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

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21 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 ...
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1answer
20 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 ...
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0answers
9 views

how do I find the plane of a 3d parametric equation.

I'm trying to figure out how to find the plane this thing is on. $$[cos(\phi)^2\times (3\times cos(\theta)^2-cos(\phi)^2)/(cos(\theta)^2+cos(\phi)^2)^2, (cos(\theta)^2-3\times cos(\phi)^2)\times ...
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1answer
16 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 ...
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13 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 ...
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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 ...
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1answer
17 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 ...
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2answers
49 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 ...
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4answers
36 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 ...
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168 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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1answer
20 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, ...
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1answer
1k views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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2answers
31 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 $y(t)=\sin(t)$.
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1answer
23 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?
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23 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 ...
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4answers
26 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 \\ ...
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1answer
373 views

Find a vector equation and parametric equations or the line in R^3 that passes through the point (1,2,-3) and is parallel to the vector u=(4,-5,1).

Find a vector equation and parametric equations or the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
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1answer
20 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 $$ y=2-\sin \frac{\pi t}{2} $$ ...
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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 ...
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2answers
31 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 ...
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2answers
255 views

Is this valid parametric equation to create control points for a helix in 3D space?

Is this a valid way to compute new points that are on a helix and if not what is it wrong? The Cartesian coordinates of each new helix control point could be described by the following ...
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0answers
23 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 ...
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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 ?
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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: ...
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0answers
32 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 ...
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0answers
45 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 ...
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1answer
36 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 ...
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1answer
17 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 ...
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2answers
41 views

Parametrised curves.

I've been working through the following question: Q1= What points on the parameterised curve $x(\theta)=\cos^2{\theta}, y(\theta)=\sin{\theta}\cos{\theta}$ correspond to the parameter values ...
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1answer
28 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
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1answer
10 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 ...
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1answer
30 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 ...
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1answer
370 views

Shortest distance between a 3D parametric surface and a point

Right now I'm working on a library for finding the distances between objects in Lua. I've had some trouble finding the distance between a point and a bounded plane. I'm using these parametric ...
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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 ...
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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 ...
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1answer
914 views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
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1answer
45 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 ...
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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 $\int_0^t||r'(u)||dt$
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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 ...
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1answer
34 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 ...
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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 ...
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3answers
20 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 - ...
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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?
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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 ...
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2answers
226 views

Finding the length of a curve?

With the information given: $$x=\frac{y^4}{8}+\frac{1}{4y^2}\,,\ \ 1 \le y \le 2$$ I must find the exact length of the curve. I use this formula to find it: ...
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2answers
42 views

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

So given this curve: $$y=\sqrt{9x-18},\ \ 2 \le x \le 6$$ And using this lovely formula: $$\int2πy\sqrt{1+(\frac{dy}{dx})^2} dx$$ This is what I get for a set up: $$\int_2^62π ...
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2answers
30 views

Find the cartesian equation from the given parametric equations

I'm tasked with converting these parametric equations into one cartesian equation. $$ x = a*sin(t) $$ $$ y = b*cos(t) $$ So I begin with my reasoning, which is potentially 100% wrong. I want to ...
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2answers
134 views

Can there be variations on the Witch of Agnesi?

The function known as "The Witch of Agnesi" can be constructed using a circle of radius $a$, and is written in Cartesian coordinates as $$f(x)=\frac{8a^3}{x^2+4a^2}$$ The family of functions that ...
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2answers
22 views

Finding cartesian equation for trigonometric parametric forms

I'm trying to find the cartesian equation for these parameteric forms: $$ x = sin\theta + 2 cos \theta \\ y = 2 sin\theta + cos\theta $$ I tried: $$\begin{align} x^2 & = sin^2\theta + ...
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1answer
35 views

Sinewave riding on sinewave help

Consider this image Top one is Cos(t), I know that. What is the equation for the second one? (sinewave within sinewave) And how would I get to the third one and then on n-amount of recursion? Mind ...