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Questions tagged [parametric]

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

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Parameterisations intuition

If I paramaterise a function like x= f(t) and y=g(t) like x=t and y=t^2 why is it that the intersection in the x,y,t plane, I only get a point? not y=$x^2$? I understand the method of eliminating the ...
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Integral representation of time, transforming a graph

I have acquired a solution to a system of differential equations in a parametric form: \begin{equation} x= x_{0}u \end{equation} \begin{equation} y=-x_{0}u+\frac{\gamma}{\beta}\ln u - \frac{C_{1}}{\...
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Drawing graphs of parametric equations

the SIR model is an epidemiological model which sorts the whole population into three subclasses. x(t)(susceptibles), y(t)(infected), z(t)(recovered), with initial populations $x(0)= N_{1},y(0) = N_{2}...
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Graphing the SIR model

the SIR model is an epidemiological model which sorts the whole population into three subclasses. x(t)(susceptibles), y(t)(infected), z(t)(recovered), with initial populations $x(0)= N_{1},y(0) = N_{2}...
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A curve is defined by the parametric equations $x=2t+\frac{1}{t^2},\; y=2t-\frac{1}{t^2}$. Find the Cartesian equation.

A curve is defined by the parametric equations $$x=2t+\frac{1}{t^2}$$ $$y=2t-\frac{1}{t^2}$$ Show that the curve has the Cartesian equation $(x-y)(x+y)^2=k$ So I understand I need to eliminate ...
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Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
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parametrize the boundary of a region

I need to parametrize the boundary of this region : $D=\{y^2+z^2\le x^2+18,x^2+y^2\le 16\}$ So It's a one-sheet hyperboloid (radius=$\sqrt{18}$)+ cylinder with radius 4 I know how to parametrize ...
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How are the steps to the solution for Arc - Length obtained?

Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...
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What is the difference between the following definitions of Vector Functions and Parametric Curves?

The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space ...
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1answer
25 views

Line Integral Work Done [closed]

I have a problem when comes to question 2, I don't know how to put this into parametric form. And I am not sure if this is a parabola. Thanks in advance.
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Length of a parametric curve formula: What does the integral represent? [duplicate]

I had a question about the formula for finding the length of a parametric curve: What does the integral represent? Here is the formula: $\int\sqrt{\bigg(\frac{dx}{dt}\bigg)^2+\bigg(\frac{dy}{dt}\bigg)...
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1answer
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Finding the graph defined by $x = \sin \theta$ and $y = 3 - 2\cos(2\theta)$

The question is as follows: Find the graph of the parametric equations defined by $$ x(\theta) = \sin \theta \\ y(\theta) = 3 - 2\cos(2\theta) $$ We are supposed to use the identity that $\...
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1answer
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Explanation on the steps of this total arc-length solution

Can someone please explain to me what rules have been used to calculate ${\bf{\dot{x}}}$ and $|{\bf{\dot{x}}}|$ in the definition for total arc-length of this problem. I've tried to calculate this by ...
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Differential equations with a parametric form

I have been exploring mathematical epidemiology for a school project and have decided to model epidemics using the SIR model (Susceptibles, Infected and Recovered), so $x(t), y(t), z(t)$ are functions ...
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2answers
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Second derivative of parametric equations

I am looking for an intuitive explanation for the formula used to take the second derivative of a parametric function. The formula is: $\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$ I ...
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1answer
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$r = ycos(\theta) - xsin(\theta) $ derivation for Hough Transform

I am trying to see how $y = xtan(\theta) + \dfrac{r}{cos(\theta)}$ is made from the graph. Also how does the derivation work if $(x, y)$ is in the different quadrant? i.e. the $\theta$ location stays ...
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Parametric equation for $T^3$ and higher dimensional tori

For the standard torus $T^2$, the parametric equation $i : [0,2\pi]^2 \to \mathbb{R}^3$ is given by $$ x(\phi, \theta) = (R + r\cos(\theta))\cos(\phi)\\ y(\phi, \theta) = (R + r\cos(\theta))\sin(\phi)...
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Convert more complex Parametric to Cartesian Equations: $x(t)=\frac 3 2(t+\frac 1 t),y(t)=2(t-\frac 1 t)$ [closed]

Trying to these functions into a cartesian equation: $$x(t)=\frac 3 2(t+\frac 1 t),y(t)=2(t-\frac 1 t)$$
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2answers
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Finding parametric equations of the tangent line to a curve of intersection

The question asks to find the parametric equations of the tangent line to the curve of intersection of the surface $z=2\sqrt{9-\frac{x^2}{2}-y^2}$ and the plane $x=2$ at the point $(2,\sqrt{3},4)$ I'...
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Find the exact length of the parametric curve: $x={e^t}+{e^{-t}}, y=5-2t,0\le t \le 3$.

