{2} $$ from (0, 2) to (1, 1). I tried the difficult paramaterization $x=t$ and obtained $$ y=2-\sin \frac{\pi t}{2} $$ ...
0
votes
1answer
18 views

Finding the initial direction of a parametric curve?

With the parameters: $$x(t)=1-\sin^2t$$ $$y(t)=2+\cos^2t$$ It starts at (1,3) and when t=pi/2 it's at (0,2), so I'm tempted to say it's going down to the right; is this correct? In general, is there ...
3
votes
2answers
37 views

Number of Curvature Maxima of a 2D Cubic Bezier curve

I am trying to prove that a standard cubic Bezier curve can only have at most 2 curvature maxima over $t \in [0,1]$. Assuming that no 3 adjacent control points are colinear, the curvature will either ...
2
votes
0answers
26 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 ...
1
vote
1answer
19 views

why can't we eliminate the parameter of straight lines in higher dimensions

Why we can't remove the parameter and find the Cartesian equation of straight lines in higher dimensions ?
0
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0answers
33 views

Parametric Equation of a parabola from the derivative of the parametric equation of a circle

Find the velocity and trajectory to throw a ball from a Ferris Wheel to a friend standing below. The Ferris Wheel has a diameter of 16 meters and its highest point is 19 meters above the ground. It ...
0
votes
0answers
49 views

Find point on rotated curve

I have a curve $f(t)$ that has been rotated through an angle $\theta$, and also have defined a given offset $Y$ from the curve origin. Using the equation $Y=x*sin(\theta)+y*cos(\theta)$ which ...
0
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1answer
26 views

From one parametric form of a curve to another one

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a(t-\tanh(t))\\y(t)=a \operatorname{sech(t)}\end{cases}$$ Is there a way to switch from these equations to the equations: ...
0
votes
1answer
37 views

Simplify a parametric equation with hyperbolic trigonometric functions

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a\cdot \operatorname{sech} (t) \\ y(t)=a\cdot(t-\tanh(t))\end{cases}$$ Now let $\theta(t)=-\arctan(\sinh(t))$ how does the ...
0
votes
1answer
18 views

Parametric equations for the tractrix

The cartesian equation of the tractrix is:$$y=\pm\left(a\cdot \operatorname{arcsech}^{-1}\left(\frac{x}{a}\right)-\sqrt{a^2-x^2}\right)$$ where $a>0$ is a real parameter and $x$ varies from $0$ to ...
1
vote
1answer
29 views

Jacobian of parametrized ellipsoid with respect to parametrized sphere

I'm not even sure how best to phrase this question, but here goes. Given $\theta$ (elevation) and $\phi$ (azimuth), the unit sphere can be parametrized as $ x = \cos(\theta)\sin(\phi) \\ y = ...
2
votes
1answer
12 views

length of intersection of parabolic cylinder and a surface

Let $C$ be the curve of intersection of the parabolic cylinder $x^2 = 2y$ and the surface $3z = xy$. Find the length of the part of $C$ from $(0, 0, 0)$ to $(6, 18, 36)$. (Hint: It may be useful to ...
0
votes
1answer
31 views

Finding the acceleration

So I am given a problem stated as: a point moves in the plane at speed 1 along the curve $y = x^2$. Find the acceleration at the point (x,y). I know that the velocity is y' = 2x, and that at a ...
0
votes
1answer
26 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
1
vote
2answers
18 views

Eliminating $\theta$ between the two expressions

How do we find the equation of this parametric curve $$2x=\cos {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos {\theta}\right)$$ $$2y=\sin {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos ...
2
votes
2answers
32 views

Create paramatric shape wihtout 'dents'

I am plotting a shape with the following equation $$\left\{ \begin{array}{c} x=r_{in} \cos(4 t)+r_{out} \cos(t)\\ y=r_{in} \sin(4 t)+r_{out} \sin( t) \end{array} \right. $$ Given various parameters ...
2
votes
1answer
46 views

Mathematical Description for Steam Rising from a Cup

I was staring at a cup of coffee I have on the desk just now. The light shines through the water vapor as they rise from the cup. The shape of the steam is not completely random, as it drift from ...
0
votes
1answer
23 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
0
votes
0answers
3 views

solution to curve limits (concept question)?

I had a question about how the limits work in that 4pi would not give the correct circle distance. I understand that if it has a radius 1 that the distance would be farther but that is only for a ...
0
votes
1answer
13 views

Finding the normals of an equation based on their parametric representation

A curve is defined parametrically by the equations $$ x = t^3 - 6t + 4, y = t - 3 + \frac{2}{t} $$ The first question, which I've partially solved, was to find the equations of the normals to the ...
0
votes
3answers
21 views

Finding cartesian equation from parametric trigonometric equations

I'm trying to find the cartesian equation of the curve which is defined parametrically by: $$ x = 2sin\theta, y = cos^2\theta $$ Both approaches I take result in the same answer: $$ y = 1 - ...
0
votes
1answer
35 views

Line Integral with Arclength Parametrization

Suppose we have an arclength parametrization of a curve in the $xy$-plane given by $x(s)$, $y(s)$ where $0 \leq s \leq L$. We want to integrate a scalar function $f(x,y)$ along this line. Since we are ...
0
votes
2answers
19 views

How do you eliminate the parameter to find a cartesian equation of the curve?

$$x=1/2cosθ$$ $$y=2sinθ$$ $$0 \le θ \le π $$ So I know the parameter that must be eliminated is θ. How should I do this? Are there trig identities that I can use?
-1
votes
1answer
13 views

Find the area of the surface obtained by rotating the curve about the x-axis?

Given this curve: $$y=\frac{x^3}{6}+\frac{1}{2x} 1/2 \le x \le 1 $$ This is what I get for my (dy/dx)^2: $$\frac{x^4+x^{-4}+2}{4}$$ I'm unsure about this. Can anyone confirm that I did it ...