Questions tagged [parametric]

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

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Do parametric equations represent the $x$-axis and $y$-axis stretching?

Do parametric equations represent the $x$-axis and $y$-axis stretching? I've just starting learning about parametric equations, and it seems to me that there are $2$ parametric equations (maybe more, ...
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2answers
28 views

What is the equation describing sin(x) rotated by an angle?

I'm talking about a curve that looks like this: I've come up with these equations, which seem to describe it: $$ \left\{\begin{aligned} x &= t \cos \alpha - \sin t \sin \alpha\\ y &= t \sin \...
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4answers
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A curve has the parametric equations $x=2t^2$ and $y=4t$. What is the value(s) of $k$ such that $y=x+k$ is a tangent to the curve?

A curve has the parametric equations $x=2t^2$ and $y=4t$. Find the value(s) of $k$ such that $y=x+k$ is a tangent to the curve. I get that you need to use differentiation to do this and I've tried ...
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27 views

Show that $c(t)=\left(f(t),t^2\right)$ lies along the graph of $h(x)=|x|$. Unclear answer book.

Calculus by michael spivak (3rd edition) chapter 12 (apendix) question 2 goes as follows Let $$f(t)= \begin{cases} t^2&,x\geq0\\ -t^2&,x\leq0 \end{cases}$$ Show that $c(t)=\left(f(t),t^...
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Push-forward Fisher information

Let $(P_\theta)_{\theta\in\Theta}$ be a family of probability distributions on some measurable space $\mathcal X$, where $\Theta$ is some open real interval. Let $\nu$ be a $\sigma$-finite measure on $...
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How to deduce the equation of tangents at $P$ and $Q$ to the curve?

Points $P(2p^2,\displaystyle\frac{1}{p})$ and $Q (2q^2,\displaystyle\frac{2}{q})$ lies on a curve. Find the gradient of $PQ$ and also the equation of line $PQ$. I’ve found the gradient of $PQ$ to be ...
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63 views

Parametric Equation help [closed]

A curve is defined by the parametric equations $$ \begin{cases} x = t^3-4,\\ y = at^2-12t \end{cases} $$ (a) Obtain expressions for $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$ in terms of $a$ ...
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Find the area enclosed by the y-axis and the parametric curve $x = t^2 - 8t + 15$; $y = e^{2t}$:

The title says it all. I tried this, since we need to find the area enclosed by the y-axis we could say that it is the line $x=0$, so now we can do this $t^2-8t+15 = (x-3)(x-5) = $ factorized form. ...
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Find the area enclosed by the line $x=5$ and the parametric curve $x=6t-t^2, y=e^{3t}$

Well, I have written the question above. This is what I tried, even though all of this is quite messed up, and I’m sorry for that. Could anybody guide me through the complete solution? Can you just ...
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Help with parametric equation

I posted this problem yesterday, but I want to make some changes regarding the questions I asked. Therefore I post it again. So here are my questions: What does $C:[0,2\pi]\rightarrow R^2$ mean. ...
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how can i rezolve the system of equations?

Solve the system of equations: $$ \left.\begin{aligned} axy+x+y &= A \\ ayz+y+z &= B \\ azx+z+x &= C \end{aligned}\right\}~~ a,A,B,C \in (0,\infty) $$
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:Find the parameter m so the equation has real solution [closed]

Recently, I have found this problem: Find the parameter $m \in \Bbb{R}$ so the equation has real solution $$\cos^2(2x)+3m^2=4m(\cos^4(x)-\sin^4(x))$$ I suppose that the answer is $\forall m \in ...
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1answer
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Find a common magnitude for two vectors such that they intersect

Trying to solve for a common magnitude of two non-parallel vectors such that they intersect. I am currently solving using a left inverse, but I am not sure if this is correct. Let $ x_1 = x_1^0 + ...
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Parametric solution of equations

Suppose we have a system of equations $$f(x; \theta)=0$$ in variables $x$ (which can be a vector), with parameters $\theta$. Sometimes, it is computationally easy to express the solutions as ...
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1answer
47 views

How to paramaterize a tractrix?

\begin{cases} & \text{ } x(t)=t - a \frac{\sinh (\frac{t}{a})}{\cosh (\frac{t}{a})} \\ & \text{ } y(t)= \frac{a}{\cosh (\frac{t}{a})} \end{cases} According to the exercise, I need to see ...
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Parametric equation of Mandelbrot curve on k-th iteration?