Find the exact length of the parametric curve: $$x={e^t}+{e^{-t}}, y=5-2t,0\le t \le 3$$ Solution: $$\frac{dx}{dt}=e^t -e^{-t}, \frac{dy}{dt}=-2$$ $$\int_{0}^{3} \sqrt{(e^t-e^{-t})^2+(-2)^2}dt$$ $$= \...
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1answer
58 views

Tough integral from a falling clock. $\int_0^{2\pi} \sqrt{g^2t^2 + 2rgt\sin(t) + r^2} {\rm d}t$

A clock is under free fall for $60$ seconds and its second hand makes exactly one revolution during that period of time. It begins at rest with its second hand facing upwards. Given that the second ...
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1answer
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Angle of ellipse's long axis and abscisa

I am reading a paper where I encountered the following equations $x=a.\cos(\omega t)$ and $y=b.\cos(\omega t + \phi)$ $\phi$ and $\omega$ are constants, t is the variable parameter. Then the ...
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What does it mean by “the consecutive points marked on the curve appear at equal time intervals but not at equal distances”?

What does it mean by the consecutive points marked on the curve appear at equal time intervals but not at equal distances ?
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1answer
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Trigonometric parametric system

I have a very specific system of two trigonometric equations $$\left( 3A^2\sin x \cos x - A \sin x \right) + \left( 3B^2\sin y \cos y - B \sin y \right) = AB \sin (x+y)$$ $$\left( 3A^2\cos^2 x - A \...
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Geodesic Dome defined parametrically

I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found this, but unfortunately it comes short of providing me the most ...
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1answer
30 views

Finding polar coordinates angle for complex numbers given cartesian form

I have the following formula for finding $\theta$ given cartesian form of complex numbers. $$\theta = \begin{cases} \tan^{-1}(\frac{y}{x}) & x \leq 0 \\ \tan^{-1}(\frac{y}{x}) & ...
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1answer
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how to make parametric equation of cube

this is my hyperbolic equation $y = x^2$ then convert to paramteric equation, like this: $x = u$ $y = u^2$ so i insert the equation into x and y Axis Generator ...
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1answer
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Given a cartesian equation get points in the plane

I hope this make sense. I'm trying to understand (I'm very newbie with curves) how could I get points from a cartesian equation. For example, given $(x^2+y^2)^2-2a^2\cdot(x^2-y^2)-a^4+c^4=0$ that is ...
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The parametric equation of a cone $z = \sqrt{x^{2} + y^{2}}$.

The given equation is - $z = \sqrt{x^{2} + y^{2}} , 0 \le z \le 1$ Let $x = r \cos t$, $y = r \sin t$ and $z = r$; where $0 \le r \le 1$ and $0 \le t \le 2 \pi$. Since $z$ is taken from $0$ to $...
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1answer
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Find the Cartesian form of the parametric equations: $x=2\sin^2(\theta)$, $y=7\cos^2(\theta)$

I am trying to find the cartesian form of the parametric expressions $x=2\sin^2(\theta)$, $y=7\cos^2(\theta)$. I have $x=2-cos^2(\theta)$ but i can't work it after that.
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Finding equation of a curve that a parametric equation of a line is tangent to

I have the line $y=x\cdot\frac{-a}{-3a^2+4}$. I want to find the curve that this line is tangent to. More info: the tangency point needs to be at coordinates $(\frac{a\left(-3a^2+4\right)}{-9a^2+16},-\...
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Finding a surface integral where S is the intersection of a cylinder with a plane

If $\vec{F}=\vec{i}+2\vec{j}+\vec{k}$ and $S$ is the intersection of the solid cylinder $x^2+y^2\le1$ with the plane $2x+y-z=1$, compute $\int\int_S\vec{F}.\vec{n} dS$ (using an upward pointing $\vec{...
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derivative of a parametric function at a point

