# 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

x

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f
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t
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y

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g
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t
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z

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h
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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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4. ### Parameterize Radial Vector of Electric Field due to Spherical Shell

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5. ### I Solution for 1st order, homogenous PDE

##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...
6. M

### I Parameterize a circle based on the contact angle with a wedge

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...
7. ### B Geodesic dome parametric formula

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8. ### Finding the parametric equation of a curve

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9. ### Calculating Line Integral in xy-Plane

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...
10. ### I Derivative and Parameterisation of a Contour Integral

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11. M

### Question about finding area using Green's Theorem

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 [/B] The Attempt at a Solution Using the parameterized...
12. ### How do I parameterize these surfaces?

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...
13. ### How to parameterize these surfaces?

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14. ### I Parameterize an offset ellipse and calculate the surface area

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15. ### How do I parameterize the intersection of these two surfaces?

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.
16. ### Parameterize a union of circles

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...
17. ### Parameterize a geodesic using one of the coordinates

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...
18. ### How to parameterize solid of revolution?

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...
19. ### Parameterize the intersection of the surfaces

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
20. ### Parameterize part of a Parabola

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...
21. ### Method to parameterize circles in R3 laying in a plane

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22. ### How to Parameterize an Ellipse with Offsets?

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.
23. ### Parameterize the curve of intersection

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...
24. ### Parameterize the circle x^2 + y^2 = r^2

parameterize the circle x^2 + y^2 = r^2 anybody pls help thanx