# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
### 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 $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...