I need to find the derivative of $$l\begin{cases} y=t\cdot \cos(t)\\ x=e^t-2t-1 \end{cases} $$ at $(0,0)$. How do I approach this? Thank you.
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1answer
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Minimum distance of curve from origin

I have a parabola $(y+5)^2 = 4x$ and I need to find its minimum distance from origin. Scientific calculators aren't allowed. I have tried : 1) Substituting parametric coordinates $(r\cos Q, r\sin Q)$...
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Divergence Theorem - Cone

Here's the question: Evaluate the surface integral $\iint _S F\cdot n \space dA$ by the divergence theorem. $ \mathit F = [xy, yz, zx]$, S the surface of the cone $x^2 + y^2 \le 4z^2, \space \space 0 ...
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1answer
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Parametric equation of a non-wrapping circle on the surface of a cylinder

I have a cylinder of radius $R$, and I wish to draw a circle of radius $r$ on its surface that does not wrap around it. So, the centre will lie somewhere on the surface of the cylinder, and the whole ...
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1answer
37 views

How can you determine if a point is inside a parametric 2D manifold?

Asume I have an arbitrary, parametric, closed, orientable, surface; a sphere, ellipsoid, closed cylinder, weird general cone.... If you only have access to the parametrization, how can you determine ...
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Smoothly merging two parametric curves

Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s \geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I ...
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Equation of line in 3d space passing in two points in a form of ax+by+cz+d=0

I'm sorry for asking probably such easy question, but need help with this.. I need to get the parameters with the equation of a straight line passing through two points in 3d space. ex: ...
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Concurrent Parametrizations

I'm currently working on a problem. I have found the parametrizations of 5 lines using position vectors, so in the form of $\overrightarrow{Q} = (1-s) \overrightarrow{B} + s \overrightarrow{T}.$ How ...
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Determine the length of the Parametric Curve given by the set of parametric equations.

I am seeking validation for my answer for the given problem below. Question: "Determine the length of the Parametric Curve given by the set of parametric equations." Parametric Equations: $x = 3 + ...
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Find the equation of the Tangent Line to the given set of Parametric Equations at given point. [closed]

I'm looking for validation for my answer to this question. Parametric Equations: $x = t^2 + 2t + 1 , y = t^3 + 7t^2 + 8t, t = -1$ For this problem I used the Point-Slope-Form formula. myAnswer:$ y =...
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Find the area of the surface formed by revolving the given curve about $(i)x$-axis and $(i)y$-axis

Q:Find the area of the surface formed by revolving the given curve about $(i)x-axis$ and $(i)y-axis$ $$x=a\cos\theta ,y=b\sin\theta,0\le\theta\le2\pi$$ About $x-$axis is, $S=2\pi\int_0^{2\pi}b\sin\...
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Parametric equations relate to $144\int_{0}^{\pi/2}\cos(t)\sin^2(t)\mathrm dt$

Parametric equations $$x=6\cos t$$ $$y=12\sin(2t)$$ $$0\le t\le 2\pi$$ The curve is symmetrical in the x axis and in the y axis. Show that $$144\int_{0}^{\pi/2}\cos(t)\sin^2(t)\mathrm dt\tag1$$ ...
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Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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parametric representation of a closed “Cylinder”

Assume I have a parametric curve $r(t) = <X(t),Y(t)>$ that defines a closed curve (like a circle or a closed B-spline) on a 2D plane. The cylinder can be defined as: $f(u,v) = <X(u), Y(u), ...
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39 views

Parametrization of a circle at (-1,-8) with Radius 9

Question: The circle centered at $(−1,−8)$ with radius $9$ can be parametrized in many ways, this still happens even if we impose the extra constraint that the circle must be traversed in the counter-...
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Is there a name for the minimal surface connecting two straight line segments in 3-dim Euclidean space?

In the $2$-dim Euclidean plane, the minimal curve that joins (and necessarily contains) two given points is a straight line. Here what is being minimized by the curve is the $1$-dim measure of the $1$...
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Reference request: Parametrizing a curve

In my university syllabus I have the following topics for which I could not find any book to study: Parametric equations, parametrizing a curve, arc length of a curve, arc length of parametric curves, ...
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1answer
21 views

Parametric Equations Finding Area Absolute Value or?

So I am given the following parametric equations. $$ y=bsin(\theta) $$ and $$ x=acos(\theta) $$ When I do the following I get a negative area. $$ \int_0^{2\pi}b\sin\theta\frac{d}{d\theta}\left(a\...