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

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Parametric Equation part A

Hi everyone I am in need of some guidance solving this parametric equation question and was wondering if you guys could give me some pointers and to see if I am doing this correctly. Here I have two ...
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1answer
67 views

Name this 2-dim Parametric Curve

is there a name for the following parametric curve? Thanks. I encountered it when playing around with the tangent lines and normal lines of an ellipse. The parameter $\theta$ is the same parametric ...
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2answers
33 views

For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
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How to graph hypocycloid on graphical calculator

I want to know how I can graph a hypocycloid using my TI-nspire calculator. I already know the parametric equations for hypocycloids which is: Parametric Equations of Hypocycloids Does anyone know ...
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18 views

Hypocycloid with an outer ellipse

I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, ...
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1answer
38 views

Solving the following parametric equation

Solve the following parametric equation: $$\frac{-(3\cos t-x)}{2\sin t-y}=-\frac{2\cos t}{3\sin t}$$ So I need to find the parametric equation of the thing in terms of $t$. But I am confused ...
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15 views

Cycloids with ellipse

I have been researching about the epitrochoids and hypotrocoids lately. I was wondering if it would be possible for us to have an ellipse rolling around a circle? If so, then how could one derive its ...
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1answer
15 views

How to find a Bezier curve without control points?

Let's say someone created a cubic Bezier curve using software and rasterised it. However, the original equation of the Bezier curve was not noted. Since we have the image of the Bezier curve, we can ...
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2answers
56 views

Why isn't the gradient vector of a parametric curve parallel to the tangent vector?

Consider a parametric curve defined by the equation: $$\mathbf{r}(t) = X(t)\mathbf{\hat{i}} + Y(t)\mathbf{\hat{j}} + Z(t)\mathbf{\hat{k}}$$ Paul's online math notes indicate that the unit tangent ...
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13 views

straight to helix transition

I am trying to get cylindrical parametric equations for a straight line to helix transition, where the straight line is the centre axis of the helix. From what I can deduce, a straight line is a helix ...
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26 views

Finding area of hypocycloids (without integration)

I have been trying to find the area of hypocycloids, I understand how to do it with integration. But the thing is I wanna find some other method for finding its area. In one of the sites online, I ...
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2answers
30 views

Another method of finding area of hypocycloids

I was finding the are the of hypocycloids. Then it struck me that apart from integration, there could be another method of finding the area of the hypocycloid with different curves. But the problem is ...
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1answer
13 views

Help writing a parametric equation from this complex polar one

A particle is moving along the curve $r=4-2\sin(\theta)$ at the moment when $\theta = t^2$. I need to write a x(t) and y(t) function that will model the particle behavior with its x position and y ...
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Hi, I have been trying to understand the derivation of a hypocycloid's parametric equation, but am stuck with one part.

I have been using someone else's answer on the same site to understand the problem: here's the link - Parametric equations for hypocycloid and epicycloid I can understand everything but the part ...
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1answer
55 views

Determine the largest area of an ellipse enclosed by the hyperbolas ($xy=1$ and $xy=-1$)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed by the ...
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22 views

cycloid of a unit-speed circle

In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, ...
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2answers
46 views

How to find the union&intersection of two lines by their equations?

I will try to be as clear as possible concerning my confusion, and I will use some examples(several ones). Case number 1. Assume two equations(in cartesian form) of two planes. $2x+2y-5z+2=0$ and $...
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1answer
29 views

find the vector equation for the intersection of a plane and sphere

$x^2+y^2+z^2=4$ and $x+y+z=3$ First I tried to parameratize: $t=x^2 \to x=\sqrt{t}$ $t=y^2 \to y=\sqrt{t}$ Then substituting those parameters into the plane to get: $z=3-\sqrt{2t}$ These three ...
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1answer
47 views

Why is a curve parameterized by arc length necessarily a unit speed curve? [duplicate]

I apologize if this is trivial but I have not been able to figure it out. For a curve $\sigma(t)$, I have a definition for arc length: $$s(t)=\int_{t_0}^t |\sigma'(t)|dt$$ We reparameterize a curve ...
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1answer
23 views

find a vector function that represents the curve of intersection of the two surfaces

The cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$ I parameterized the plane and put it into vector form: $t=1+y \to y=t-1$ $z= 1+t-1 \to z = t$ $y= t-1$, $z=t$ Since I'm finding the intersection, ...
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31 views

Techniques for finding functions with known values and derivatives at two points

I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$. What ...
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1answer
16 views

How does using l1 norm or l2 norm for parametric spline affect the shape

When constructing a parametric cubic spline in three dimensions, I get three splines x(h), y(h) and z(h). When calculating the parameter h I would intuitively use the l2 norm between each successive ...
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1answer
24 views

At what points does the curve intersect with the paraboloid?

$r(t) = ti+(2t-t^{2})k$ intersect the paraboloid $z = x^2 + y^2$ What am I missing here? Can I get some hints that lead me as to what I need to do here? I haven't the faintest idea where to start. I ...
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1answer
28 views

Surface integrals-Important parametrization of a surface

As we know, an elipse is parametrized as $x=ar\cos(\theta)$ and $y=b r\sin(\theta)$, where $r$ is the radius and $a,b$ are some constants. Well, my question is, how shall I parametrize the surface $z=...
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1answer
33 views

Baricenter of a region bounded by a parametric curve

I just want to ask if there exists a general rule to get the baricenter of a region bounded by a parametric curve?
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2answers
28 views

Find the limit of the vector function

$lim_{t\to\infty} \Big(te^{-t},\frac{t^3+t}{2t^3-1},tsin(\frac{1}{t})\Big)$ a) $lim_{t\to\infty} te^{-t} = \infty \times 0$ $lim_{t\to\infty} 1e^{-t}+-e^tt = 0+(0\times\infty)$=undefined, and ...
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1answer
25 views

Curve of intersection, value for parameter

This is for a line integral. Parametrize the curve of intersection: \begin{align*} S_1: x^2+4y^2 + z^2 &= 4a^2, y<0\\ S_2: x+2y &= 0 \end{align*} Orientation from $(0,0,-2a)$ to $(0,0,2a)$....
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3answers
31 views

Find a Parametrization for the following curve? Help?

