What is Parametric: Definition and 673 Discussions
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object.For example, the equations
x
=
cos
t
y
=
sin
t
{\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}
form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors:
(
x
,
y
)
=
(
cos
t
,
sin
t
)
.
{\displaystyle (x,y)=(\cos t,\sin t).}
Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).
Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.
My work so far:
I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)
Hi,
I am having problems with task b
I then defined the velocity vector and the acceleration vector as follows
##dot{\textbf{r}}'(t) = \frac{1}{||\dot{\textbf{r}}(t)||} \left(\begin{array}{c} \dot{r_1}(t) \\ \dot{r_2}(t) \end{array}\right)##
and
##ddot{\textbf{r}}'(t) =...
So this might be long question that requires some literature review but I will try condense it as much as possible such that hopefully I can get some help without the reader having to review the related paper.
So I will start off by saying that I am involved in a honours thesis in which I need...
This is a solution to a problem inspired by another thread. It is posted here to separate it from the multiple choice question which was the subject of that thread. A parametric solution for the trajectory can be found quite easily if the motion is modeled as a particle with charge ##q##...
My take;
Part (a);
##\dfrac{dy}{dx}=\dfrac{1}{t}##
therefore,
##y-2at=\dfrac{1}{t}(x-at^2)##
##ty-2at^2=x-at^2##
##ty=x+at^2## implying that ##T## has co-ordinates ##(-at^2,0)##.
##SP=\sqrt{(a-at^2)^2+(0-2at)^2}##
##SP=\sqrt{4a^2t^2-2a^2t^2+a^2t^4+a^2}##
##SP=\sqrt{a^2t^4+2a^2t^2+a^2}##...
My take;
##\dfrac{dy}{dx}=\dfrac{-1}{t^2}⋅\dfrac{1}{2t}=\dfrac{-1}{2t^3}##
The equation of the tangent line AB is given by;
##y-\dfrac{1}{t}=\dfrac{-1}{2t^3}(x-t^2)##
##ty=\dfrac{-1}{2t^2}(x-t^2)+1##
At point A, ##(x,y)=(3t^2,0)##
At point B, ##(x,y)=(0,1.5t)##...
hmmmmm nice one...boggled me a bit; was trying to figure out which trig identity and then alas it clicked :wink:
My take;
##x=(\cos t)^3 ## and ##y=(\sin t)^3##
##\sqrt[3] x=\cos t## and ##\sqrt[3] y=\sin t##
we know that
##\cos^2 t + \sin^2t=1##
therefore we shall have...
For part (a) i have two approaches;
We can have,
##\dfrac{dy}{dx}=\dfrac{dy}{dt}\cdot\dfrac{dt}{dx}##
##\dfrac{dy}{dx}=-\dfrac{2}{x^2}##
##\dfrac{dy}{dx}\left[x=\frac{1}{p}\right]=-2p^2##
Therefore,
##p(y-2p)=-2p^3x+2p^2##
##py=-2p^3x+4p^2##
##y=-2p^2x+4p##The other approach to this is;
since...
I'm looking at the following web page which looks at rendering bezier curves.
GPU Gems 3 - Chapter 25
Paper on same topic
The mathematics is quite interesting, I was interested to know what the F matrix would look like for for a linear bezier equation. The maths for the quadratic case is in...
Let P (1, 2, 3), Q (2, 3, 1), and R (3, 1, 2).
(a) Derive the parametric equations for the line that passes through P and Q without resorting
to the known formula.
(b) Derive the equation of the plane that passes through the points P, Q, and R without
resorting to the known formula.
(c) Find the...
I have found the turning point. I want to ask how to check the nature of the turning point.
My idea is to change the equation into cartesian form then find the second derivative and put the ##x## value of the turning point. If second derivative is positive, then it is minimum and if the second...
My interest on this question is solely on ##10.iii## only... i shared the whole question so as to give some background information.
the solution to ##10.iii## here,
now my question is, what if one would approach the question like this,
##\frac {dy}{dx}=\frac{t^2+2}{t^2-2}##
we know that...
