# What is Parametrization: Definition and 89 Discussions

In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters".Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.
For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed origin. If x, y, z are the coordinates of the point, the movement is thus described by a parametric equation

x

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y

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z

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{\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t)\\z&=h(t),\end{aligned}}}
where t is the parameter and denotes the time. Such a parametric equation completely determines the curve, without the need of any interpretation of t as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters t and u.

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1. ### I Example of SO(2) being not simply connected

The example goes like this: The group SO(2) is specified by angles ##\theta##. Let's parametrize a path by ##0 \leq t \lt 1## and consider the path ##\theta (t) = 2 \pi t##. Then it says, "There is no smooth function ##\theta (t,u)## for ##0 \leq u \leq 1##, such that ##\theta (t,0) = \theta...
2. ### A Could a CPL parametrization be included into Brans-Dicke model?

I have studied up to now about forecasts to constrain cosmological parameters in the context of CPL( Chevallier-Polarski-Linder ) parametrization with w_0, ,w_a parameters in equation of state for cosmic fluid. For this, I have used Matter power spectra ("fake data") generated by CAMB and CLASS...
3. ### A Feynman parametrization integration by parts

How can i move from this expression: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(k-k_{f}))^3} \frac{1}{(1+i(k-k_{i}))^3}$$ to this one: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+|k-k_{i}|^2)^2} \frac{1}{(1+|k-k_{f}|^2)^2}$$ using Feynman parametrization (Integration by...
4. ### I When to use Feynman or Schwinger Parametrization

I had been doing some calculations involving propagators with both a quadratic and a linear power of loop momentum in the denominator. In the context of HQET and QCD with strategy of regions. The texts which I am following sometimes tend to straightaway use Schwinger and I am just wondering if...
5. ### I Geodesics with arbitrary parametrization

Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not? My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)##...
6. ### One-parameter parametrization of a unit circle in R^n

I tried to looking at lower-dimensional cases: For ##n=2## we have $$(x(t),y(t))=(cos(t),sin(t))$$ For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to $$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$ It seems...
7. ### I Understanding Geodesic Parametrization on a Sphere

Let us consider a sphere of a unit radius . Therefore, by choosing the canonical spherical coordinates ##\theta## and ##\phi## we have, for the differential length element: $$dl = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2}$$ In order to find the geodesic we need to extremize the...
8. ### A Natural parametrization of a curve

Hello, I need the natural parametrization or a geodesic curve contained in the surface z=x^2+y^2, that goes through the origin, with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with "a" constant, expressed as a function of the arc length, i.e., I need r(s)=r(x(s),y(s)). Thank...
9. ### Calculating crossproduct integral, Parametrization

i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right? The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...
10. ### I Purpose of parametrization

To describe the equation of a line, in 2 dimensions, we need a (point on the line + slope to measure slantiness) or two points. Another way: The trajectory of a moving point along the line. Suppose that the moving point initially is at a point of know coordinates r0=(x(t=0), y(t=0), z(t=0)) and...
11. ### I Triad T, N, B and path parametrization

Hello, In 3D, the trajectory, which is a curve, represents all the points that an object occupies during its motion. Given a certain basis (Cartesian, cylindrical, spherical, etc.), the instantaneous position of the moving object, relative to the origin, along its trajectory can always be...
12. ### I Parametrization manifold of SL(2,R)

I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...
13. ### B Parametrization of the 3-sphere

The equation is $$\|\left(\begin{array} &a\\b\\c\\d\end{array}\right)\|^2=1$$ I was wondering if the number of parameters is 6 and not 3, since we can consider rotations in the differents planes : we choose 2 directions among 4 hence $$C^4_2=6$$ possibilities ?
14. ### MHB Which is the parametrization σ?

