# What is Parametric form: Definition and 23 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. ### I Surfaces wrapped around a cylinder for 3d printing

Hi All, I am looking to determine how these Vases where modeled using maths on this webpage https://www.3dforprint.com/3dmodel/sine-wave-vase-generator/2116. It looks like the surface is parametrically defined and wrapped around a cylinder. Interestingly he mentions "Sine waves combine to...
2. ### 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...
3. ### I Area ellipse: parametric form, angles and coincidences

Dear all, I have a question regarding the computation of the area of an ellipse. The parametric form of the ellipse with axes a and b is $$x(t) = a\cos{(t)}, \ \ \ y(t) = b\sin{(t)}$$ Using this to evaluate the area of the ellipse, usually one takes one halve or one quarter of the ellipse...
4. ### I Is (u,v) = (x square - x, x+1) a Parametric Form of a Parabola?

Hello. How can I verify that (u,v) = (x square - x, x+1) is a parametric form of a parabola? Thank you!
5. ### MHB Vector or Parametric Form of the Equation of a Plane P

I am reading David Poole's book: Linear Algebra: A Modern Introduction (Third Edition) ... I have a basic (and probably simple) question regarding Poole's introductory discussion of the vector or parametric form of the equation of a plane \mathscr{P} (page 38, Section 1.3 Lines and Planes) ...
6. ### Describing elliptic orbit as a parametric function

Hi PF I've beent rying to model the lunar orbit around the sun (cardioide) as a parametric function, but have run into a problem. f(t) = r(t) : x = a cos(ωt) y = b sin(ωt) z = k t The angular frequency ω as well as the distance from to the center varies around the orbit. Is...
7. ### Finding normal and tangential acceleration

Homework Statement A point moves along the curve y = x3 + x such that the vertical component of velocity is always 3. Find the tangential and normal components of acceleration at the point P(2,10). Homework Equations Tangential Acceleration - aT(t) = v(t) ⋅ a(t)/ magnitude of velocity vector...
8. ### Finding a parametric form and calculating line integrals.

Homework Statement Let C be the straight line from the point r =^i to the point r = 2j - k Find a parametric form for C. And calculate the line integrals ∫cV*dr and ∫c*v x dr where v = xi-yk. and is a vector field Homework EquationsThe Attempt at a Solution For parametric form (1-t)i + (2*t)j...
9. ### Finding the distance between two parametric lines

1. Write down the equation for the line in 3D through the point a=(1,2,4), parallel to the line r=(1,-5,0)+λ(1,2,2). Then, find the distance between these lines. 2. 3. Lets say, b= (1,2,2). b is parallel to given line, so it must also be parallel to the new line. My guess is that the equation...
10. ### Explicit form to parametric form

Hi there, I have the plane in an explicit form: 12x + 2y – 20z = -56 How do I make it in parametric form? Thanks in advance
11. ### How Do You Convert Equations Into Vector and Parametric Forms?

Homework Statement I have three questions regarding vectors in parametric/circle form. I understand that there is a starting point and a direction vector, but I just can't seem to get my head around this :confused: Homework Equations 1. Rewrite y=3x-1 in vector form. 2. Rewrite...
12. ### Finding a point where a line in parametric form meets a plane

Homework Statement Here is the problem: x=y-1=2z and the equation of the plane is 4x-y+3z=8 Homework Equations The Attempt at a Solution Ya so i got the normal line to be <1,1,-1/2> but i do not know where to go from here? help please?
13. ### Finding the parametric form of a tangent line vectors

Homework Statement Find the parametric form for the tangent line to the graph of y=2x2−5x+3 at x=2 is Homework Equations I have no clue! The Attempt at a Solution I found the tangent line to be y=3x-5 I know that the answer has to be in the form... <x0,y0>+t<x1-x0,y1-y0> I...
14. ### Equation of a plane given point and line in parametric form

Homework Statement Find the equation of the plane that contains P=(-1,0,1) and r(t)=<3t,t,8> Homework Equations The Attempt at a Solution n * <r-r0>=0 n * <t+2, 2t, 3t> = 0 I distributed the n, adding the terms and obtained: 1/t = -2n(n+2n+3n) Clearly, I've done...
15. ### What is the Limit of (a^t + b^t)/2^(1/t) as t approaches infinity?

Homework Statement Calculate lim{t-> + \infty} ( \frac{a^t + b^t)}{2}) ^ {1/t} Homework Equations The Attempt at a Solution lim _{t-> + \infty} ( \frac{a^t + b^t)}{2}) ^ {1/t} = lim_{t-> + \infty} ( a^t ( \frac{1 + b^t/a^t)}{2}) ^ {1/t} = lim_{t-> + \infty} ( a ( \frac{1 +...
16. ### Linear Algebra (Parametric Form)

Homework Statement L1 : x = (0, 1, 2) + s(1, 0, 2) L2 : x = (4, 2, c) + t(−2, 0, d) If c = 5 & d = 0, find the point P on L1 and Q on L2 so that the distance between P & Q is the smallest possible. Homework Equations the point of intersection? The Attempt at a Solution...
17. ### Second derivative from parametric form

If x and y are defined in terms of a third vatiable say t , then to find d2y/dx2 , we cannot find d2y/dt2 and d2x/dt2 and divide them to get d2y/dx2 , i am unable to fingure out the reason for this !
18. ### How Does Parametric Form Differ from Standard Graph Representation?

How is the parametric form of the graph of an equation different from its standard graph and what the dots in the parametric form of a graph represent.
19. J

### What are the Eigenvalues and Eigenvectors of a Matrix?

Homework Statement Let: a matrix be: -5 -0.5 -0 -8 Find an invertible P and a diagonal D such that PDP(inverse) Homework Equations DET( (I)Lamda-A))= 0 for Eigenvalues The Attempt at a Solution when y=0 at the end matrix for finding the...
20. ### Converting from Cartesian to Parametric form

[SOLVED] Converting from Cartesian to Parametric form Homework Statement Find a parametric vector equation of for the plane in R^3 having cartesian equation 4y + 5z = - 6 Homework Equations None The Attempt at a Solution What I did was, first I turned the equation into 4x +...
21. ### How Do You Convert Parametric Equations to Algebraic Form?

X1 T = 10T Y1 T = 100 + (.5 * -9.8T^2) X2 T = 100 - 12.3 T X2 T = 0 How do I put this into algebraic form? it seems easy but I just can't get it. Do you simply add the X and Y components? If so what do x and y each stand for?? Does it have something to do with sine and cosine? =/
22. ### Changing from parametric form to algebraic form

X1 T = 10T Y1 T = 100 + (.5 * -9.8T^2) X2 T = 100 - 12.3 T Y2 T = 0 How do I put this into algebraic form? it seems easy but I just can't get it.
23. ### Reduced a big matrix, now the parametric form is not right, :\

Hello everyone I did the following problem: Click http://img220.imageshack.us/img220/8486/untitled1copy4oq.jpg to view the problem and my answer. The row reduced form is: 1 5 0 0 -7 6 -7 0 0 1 0 -1 1 -1 0 0 0 1 - 2 -4 8 Any help would be great