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

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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.
1
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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 ...
-3
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0answers
10 views

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 ...
0
votes
1answer
37 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 ...
0
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0answers
17 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, ...
0
votes
1answer
400 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
0
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2answers
2k views

Parametrization for the curve on cylinder $y = 7 - x^4$ 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 ...
3
votes
3answers
632 views

Distance between point and a spiral

I'm trying to work out an algorithm where, given the equation for a spiral in polar coordinates, $r(\theta)$, and a point rectilinear coordinates, $P(x,y)$, I can work out the minimum distance between ...
0
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0answers
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 ...
1
vote
1answer
14 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 ...
2
votes
2answers
55 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 ...
0
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0answers
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 ...
0
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0answers
25 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 ...
36
votes
3answers
2k views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two ...
2
votes
2answers
4k 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 $r⃗ (7)=(1,4,7)$, ...
0
votes
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 ...
1
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0answers
33 views

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 ...
0
votes
1answer
54 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 ...
1
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0answers
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, ...
1
vote
2answers
45 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 $...
0
votes
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 ...
0
votes
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 ...
1
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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, ...
0
votes
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 ...
1
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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?
0
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0answers
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 ...
0
votes
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=...
0
votes
2answers
756 views

Reverse direction of parametric equation

For the graph $y = \sqrt{x}$ the normal parametric equations would $x = t^2$ and $y = |t|$. However, the direction for that graph would be going from infinity to zero when $t \leq 0$ and zero to ...
0
votes
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 ...
18
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6answers
29k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
0
votes
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 $(0,0,2a)$....
1
vote
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 ...
0
votes
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 $$...
1
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0answers
20 views

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 \...
-1
votes
2answers
14 views

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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0answers
12 views

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 ...
1
vote
1answer
565 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
1
vote
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 $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...
1
vote
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 (\...
0
votes
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.
0
votes
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\...
0
votes
3answers
403 views

Line integral around intersection of sphere and plane

The unit sphere is intersected by the plane x + y = 1. Find the line integral of F = around the intersection. $\int\int\nabla$x$F\cdot$ n dA the unit normal vector is easily found by looking at the ...
0
votes
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. ...
1
vote
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)...
0
votes
1answer
679 views

Parametrization of the intersection of a cone and plane.

EDITED with new progress updates. As the title states, I'm trying to parametrize the intersection of a cone and a plane. The equations are: $z^2 = 2x^2+2y^2$ and $2x+y+3z=4\implies z=\frac{1}{3}(4-...
0
votes
0answers
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} \...
0
votes
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 ...
0
votes
0answers
11 views

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. ...
2
votes
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 ...
0
votes
0answers
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) = ...