Hey! :o I want to show that $\iint_{\Sigma}(\nabla\times f)\cdot d\Sigma=0$ for the function $f(x,y,z)=(1,1,1)\times g(x,y,z)$ when $\Sigma$ is the surfcae that is defined by the relations $x^2+y^2+z^2=1$ and $x+y+z\geq 1$. I have done the following: Let $g(x,y,z)=(g_1, g_2, g_3)$. Then...
15. ### Parametrize the Curve of Intersection

Hi everyone! I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a...
16. ### MHB Parametrization Of Complex Curves .... Mathews And Howell, Example 1.22 .... ....

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 1.6 The Topology of Complex Numbers ... I need help in fully understanding a remark by M&H ... made just after Example 1.22 ... Example...
17. ### I Surface parametrization and its differential

I will use an example: -The surface is given by the intersection of the plane: y+z=2 -And the infinite cilinder: x2+y2<=1 We want to parametrize this surface, it could be done easily with: x=r cosθ y=r sin θ z=2 - r cos θ Then this surface could be written using vector notation: S= r...
18. ### I Contour integration - reversing orientation

I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...
19. ### I Parametrization and kinematics

Hello Forum, In kinematics, the important variables are the velocity v, the acceleration a, and the object's position x. These variables are usually presented as functions of time: x(t), v(t) and a(t). The acceleration can either be constant, or vary with time, i.e. a(t), or vary with position...
20. ### Parametrization for Surface F and Area A

An area A in the (x,y) plane is limited by the y-axis and a parabola with the equation x=6-y^2. Further, is a surface F given by the part of the graph for the function h(x,y)=6-x-y^2 which satisfies the conditions x>=0 and z>=0. Determine a parametrization for A and for F. So far I've got the...
21. ### I Parameterize an offset ellipse and calculate the surface area

I'm given that: S is the surface z =√(x² + y²) and (x − 2)² + 4y² ≤ 1 I tried parametrizing it using polar coordinates setting x = 2 + rcos(θ) y = 2rsin(θ) 0≤θ≤2π, 0≤r≤1 But I'm not getting the ellipse that the original equation for the domain describes So far I've tried dividing everything...
22. ### MHB Find a parametrization of the following level curves

Hello! (Wave) Is $r(t)=(t^2,t^4)$ a parametrization of the parabola $y=x^2$? I have written the following: For $x(t)=t^2$ and $y(t)=t^4$ we have that $y(t)=t^4=(t^2)^2=x^2(t)$, so $r(t)=(t^2,t^4)$ a parametrization of the parabola $y=x^2$. Is it right? How could we say it more formally...
23. ### Parametrization of implicit curve

Homework Statement y^2 + 3x - x^3 = C, C\in\mathbb{R}\setminus\{0\} Homework EquationsThe Attempt at a Solution Keeping in mind that ##\cos ^2\alpha + \sin ^2\alpha = 1## I would go about it \left (\frac{y}{\sqrt{C}}\right )^2 + \left (\frac{\sqrt{3x-x^3}}{\sqrt{C}}\right )^2 = 1 would then...
24. ### Curve integral, singularity, and parametrization

Well, it's physics friday! (carpe diem etc, what else) :) 1. Homework Statement I present to you this (not so) pleasant expression that seemingly appeared on a page out of nowhere. \vec{F}(r, \theta, \varphi) = \frac{F_0}{ar \sin\theta}[(a^2 + ar \sin\theta \cos\varphi)(\sin\theta \hat{r} +...
25. ### Torus parametrization and inverse

I've been looking at the torus parametrization $$\phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u)$$ with $$a>0, r\in(0,a)$$. I want to invert this map to get a chart map for the torus. Can anyone give me a hand with this? Thanks!
26. ### Parametrization of a curve(the intersection of two surfaces)

Homework Statement I am looking to find the parametrization of the curve found by the intersection of two surfaces. The surfaces are defined by the following equations: z=x^2-y^2 and z=x^2+xy-1 Homework EquationsThe Attempt at a Solution I can't seem to separate the variables well...
27. ### Relativity Can't remember where I read this (when using the proper-time parametrization)

A while ago, I read a proof in a book on GR that when using the proper-time parametrization, the two conditions ## \delta \int_{\lambda_1}^{\lambda_2} \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} d\lambda=0## and ## \delta \int_{\lambda_1}^{\lambda_2} (-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu)...
28. ### Is this a valid parametrization of the torus?