Find the parametrization for: The lower half of the parabola $y^2=x-1$ So here's what I did: I put in $$x=t$$ Then I solved for $y$ $$y^2=t-1$$ $$y=\sqrt {t-1}$$ So the parametrization would be $$...
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Set up an integral that is obtained by rotating the given curve about the $x$ axis and find the surface area

Set up an integral that is obtained by rotating the given curve about the $x$ axis and find the surface area. $$ \begin{aligned} x &= t\sin(t) & y &= t\cos(t) & 0 &\leq t \leq \...
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2answers
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Find an equation tangent to the curve at the point corresponding to the given value of the parameter

$x = 1 +4t -t^2$, $y = 2 - t^3$, at $t=1$ $\frac{dy}{dx}$ $= \frac{-3}{2}$ at t = 1. Where do I go from here?
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Paramerization for compact rational knots of degree 6?

The algorithm computes but it computes rational function of degree 8. I am interested in rational knotted functions of degree 6. Perhaps relevant publications here on non-compact curves of degree 6 ...
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2answers
22 views

Eliminate the parameter given $x = \tan^{2}\theta$ and $y=\sec\theta$

$x = \tan^{2} (\theta)$ and $y = \sec (\theta)$ knowing that $\tan^{2} (\theta) = (\tan (\theta))^2 = \dfrac{\sin^{2}\theta}{\cos^{2}\theta}$ and that $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...
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2answers
32 views

eliminate the parameters

Given: $x = \frac{1}{2} \cos(\theta)$ and $y = 2\sin(\theta)$ Part a) solving the first one for theta: 1) multiply both sides by $2$: $$2x = \cos(\theta)$$ 2) divide both sides by $\cos (\...
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1answer
35 views

Help with arc length

I have a curve defined by parametric equations $$y=a\sin^{5}t \\ x=a\cos^{5}t, $$ where $t\in(0;2\pi)$. I solve it by well-known formula: $L=\int\limits_{0}^{2\pi}\sqrt{(y')^2+(x')^2}dt$. $$x'=-5a\...
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2answers
26 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
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1answer
46 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
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1answer
16 views

Last Step of a Parametric to Cartesian Conversion

I need to figure out how to combine the (4) line to make the t=y-z/R+x and I just don't have any ideas. I'm sorry if these seem basic but I'm 16 and struggling through a topic I've never done before. ...
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31 views

Find an equation to the tangent line to the curve at the given point

\begin{align} x &= \cos t + \cos 2t, & y &= \sin t + \sin 2t, & \left(−1, 1\right) \end{align} Using the above information I found that $\;\frac{dy}{dx}\;$ is: \begin{align} \...
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2answers
39 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
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I need input and help understanding how the formula for x arises in a cycloid that is parameterized with theta with the cusp at the origin

Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. ...
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28 views

Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
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2answers
56 views

Find curve parametrization

I am asked to find the work of $f(x, y, z) = (x, z, 2y)$ through the curve given by the intersection of two surfaces. I have been doing a series of exercises on this and my question has simply to do ...
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1answer
47 views

Find $\frac{dy}{dx}$ when $t=0$ for $\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$

A dot is moving on a grid following this rule: $$\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$$ I need to find $\frac{dy}{dx}$ when $t =0$. It seems like I should use implicit ...
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1answer
40 views

Parameterise linear combination of cosines

How do I parameterise the following implicit surface? $$ \cos x + \cos y + \cos z = 0 $$ Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. ...
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1answer
22 views

Sampling a curve (parametric)

I am working with parametric curves, I need to find the maximum curvature of these curves. I know the starting point, ending point and length of a curve. I want to use sampling method to know the ...
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2answers
32 views

Intersection between roses with given polar equations

$ r_1 \ = \ 4 \sin(3 \theta) \ $ and $ \ r_2 \ = \ 3 \cos(3\theta) \ $ a) find the solutions to the system using polar coordinates I was able to solve this by setting $ \ r_1 \ $ and $ \ r_2 \ $ ...
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1answer
69 views

Which way should you run from the lions?

This is a fun problem that I saw somewhere on the internet a long time ago: Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion ...
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1answer
22 views

Maximizing this parametric expression with a certain range of integer inputs

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the expression: $b^s (a^{t-s} - b^{t-s})$ My intuition says this should ...
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0answers
13 views

Maximum of this parametric expression

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the following expression: $b^s~(a^{t-s}-b^{t-s})$ My intuition says this ...
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1answer
64 views

Solving a Diophantine equation in three variables as a parametric equation in one variable

Let’s say that $a$, $b$, and $c$ are integers such that $$(b^2+2)^2=(a^2+2c^2)(bc-a). \tag{$\star$}$$ By brute force search, I think I’ve discovered that $$(a,b,c)=(5d+1,3d+1,d+2), \qquad d=\dots,-2,-...