Questions tagged [parametric]
For questions about parametric equations, their application, equivalence to other equation types and definition.
1,978
questions
0
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0answers
5 views
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, ...
-1
votes
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:
$$
\left\{\begin{aligned}
x &= t \cos \alpha - \sin t \sin \alpha\\
y &= t \sin \...
0
votes
4answers
24 views
A curve has the parametric equations $x=2t^2$ and $y=4t$. What is the value(s) of $k$ such that $y=x+k$ is a tangent to the curve?
A curve has the parametric equations $x=2t^2$ and $y=4t$. Find the value(s) of $k$ such that $y=x+k$ is a tangent to the curve.
I get that you need to use differentiation to do this and I've tried ...
0
votes
1answer
27 views
Show that $c(t)=\left(f(t),t^2\right)$ lies along the graph of $h(x)=|x|$. Unclear answer book.
Calculus by michael spivak (3rd edition) chapter 12 (apendix) question 2 goes as follows
Let
$$f(t)=
\begin{cases}
t^2&,x\geq0\\
-t^2&,x\leq0
\end{cases}$$
Show that $c(t)=\left(f(t),t^...
0
votes
0answers
20 views
Push-forward Fisher information
Let $(P_\theta)_{\theta\in\Theta}$ be a family of probability distributions on some measurable space $\mathcal X$, where $\Theta$ is some open real interval. Let $\nu$ be a $\sigma$-finite measure on $...
0
votes
0answers
33 views
How to deduce the equation of tangents at $P$ and $Q$ to the curve?
Points $P(2p^2,\displaystyle\frac{1}{p})$ and $Q (2q^2,\displaystyle\frac{2}{q})$ lies on a curve.
Find the gradient of $PQ$ and also the equation of line $PQ$.
I’ve found the gradient of $PQ$ to be ...
-1
votes
1answer
63 views
Parametric Equation help [closed]
A curve is defined by the parametric equations
$$
\begin{cases}
x = t^3-4,\\
y = at^2-12t
\end{cases}
$$
(a) Obtain expressions for $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$ in terms of $a$ ...
0
votes
2answers
33 views
Find the area enclosed by the y-axis and the parametric curve $x = t^2 - 8t + 15$; $y = e^{2t}$:
The title says it all. I tried this, since we need to find the area enclosed by the y-axis we could say that it is the line $x=0$, so now we can do this $t^2-8t+15 = (x-3)(x-5) = $ factorized form. ...
0
votes
3answers
58 views
Find the area enclosed by the line $x=5$ and the parametric curve $x=6t-t^2, y=e^{3t}$
Well, I have written the question above. This is what I tried, even though all of this is quite messed up, and I’m sorry for that. Could anybody guide me through the complete solution? Can you just ...
0
votes
1answer
18 views
Help with parametric equation
I posted this problem yesterday, but I want to make some changes regarding the questions I asked. Therefore I post it again.
So here are my questions:
What does $C:[0,2\pi]\rightarrow R^2$ mean. ...
1
vote
1answer
40 views
how can i rezolve the system of equations?
Solve the system of equations:
$$ \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)
$$
-1
votes
2answers
35 views
:Find the parameter m so the equation has real solution [closed]
Recently, I have found this problem:
Find the parameter $m \in \Bbb{R}$ so the equation has real solution
$$\cos^2(2x)+3m^2=4m(\cos^4(x)-\sin^4(x))$$
I suppose that the answer is $\forall m \in ...
2
votes
1answer
22 views
Find a common magnitude for two vectors such that they intersect
Trying to solve for a common magnitude of two non-parallel vectors such that they intersect. I am currently solving using a left inverse, but I am not sure if this is correct.
Let $ x_1 = x_1^0 + ...
0
votes
0answers
7 views
Parametric solution of equations
Suppose we have a system of equations
$$f(x; \theta)=0$$
in variables $x$ (which can be a vector), with parameters $\theta$. Sometimes, it is computationally easy to express the solutions as ...
0
votes
1answer
47 views
How to paramaterize a tractrix?
\begin{cases}
& \text{ } x(t)=t - a \frac{\sinh (\frac{t}{a})}{\cosh (\frac{t}{a})} \\
& \text{ } y(t)= \frac{a}{\cosh (\frac{t}{a})}
\end{cases}
According to the exercise, I need to see ...
0
votes
1answer
19 views
Parametric equation of Mandelbrot curve on k-th iteration?
Is there a way to write the parametric equation of the Mandelbrot-set's boundary curve at every $k^{th}$ iteration?
4
votes
1answer
46 views
relationship between derivative of vector valued function and gradient/partial derivatives
From This article I understand that if I have a function defined using parametric equations like this
$$
(1) \quad \quad \quad{{\mathbf{r}\left( t \right) = f\left( t \right)\mathbf{i} + g\left( t \...
-2
votes
0answers
25 views
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 ...
0
votes
0answers
14 views
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
$$\begin{align}x &= (R+ r\cos(\phi))\cos(\theta) \\
y &= (R+ ...
0
votes
0answers
21 views
An ellipse in (x,y)-plane, an ellipse in (y,z)-plane and two initial value conditions - make parametric equation
I have the following problem:
Two ellipses in respectively the $(x,y)$-plane and $(y,z)$-plane is given:
$E_{xy}=\frac{(x-5)^2}{3^2}+\frac{y^2}{5^2}=1$
$E_{yz}=\frac{y^2}{2^2}+\frac{(z-5)^...