I am reading "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and on page 156 he gives the following parameterization of the torus x(u,v) = ((a + r cos u )cos v, (a + r cos u)sin v, r sin u) 0 < u < 2*pi, 0 < v < 2*pi Doesnt this leave out some of the torus,? I know that...
29. ### Parametrization of Witch of Agnesi

Homework Statement The question is completely described in the photo. Homework Equations Trigonometric translation properties The Attempt at a Solution The problem is in two dimensions, so I'm ignoring the z coordinates. For a circle centered at (0,a), the position vector of P is ##(a##...
30. ### Can the Thickness of a 3D Spiral Curve be Defined Parametrically?

Hello! There is a parametric way of defining a spiral curve: z = a*t; x = r1*cos(w*t) y = r2*sin(w*t). Is there a way to define the thickness of spiral?
31. ### Poincare Transformations: Parametrization-Independent

Well if I have a worldline given by x^{\mu}(\tau) And I want to make a Poincare transformation: x^{\mu} (\tau) \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}(\tau) + a^{\mu}. I have one question,why can't a, \Lambda explicitly depend on \tau? that is to have: x^{\mu}(\tau) \rightarrow...
32. ### Find the arc length parametrization of a curve

Homework Statement Find the arc length parametrization of the curve r = (3t cost, 3tsint, 2sqrt(2)t^(3/2) ) . Homework Equations s(t)=integral of |r'(t)| dt The Attempt at a Solution I was able to get the integral of the magnitude of the velocity vector to simplify to: s(t) = integral of...
33. ### MHB Finding the parametrization of the curve

A particle moves along the curve 9x^2 + 16y^2 = 144 a)Find a parametrization of the curve which corresponds to the particle making one trip around the curve in a clockwise direction starting at (4,0) so I know that cos^2t + sin^2t = 1 which is a circle. I also know that x^2 + y^2 = 1 is a...
34. ### Integral after alpha parametrization

I have been stuck on the following double integral for some time: ∫(0 to inf) dα1 ∫(0 to inf) dα2 a1^(n1) * exp(-i (α1+α2) m^2) * (α1+α2)^(n2) which arose after using alpha paremetrization on a Feynman integral. I was advised by my supervisor to use the substitution α1 = 1/2 (t+u) and α2 = 1/2...
35. ### MHB Parametrization of a Reduced Matrix

I'm facing some doubts regarding the parametrization of a given matrix. Let's say, the following matrix is reduced. From: $\begin{bmatrix}0 & 2 & -8\\0 & 2 & 0\\0 & 0 & 2\end{bmatrix}$ To: $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$ To Parametrize that I would do the...
36. ### Affine parametrization for null geodesic?

The geodesic equation for a path X^\mu(s) is: \frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0 where U^\mu = \frac{d}{ds} X^\mu But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter s must be...
37. ### Finding the Arc Length Parameterization of a Vector Function

Homework Statement Find the arc length parameterization of r(t) = <(e^t)sin(t),(e^t)cos(t),10e^t>The Attempt at a Solution so I guess i'll start by taking the derivative of r(t)... r'(t) = <e^t*cos(t) + e^t*sin(t), -e^t*sin(t) + e^t*cos(t), 10e^t> ehh... now do I do ds = |r'(t)|dt and...
38. ### Find appropriate parametrization to find area bounded by a curve

Problem: Use an appropraite parametrization x=f(r,\theta), y=g(r,\theta) and the corresponding Jacobian such that dx \ dy \ =|J| dr \ d\theta to find the area bounded by the curve x^{2/5}+y^{2/5}=a^{2/5} Attempt at a Solution: I'm not really sure how to find the parametrization. Once I...
39. ### Parametrization of uniformly distributed n dimensional states