Is there a way to write the parametric equation of the Mandelbrot-set's boundary curve at every $k^{th}$ iteration?
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1answer
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relationship between derivative of vector valued function and gradient/partial derivatives

From This article I understand that if I have a function defined using parametric equations like this $$ (1) \quad \quad \quad{{\mathbf{r}\left( t \right) = f\left( t \right)\mathbf{i} + g\left( t \...
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Parametric form of a triangle

I was thinking whether there was any parametric form for a triangle involving the use of trigonometric functions, more specifically for an equilateral triangle(?) In this question, they used an ...
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Parametric equation of surface of a torus in toroidal co-ordinate

What would be the parametric equation for a surface in toroidal coordinate whose parametric equation in polar ordinates would be $$\begin{align}x &= (R+ r\cos(\phi))\cos(\theta) \\ y &= (R+ ...
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An ellipse in (x,y)-plane, an ellipse in (y,z)-plane and two initial value conditions - make parametric equation

I have the following problem: Two ellipses in respectively the $(x,y)$-plane and $(y,z)$-plane is given: $E_{xy}=\frac{(x-5)^2}{3^2}+\frac{y^2}{5^2}=1$ $E_{yz}=\frac{y^2}{2^2}+\frac{(z-5)^...
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1answer
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2nd derivative of function with 3 parametric variables?

For the function $f(x,y,z)$, where $x,y,z$ are all functions of $t$, what is the 2nd derivative of $f$ with respect to $t$? I know that the 1st derivative is $\frac {df}{dt} = \frac{\partial f}{\...
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Radius of curvature for a cycloid given parametric equations

I have been given $x=a(\theta - \sin\theta)$ and $y=a(1-\cos\theta)$ I need to prove the radius of curvature is = $2\sqrt2a(1-\cos\theta)^{1/2}$ $x'=a(1-\cos\theta)$ and $y'=a\sin\theta.$
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1answer
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Parametric equation of a line knowing the trajectory of the center of a ball rolling on it

I have a parametric curve (in polar coordinates) that describes the trajectory of the center of a rolling ball. This ball (assimilated as a circle) rolls smoothly along a relief. I need to get an ...
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2answers
54 views

Why is discriminant finding wrong solution? [closed]

I have this equation $(1-9y^2)x^2 + (54y^2 - 18 y)x -10 +54y -81y^2=0$ and I want to find the y for which it has only one solution, so I use discriminant $$D=(54y^2 - 18 y)^2-4(1-9y^2)(-10 +54y -...
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1answer
59 views

Impossible problem to solve manually?

Here is the attached problem: [![enter image description here][1]][1] I have solved part a) and can do c and other also.. But for that I need to solve b) this is where I got stuck. I don't know how ...
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9 views

Trying to find an equation of a curve which is made by joining vectors in a particular vector field

Consider the equation dy/dx = cx/y, where c is a real number. Now, consider the points on a Cartesian graph and some vectors associated with each point in such a way that the gradient of each vector ...
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Is a joint random variable $(X,Y)$ similar to a parametric representation/vector valued function?

Question/Request: Can you please verify if the following analogy would be a good way to understand/introduce multiple random variables? There may be some caveats but if it is getting to the intuitive ...
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Determine the surface integral and the mass of $f$

I have the following problem: An area $B$ in the $(x,y)$-plane is given by the parametric equation: $\begin{bmatrix} x \\ y \end{bmatrix}= \begin{bmatrix} v^4 \cdot \mbox{sin(u)} \cdot \mbox{...
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How to find $y'(x=1)$ in parametric equation?

Given parametric equation \begin{align} x(t)&=a-b(2t+\sin(2t))\\ y(t)&=b(1+\cos(2t)) \end{align} and $y(x=0)=1$. Find $y'(x=1)$. This is my effort. \begin{align} \dfrac{dy}{dx}&=\dfrac{\...
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56 views

Make parametric equation for closed space

I have the following problem. Let $a$ be a positive number. A curve $K$ in the $(x,z)$-plane is given by the parametric equation: $$r(u)=(2 \mbox{sinh(u)},\mbox{cosh(u)}), u \in[0,a]$$ Let $A$ ...
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Counting points of intersection of a hypocycloid and epicycloid

For $k$ a rational number, say $k=p/q$ expressed in simplest terms, the hypocycloid $$\begin{align} x&= r(k-1) \cos\theta+r \cos (k-1)\theta \\ y&= r(k-1) \sin\theta-r \sin (k-1)\theta \end{...
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Parametric Equations for Circles [closed]

Question - A circle has the equation: $$(x-3)^2 + (y+1)^2 = 16$$ Find parametric equations to describe the circle given that: (a) $x = 3 + 4\cos t$ (b) $x = 3 - 4\sin t$ Much appreciated if ...
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General equation for a cylinder of thickness T

I know the equation for a cylinder of radius $R$ centered on the $z$-axis is $F(x,y,z) = x^2 + y^2 - R^2$, which makes intuitive sense by considering the z-axis as the line $x^2 + y^2 = 0$ (only ...
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Clarification on parametric formula for 'uniformly accelerated curves'

I am looking for an example how to use parametric formula for 'uniformly accelerated curves' derived at MO for a particular case. Actually it would be a curve $\bigl(x(t),y(t)\bigr)$ that satisfies an ...
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3answers
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Finding the formula for a parabola from the control points of a bezier