I have no problem in following the literature on this, i find it pretty easy. My concern is on the derived function, i think the textbook is wrong, it ought to be,
##S^{'}(t)##=##\frac {4t} {\sqrt{1+4t^2}}=0## is this correct? if so then i guess i have to look for a different textbook to use...
ok this is pretty straightforward to me, my question is on the order of differentiation, i know that:
##\frac {d^2y}{dx^2}=####\frac {d}{dt}.####\frac {dy}{dx}.####\frac {dt}{dx}##
is it correct to have,
##\frac {d^2y}{dx^2}=####\frac {d}{dt}##.##\frac {dt}{dx}##.##\frac {dy}{dx}##?
that is...
Solution:
The point of tangency of the string moves around the circle at ##2\pi## radians per second. First, we compute the position of the point of tangency of the string with the bobbin. Because this is simply a revolution around a circle of radius 10, the parameterization of the point of...
It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
Since ##h## and ##k## are constants:
$$\frac{h}{k}\cdot \int \frac{1}{y(h-y)} \ dy$$
Now, rewriting the integrating function in terms of coefficients ##A## and ##B##:
$$\frac{1}{y(h-y)}=\frac{A}{y}+\frac{B}{h-y}\rightarrow B=A=\frac{1}{h} \rightarrow$$
$$\frac{1}{h}\int \frac{1}{y}\ dy +...
hi guys
i was trying to solve this differential equation ##\frac{d^{2}y}{dt^{2}}=-a-k*(\frac{dy}{dt})^{3}## in which it describe the motion of a vertical projectile in a cubic resisting medium , i know that this equation is separable in ##\dot{y}## but in order to solve for ##y## it becomes...
Hello,
I have a question about the creation of the Bell's entanglement state ##1/\sqrt{2} (|HH> + |VV>)##using type I BBO crystals (https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion).
Two crystals are put orthogonal to each other and each of them emits a photon pair...
I want to ask about the solution. The solution divides region R into two parts: curved part and triangle. The triangle is obtained by drawing line ##x=5##. Let say line ##x=5## cuts x-axis at point A so the triangle is PAQ
For the curved part:
$$\int_{-1}^{2} (3+3t) ~2t~ dt$$
My question:
Why...
$\tiny{311.1.5.19}$
find the parametric equation of the line through a parallel to b.
$a=\left[\begin{array}{rr}
-2\\0
\end{array}\right],
\, b=\left[\begin{array}{rr}
-5\\3
\end{array}\right]$
ok I know this like a line from 0,0 to -5,3 and $m=dfrac{-5}{3}$
so we could get line eq with point...
For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##.
Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how?
Since...
In the book "The Variational Principles of Mechanics" by Cornelius Lanczos, the following statement is made about a lagrangian ##L_1## where time is given as an dependent parameter, and a new parameter ##\tau## is introduced as the independent variable, see (610.3) and (610.4) pg. 186,187 Dover...
$\tiny{1.5.12}$
Describe all solutions of $Ax=0$ in parametric vector form, where $A$ is row equivalent to the given matrix.
RREF
$A=\left[\begin{array}{rrrrrr}
1&5&2&-6&9& 0\\
0&0&1&-7&4&-8\\
0& 0& 0& 0& 0&1\\
0& 0& 0& 0& 0&0
\end{array}\right]
\sim \left[\begin{array}{rrrrrr}
1&5&0&8&1&0\\...
Describe all solutions of $Ax=b$ in parametric vector form, where $A$ is row equivalent to the given matrix.
$A=\left[\begin{array}{rrrrr}
1&-3&-8&5\\
0&1&2&-4
\end{array}\right]$
RREF
$\begin{bmatrix}1&0&-2&-7\\ 0&1&2&-4\end{bmatrix}$
general equation
$\begin{array}{rrrrr}
x_1& &-2x_3&-7x_4...
Write the solution set of the given homogeneous systems in parametric vector form.
$\begin{array}{rrrr}
-2x_1& +2x_2& +4x_3& =0\\
-4x_1& -4x_2& -8x_3& =0\\
&-3x_2& -3x_3& =0
\end{array}\implies
\left[\begin{array}{rrrr}
x_1\\x_2\\x_3
\end{array}\right]
=\left[\begin{array}{rrrr}...
I'll write my procedure:
$$\lim_{x\to\infty}[\frac{(a-2)x^3+x^2}{ax^2+6x+1}]\rightarrow\frac{x(a-2)}{a}\in \mathbb{R}$$
And now, assumed that everything's correct, how do I assign ##a## a value for having that limit finite and ##\in \mathbb{R}##, and so an horizontal asymptote?