Any two dimensional state can be written as: |\phi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle where 0\leq\theta\leq\pi and 0\leq\phi\leq 2\pi, and 0\leq\theta\leq\pi. To pick one such state uniformly at random it suffices to draw \phi at random from its...
40. ### Surface integrals and parametrization

An area A in the xy-plane is defined by the y-axis and by the parabola with the equation x=6-y^2. Furthermore a surface S is given by that part of the graph for the function h(x,y)=6-x-y^2 that satisfies x>=0 and z>=0. I have to parametrisize A and S. Could this be a...
41. ### Parametrization of a Portion of a Sphere

Homework Statement Let ##S## be the portion of the sphere ##x^2 + y^2 + z^2 = 9##, where ##1 \le x^2 + y^2 \le 4## and ##z \ge 0##. Give a parametrization of ##S## using polar coordinates. Homework Equations The Attempt at a Solution ##1 \le r \le 2 \\ 0 \le \theta \le 2\pi \\ 0...
42. ### Parametrization of plane curves

1. The function f(x,y) = x + y 2. The area A is formed by the lines : x = 0 and x = pi/4 and by the graphs : x + cos(x) and x + sin(x) 3. I have to parametrize A 4. 'Formula' : r(u,v) = (u,v*f(u)+(1-v)*g(u)) Could this be a parameterization of A : assuming f(u) = u+ cos(u)...
43. ### Solve SU(2) Parametrization Problem w/ Westra's PDF

I have a trivial mathematical problem with SU(2) parametrization. In www.mat.univie.ac.at/~westra/so3su2.pdf , section 3, there is a sentence starting with "We first assume b = 0 and find then(...)". My question is: doesn't assuming that b = 0 reduce generality of our parametrization? If not, why?
44. ### Greens theorem and parametrization

Hello. I just wonder if anybody know if there are any rules, when to use parametrization to greens theorem in a vector line integral over a plane. Becouse, it seems sometimes, you have to parametrizice, and other places you dont. I get confused.
45. ### Parametrization of a regular planar polygon with an arbitrary number of sides

I was wondering if anyone knew of a common technique for parametrizing a regular polygon with an arbitrary number of sides. I figured such a problem would be easy or at least be well documented online, but that doesn't seem to be the case. I started by assuming that the polygon was centered...
46. ### Find a parametrization of the vertical line passing through the point

Homework Statement Find a parametrization of the vertical line passing through the point (7,-4,2) and use z=t as a parameter. Homework Equations r(t) = (a,b,c) + t<x,y,z> The Attempt at a Solution I used (7,-4,2) as (a,b,c) (the point) and used <0,0,1> for the vector since it had...
47. ### Question about parametrization and number of free variables

Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of...
48. ### Parametrization of a line formed by 3 points

Homework Statement Find a parametrization of the equation of the line formed by the points A, B, and P. A(2,-1,3) B(4,3,1) P(3,1,2)Homework Equations x=x_0+v_1*t y=y_0+v_2*t z=z_0+v_3*tThe Attempt at a Solution Alright, so, I've already determined that P is equidistant from the points A and...
49. ### Parametrization question for my Intro. to Higher Math Class

Two objects A and B are traveling in opposite direction on a straight line. At t=0 A and B are at positions P(A)=(-40, -20) and P(B)=(190, 980), respectively. If additionally, their paths are parameterized by directions V(A)=(3,5) and V(B)=(-24, -40), respectively. Then, a) find the point...
50. ### Rotation parametrization of alignment of two vectors

Hello, I'm looking for an appropriate rotation representation for the following situation. I have two (always non-zero) vectors, v1, v2, that may or may not be parallel. The rotation relating the two vectors is obviously non-unique having one degree of freedom, parametrized by p. So my...