I'm trying to write a program that can detect whether the mouse is inside the parabola defined by three Bezier control points. I first tried doing this by looping through each line in the "string art"...
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How would I express the following parametric pair in Cartesian form? $\bigl(4\sin(4t), 3\sin(3t)\bigr)$

Since I know that $\sin(4t) = 4\cos^3t\sin t-4\cos t\sin^3t$ and $\sin(3t) = 3\sin t-4\sin^3 t$ (or at least I think I do), I think I'm halfway there.
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Calculating the center of mass of a space bound by two parametric curves

Is there a way to calculate the centre of mass of a space bound by two general parametric curves. To be more specific, if you have an outer curve defined by: $X_o(t)$ and $Y_o(t)$, and an inner curve ...
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1answer
38 views

Solving ODE to find the unit tangent vector

A parametric curve is described by the following equations $$\frac{dx}{dt} = x, y = \cos(t), z = \sin(t)$$ and passes through $(1,1,0)$ when $t = 0$. By solving the ODE for $x(t)$, or otherwise, ...
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Problem Solving This Question?

A parametric curve is described by the following equations $\dfrac{\text{d}x}{\text{d}t} =x$, $y=\cos(t)$, $z=\sin(t)$, and passes through ⟨1, 1, 0⟩ when $t = 0$. By solving the ODE for $x(t)$, or ...
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For $v_0=(1,0,0)$ and unit vectors $v_1$ and $v_2$ in $\mathbb{R}^3$, what is this set? $\{(x,y,z):x=v_0\cdot v_1,y=v_0\cdot v_2,z=v_1\cdot v_2\}$ [closed]

Let $v_{0}=(1,0,0)$. Then what does $$\left\{ \left(v_{0}\cdot v_{1},v_{0}\cdot v_{2},v_{1}\cdot v_{2}\right):v_{1},v_{2}\in\mathbb{R}^{3} \text{are unit vectors}\right\} $$ look like geometrically ...
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1answer
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Coordinate Curves of a Parametrization

If I have a paraboloid $z = 1 - x^{2} - y^2$ with a parametrization of $\phi(r,\theta) = (rcos\theta, rsin\theta, 1 - r^2)$ I believe to prove $\phi$ as a parametrization I need $\phi$ to be ...
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2answers
102 views

Finding a function for linear growth that gradually plateaus

I need to draw a very particular kind of line (approximating it using a logarithmic curve is not going to be sufficient). Any help would be very gratefully appreciated! https://i.imgur.com/qcgqnEs....
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How to hand sketch the parametric curve $y=\sin3t$ and $x=\cos t$?

I tried to eliminate the variable $t$ and I've got $y = \sin(t)(4x^2-1)$ I know what it looks like by plotting with computer. but how do i do it by hand?
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How do I analyze the correlation of set of points to this complicated parametric equation?

I am trying to quantify the correlation of a given set of points to the set of points defined by a parametric equation, as well as find the best $l$ to fit the points. However, I am unsure of how to ...
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1answer
37 views

Find the points where the curve $r(t) = \langle{t, t^2, t^3\rangle}$ intersects the surface $zx = 13y - 36$

1) Rearrange the equation to : 13y - zx = 36 2) To find the point of intersection, we plug the parametric equations into equation for the plane; 13 (t^2) - 1(t^3) x 1(t) = 36 13t^2 - ( t^3 x t ) = ...
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1answer
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Finite Element boundary normal vector

I have a finite element and the meshs coordinates can be described using isoparametric shape functions as: $x(\xi, \eta) = \sum _i N_i(\xi, \eta)x_i$, $y(\xi, \eta) = \sum_i N_i(\xi, \eta)y_i$ I ...
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2answers
23 views

Translate on the horizontal axis the graph of a second degree polynomial

I'm trying to find the parametric second degree polynomial that would allow me to translate the its graph on the horizontal axis. Basically the equivalent of increasing or decreasing ...
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23 views

Using rotational sweeping, define by parametric functions x(u,v), y(u,v), z(u,v), u,v Î[0,1] the surface displayed in Figure Q1.

I am having difficulty visualizing. Can someone help me? The figure can be seen by clicking on the underlined Figure 1 Figure 1
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1answer
28 views

From parametric to symmetric form

I'm studying parametric equations and the manual I'm following does not explain how it goes from parametric to symmetric form. The problem is: $$ x(t)=3\cos^2 t\\ y(t)=3\sin^2 t $$ I only know that ...
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1answer
30 views

Verify function is reparameterization

Let C be the right half of the circle $\{ z \in \mathbb{C} : |z| = 2\}$. Consider the parameterizations $$ \sigma(t) = 2e^{i\theta}, \frac {-\pi}{2} < \theta < \frac {\pi}{2} $$ and $$ \delta(t)=...

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