This is the code line that i used to generate the following graphs
ParametricPlot3D[{{1 + Cos[t], Sin[t],
2*Sin[t/2]}, {2 *Cos[t]*Sin[\[Phi]], 2*Sin[t]*Sin[\[Phi]],
2*Cos[\[Phi]]}}, {t, 0, 2 \[Pi]}, {\[Phi], 0, \[Pi]/2},
PlotStyle -> {Directive[Green, Thickness[0.025]], Yellow}...
I'm fascinated by the delayed-choice quantum eraser (DCQE) experiment from Kim et al. 1999.
As I understand from the paper, the observer at the signal beam detector d0 (the screen) never sees an interference pattern, but the "lump" sum of all possible outcomes at the idler photon detectors...
Hi, the above image is from the Line Integral Convolution paper by Cabral and Leedom. However, I am having a hard time implementing it, and I am quite certain I am misreading it. It is supposed to give me the distances of the lines like in the example below, but I am not sure how it can. First...
11.1 Parametric equations and a parameter interval for the motion of a particle in the xy-plane given. Identify the paritcals path by finding a Cartestian equation for it $x=2\cos t, \quad 2 \sin t, \quad \pi\le t \le 2\pi$
(a) Identify the particles path by finding a Cartesian Equation the...
for my formatting, (dot) implies a single time derivative with respect to the variable
Kinetic Energy = T = (1/2) m (x(dot)2 +y(dot)2 + z(dot)2
Plug in respective values for x y and z -> T= (1/2) m (a2 α2sin2(αλ) λ(dot) +a2 α2cos2(αλ) λ(dot) + b2λ(dot)
After canceling out Sin and cos ->...
I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
Homework Statement
Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height.
Homework Equations
I will include some pictures to show what I'm doing
The Attempt at a...
Homework Statement
Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5)
Homework EquationsThe Attempt at a Solution
Finding the scalar equation:
Ax+By+Cz+D=0
3x-4y+6z+D=0
3(9)-4(2)+6(-5)+D=0
-11+D=0
D=11...
Homework Statement
[/B]
Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1).
Homework EquationsThe Attempt at a Solution
First I find the direction vector for PQ.
PQ=Q-P = (2,7,1)-(-3,5,2)
=[2-(-3),7-5,1-2]
=5,2,-1
PQ= (5,2,-1)
Now I...
Homework Statement
A curve is defined by the parametric equations ##x=t^3+1## and ##y=t^2+1##.
Show that ##\frac{\frac{d^2y}{dx^2}}{\left(\frac{dy}{dx}\right)^4}## is a constant.
Homework EquationsThe Attempt at a Solution
So you differentiate both equations wrt ##t## then apply the chain rule...
Homework Statement
Compute the flux of a vector field ##\vec{v}## through the unit sphere, where
$$ \vec{v} = 3xy i + x z^2 j + y^3 k $$
Homework Equations
Gauss Law:
$$ \int (\nabla \cdot \vec{B}) dV = \int \vec{B} \cdot d\vec{a}$$
The Attempt at a Solution
Ok so after applying Gauss Law...
Homework Statement
The degenerate parametric amplifier is described by the Hamiltonian:
$$H=\hbar \omega a^\dagger a-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)$$
Where ##a## and ##a^\dagger## as just the operators of creation and anhiquilation and ##\chi## is just a...
Hi PF!
Given a vector field ##\vec f## in spherical coordinates as a function of a single parameter ##s##, shown here as
$$\vec f(s) = f_r(s) \hat r + f_\theta(s) \hat \theta + f_\phi(s) \hat\phi$$
where here subscripts do not denote partial derivatives, but instead are used to define...
Suppose a baseball is hit 3 feet above the ground, and that it leaves the bat at a speed of 100 miles an hour at an angle of 20° from the horizontal.
I've got the parametric equations in terms of x and in terms of y, and I have values plotted and a graph sketched. My question is in regards to...
Hi,
The main question revolves around the Rhodonea curve AKA rose curve. The polar equation given for the curve is r=cos(k). The parametric equation is = cos(k(theta)) cos (theta), = cos(k(theta)) sin(theta) . Can anyone show me the conversion from the general parametric form to the general...