1
vote
1answer
24 views
2nd derivative of function with 3 parametric variables?
For the function $f(x,y,z)$, where $x,y,z$ are all functions of $t$, what is the 2nd derivative of $f$ with respect to $t$?
I know that the 1st derivative is
$\frac {df}{dt} = \frac{\partial f}{\...
0
votes
0answers
10 views
Radius of curvature for a cycloid given parametric equations
I have been given $x=a(\theta - \sin\theta)$ and $y=a(1-\cos\theta)$
I need to prove the radius of curvature is = $2\sqrt2a(1-\cos\theta)^{1/2}$
$x'=a(1-\cos\theta)$ and $y'=a\sin\theta.$
1
vote
1answer
27 views
Parametric equation of a line knowing the trajectory of the center of a ball rolling on it
I have a parametric curve (in polar coordinates) that describes the trajectory of the center of a rolling ball. This ball (assimilated as a circle) rolls smoothly along a relief. I need to get an ...
0
votes
2answers
54 views
Why is discriminant finding wrong solution? [closed]
I have this equation $(1-9y^2)x^2 + (54y^2 - 18 y)x -10 +54y -81y^2=0$ and I want to find the y for which it has only one solution, so I use discriminant
$$D=(54y^2 - 18 y)^2-4(1-9y^2)(-10 +54y -...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
18 views
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 ...
0
votes
0answers
19 views
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{...
1
vote
0answers
44 views
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{\...
0
votes
2answers
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 $A$ ...
0
votes
0answers
34 views
Counting points of intersection of a hypocycloid and epicycloid
For $k$ a rational number, say $k=p/q$ expressed in simplest terms, the hypocycloid
$$\begin{align}
x&= r(k-1) \cos\theta+r \cos (k-1)\theta \\
y&= r(k-1) \sin\theta-r \sin (k-1)\theta
\end{...
0
votes
2answers
32 views
Parametric Equations for Circles [closed]
Question - A circle has the equation:
$$(x-3)^2 + (y+1)^2 = 16$$
Find parametric equations to describe the circle given that:
(a) $x = 3 + 4\cos t$
(b) $x = 3 - 4\sin t$
Much appreciated if ...
0
votes
2answers
22 views
General equation for a cylinder of thickness T
I know the equation for a cylinder of radius $R$ centered on the $z$-axis is
$F(x,y,z) = x^2 + y^2 - R^2$, which makes intuitive sense by considering the z-axis as the line $x^2 + y^2 = 0$ (only ...
0
votes
0answers
12 views
Clarification on parametric formula for 'uniformly accelerated curves'
I am looking for an example how to use parametric formula for 'uniformly accelerated curves' derived at MO for a particular case.
Actually it would be a curve $\bigl(x(t),y(t)\bigr)$ that satisfies an ...
0
votes
3answers
39 views
Finding the formula for a parabola from the control points of a bezier
I'm trying to write a program that can detect whether the mouse is inside the parabola defined by three Bezier control points. I first tried doing this by looping through each line in the "string art"...
0
votes
1answer
25 views
How would I express the following parametric pair in Cartesian form? $\bigl(4\sin(4t), 3\sin(3t)\bigr)$
Since I know that $\sin(4t) = 4\cos^3t\sin t-4\cos t\sin^3t$ and $\sin(3t) = 3\sin t-4\sin^3 t$ (or at least I think I do), I think I'm halfway there.
0
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0answers
39 views
Calculating the center of mass of a space bound by two parametric curves
Is there a way to calculate the centre of mass of a space bound by two general parametric curves. To be more specific, if you have an outer curve defined by: $X_o(t)$ and $Y_o(t)$, and an inner curve ...
2
votes
1answer
38 views
Solving ODE to find the unit tangent vector
A parametric curve is described by the following equations
$$\frac{dx}{dt} = x, y = \cos(t), z = \sin(t)$$ and passes through
$(1,1,0)$ when $t = 0$. By solving the ODE for $x(t)$, or otherwise,
...
0
votes
2answers
47 views
Problem Solving This Question?
A parametric curve is described by the following equations $\dfrac{\text{d}x}{\text{d}t} =x$, $y=\cos(t)$, $z=\sin(t)$,
and passes through ⟨1, 1, 0⟩ when $t = 0$. By solving the ODE for $x(t)$, or ...
2
votes
0answers
87 views
For $v_0=(1,0,0)$ and unit vectors $v_1$ and $v_2$ in $\mathbb{R}^3$, what is this set? $\{(x,y,z):x=v_0\cdot v_1,y=v_0\cdot v_2,z=v_1\cdot v_2\}$ [closed]
Let $v_{0}=(1,0,0)$. Then what does $$\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 ...
0
votes
1answer
26 views
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 ...
2
votes
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....
0
votes
1answer
32 views
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?
0
votes
1answer
30 views
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 ...
0
votes
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 ) = ...
0
votes
1answer
16 views
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 ...
0
votes
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 ...
0
votes
0answers
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
0
votes
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
$$
I only know that ...
1
vote
1answer
30 views
Verify function is reparameterization
Let C be the right half of the circle $\{ z \in \mathbb{C} : |z| = 2\}$. Consider the parameterizations $$ \sigma(t) = 2e^{i\theta}, \frac {-\pi}{2} < \theta < \frac {\pi}{2} $$ and $$ \delta(t)=...
