# 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.

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1. ### Parametric curve question (determining unknown point)

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! :)
2. ### Inner product between velocity and acceleration is zero (parametric)

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) =...
3. ### Help with understanding of RF theory-Kinetic inductance parametric amp

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...
4. ### Finding a Parametric Solution for Particle Trajectory in Magnetic Field

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##...
5. ### Solve this problem that involves parametric equations

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}##...
6. ### Solve the given problem involving parametric equations

My take; ##y=\dfrac{c^2}{x}## ##y+x\dfrac{dy}{dx}=0## ##\dfrac{dy}{dx}=\dfrac{-y}{x}## ##y-\dfrac{c}{t}=-\dfrac{y}{x}(x-ct)## ##yt-c=-\dfrac{yt}{x}(x-ct)## ##xyt-cx=-yt(x-ct)## ##c^2t-cx=-cx+yct^2## ##c^2t-cx=-cx+ytct## ##c^2t-cx=-cx+c^2t## ##⇒-cx=-cx## ##⇒cx=cx## Therefore it...
7. ### Prove that PA=2BP in the problem involving parametric equations

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)##...
8. ### Find the Cartesian equation given the parametric equations

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...
9. ### Solve the given parametric equation

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...
10. ### I Linear Bezier curve projection

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...
11. ### MHB -12.5.2 Find Parametric eq for line segment from (-2,18,31) to (11,-4,48)

Find Parametric eq for line segment from (-2,18,31) to (11,-4,48) ok not sure how to start on this the book example is in the spoiler
12. ### MHB Parametric Eqs: Find Line & Plane, Find Triangle Area

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...
13. ### Checking nature of turning point of parametric equation

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...
14. ### Find the Cartesian equation of a curve given the parametric equation

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...
15. ### Finding the relative extrema for a speed function using parametric curves

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...
16. ### Finding the second derivative of a given parametric equation

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...
17. ### How Does the Point of Tangency Move in Circular Motion?

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...
18. ### Finding the convergence of a parametric series

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...

49. ### B Parametric Equations- Ball travel

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...
50. ### B Conversion of parametric form to polar for the rose curve

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...