What is Parameterize: Definition and 24 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
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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.
Here it is the image of the statement:
As I mentioned in the "relevant equations" section, my approach to solving this exercise involves calculating the difference between the centers of mass of the square and the triangle.
Starting with calculation of center of mass for the square.
Starting...
Homework statement:
Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ.
Relevant Equations: Gauss' Law
$$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$
My Attempt:
By using the spherical symmetry, it is fairly obvious...
##u_t + t \cdot u_x = 0##
The equation can be written as ##<1, t, 0> \cdot <d_t, d_x, -1>## where the second vector represents the perpendicular vector to the surface and since the dot product is zero, the first vector must necessarily represent the tangent to the surface. We parameterize this...
Hi PF!
Given a 2D plane, the following is a parameterization of a circular arc with contact angle ##\alpha## to the x-axis: $$\left\langle \frac{\sin s}{\sin\alpha},\frac{\cos s - \cos\alpha}{\sin\alpha} \right\rangle : s \in [-\alpha,\alpha]$$
However, I am trying to parameterize a circle...
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
Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j.
Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution
I have to write something...
As part of the work I'm doing, I'm evaluating a contour integral:
$$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$
along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
Homework Statement
Use Green's Theorem to find the area of the region between the x-axis and the curve parameterized by r(t)=<t-sin(t), 1-cos(t)>, 0 <= t <= 2pi
Attached is a figure pertaining to the question
Homework Equations
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The Attempt at a Solution
Using the parameterized...
Homework Statement
Parameterize ##S={ S }_{ 1 }\bigcup { { S }_{ 2 } } ##, where ##S_1## is the surface with equation ##x^2+y^2=4## bounded above by the graph of ##2y+z=6## and below by the ##xy## plane. ##S_2## is the bottom disk
Homework EquationsThe Attempt at a Solution
##{ S }_{ 1...
Homework Statement
Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0##
Homework Equations
Parameterizations:
##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le...
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...
Homework Statement
Parameterize the curve of intersection of the two surfaces:
x^2+y^2+z^2=14
z=y^2-x^2
Homework EquationsThe Attempt at a Solution
I tried manipulating the equations above but can't seem to get a nice parameterization which I can use to do the rest of the (calculus) problem.
Homework Statement
Let C=\lbrace(x,y) \in R^2: x^2+y^2=1 \rbrace \cup \lbrace (x,y) \in R^2: (x-1)^2+y^2=1 \rbrace . Give a parameterization of the curve C.
The Attempt at a Solution
I'm not sure how valid it is but I tried to use a 'piecewise parameterisation', defining it to be...
I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so...
Homework Statement
Here is the surface I need to parameterize. It is a solid of revolution.
Homework Equations
The Attempt at a Solution
So since its a piecewise function, I can define it as follows
(x-2)^2 + z^2 = 1, 1<x<2
z = -x+3, 2<x<3
z = x-3, 2<x<3
I know...
Parameterize the intersection of the surfaces z=x^2-y^2 and z=x^2+xy-1
What's getting me stuck on this problem is the xy. I set x=t
z=x^2-y^2
z=t^2-y^2
z=x^2+xy-1
t^2-y^2=t^2+ty-1
y^2=1-ty
Thats as far as of come, I'm stuck on this
Homework Statement
Find a parametric equation for a part of a parabola.
Given:
y=-2x2
initial point: (-2,-8)
terminal point: (1,-2)
Homework Equations
x(t)=a+t(c-a)
y(t)=b+t(d-b)
The Attempt at a Solution
x(t)=-2+t(1-(-2))
=3t-2
y(t)=-8+t(-2-(-8))
=6t-8...
Homework Statement
In general how do i parametrize a circle of radius r at centre (a,b,c) laying on a plane? E.g. (x + y + z = 6)
Homework Equations
The Attempt at a Solution
How do I parameterize the following?
x^{2}/a^{2} + y^{2}/b^{2} -2x/a -2y/b = 0
I tried letting x =t or some other parameters but found it difficult to solve for y.
Homework Statement
Parameterize the curve of intersection of the cylinder x^2 + y^2 = 16 and the plane x + z = 5
Homework Equations
The Attempt at a Solution
i think i must first parameterize the plane
x = 5t, y = 0, z = -5t
then i think i plug those into the eq. of the...