Parametrization of lines, surfaces and solids

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Surface Plot of $z=x^2-y^2$
Fields: Geometry, Calculus, and Algebra
Image Created By: Matlab, Graphing Calculator

Surface Plot of $z=x^2-y^2$

This is the Surface Plot of $z=x^2-y^2$. However, this is not a parameterization. Instead, this is the explicit representation of the surface, that is expressing one in terms of two other variables. A loose parameterization of the surface will be $(x,y,z)=(x,y,x^2-y^2)$. This is however is not a parameterization in the strictest sense. A parameter is a variable that does not appear in the new expression. Therefore, a few of the more appropriate ones will be $(x,y,z) = (u,v,u^2-v^2)$ or $(\frac{pq}{2},\frac{p-q}{2}, pq)$ or $(\sin t\cos s, \cos s, \cos^2 t \cos^2 s)$.

Basic Description

Generally, a parameterization of something is simply the description of that something. For example, if you want to describe Barack Obama, you must decide which aspect (the parameter) you want to use. Say we want to use the parameter of occupation, then he is the $44^{th}$ President of the United States and that is one description (parameterization) of him. Or, we want to use the parameter of gender, then he is a male and that is another description (parameterization) of him. However, by definition, parameterization has to be a complete and relevant specification of the something that is being parameterized. Therefore, the former of the parameterizations fits the definition as the ordinal number 44 specifies Barack Obama exclusively but the latter does not fit as gender does not fit the definition.

In mathematics, parameterization is the specification of an object such as curve, surface, etc., by means of one or more variables which are allowed to take on values in a given specified range as shown in the previous example by expressing x, y and z in term of p and q, or s and t, or u and t which are the parameters. A parameter does not appear in the picture, that is, it is not being graphed. In other words, it is a behind-the-scene variable that describes the variable that is being graphed. In addition, a parameterization has to completely and exlusively specify the object.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Vector and Geometry

Lines and Curves

Motion with constant velocity
[[Image:Lineaccel [...]

Lines and Curves

Motion with constant velocity
Accelerated motion

More specifically in the case of lines, curves and planes, a parameterization is a description of a set of points in terms of another variable.

For instance, imagine that you are a particle in a 3 dimensional space, then your coordinates,$(x,y,z)$ are described by a set of numbers such as$(1,2,3)$ with respect to the origin and you could move freely in any direction in any manner of your choosing. Now, suppose you are at the origin and decide to move in the positive $x$-direction only. As you move, your $x$ coordinate will change as your position changes.

However, the manner in which you are moving can vary. For example, you could be moving with a constant velocity, $v$ and your position will be $(vt,0,0)$ with $t$ stands for the time elapsed as shown by the animation on the left. Or, you could be moving with constant acceleration, $a$ and an initial velocity 0 feet per sec, and therefore, your position will be $(\frac{1}{2}at^2,0,0)$ as show by the animation on the right. The particle on the left moves with constant velocity while the one on the right moves faster as times goes on. You would notice that however the particle moves, it is forever on the x-axis and therefore, both $(vt,0,0)$ and $(\frac{1}{2}at^2,0,0)$ are both parameterizations of a segment of the positive x-axis, in both cases with the parameter being time $t$ with $t\ge 0$. As illustrated here, time($t$)is the parameter that describes how your position($x$)changes.

Let $\vec r$ be the point on a line $l$, and let $\vec a$ be any point on the line and $\vec d$ be the directional vector on the line. Then

$l: \vec r = \vec a + k \vec d \qquad k\in \mathbb{R}$

In the diagram to the left, the purple arrow is $\vec r$, the yellow arrow is $\vec a$ and the red arrow is $\vec d$.

Polar Coordinates

Any points in $\mathbb{R^2}$ can be expressed in terms of $(r,\theta)$. For example, Circles can be parameterized by Polar Coordinates (x,y)=(r\cos theta,r\sin theta). It is important to realize that path and curve are different things. Whenever you are thinking about a path, it is always helpful to imagine yourself as a particle and visualize how it will move under the constrains given by the equations. The trail it leaves behind is the path and hence, one parameterization of the curve which is the set of the points. It is worth noticing that for a certain curve, say the line x-axis, there are many parameterizations of it as shown in the very first example. Similarly for the plane, there are infinitely many parameterizations. Hence, there is no unique way of parameterization. Another example that illustrates this point is the circle lying on x-y plane. The curve is $x^2+y^2=r^2$ where r is the radius of the circle. But there are many ways to parameterize it. One is $[cos(\theta),sin(\theta),0] \qquad 0<\theta<\pi$ . Another is $[-cos(\theta),sin(\theta),0] \qquad 0<\theta<\pi$. The difference is that the first path is counterclockwise while the latter is clockwise, and that the two parametrizations start at different points.

Counter-clockwise
Clockwise

Similarly, if you decide to go in a circle in x-y plane (the horizontal plane) and at the same time, move in the z direction with a constant velocity, you will trace out a spiral parameterized by time $(x,y,z)=(\cos t, \sin t, v)$ where $v$ is the constant velocity. To extend this analogy further,you could decide to go in all directions and your displacement $\vec r=[f(t), g(t), h(t)]$ with all components as functions of time. Using calculus, your velocity vector $\vec v=\vec r\ '=[f'(t), g'(t), h'(t)]$ and acceleration vector $\vec a=\vec v\ '=[f''(t), g''(t), h''(t)]$. However, this is not within the scope of this page and please refer to future pages on multivariable calculus.

Surfaces

Cylindrical Coordinates

A cylindrical surface with radius 1 and heigh 1 is probably the simplest volume to start with. Using cylindrical coordinates, then the parameterization is $[\cos\theta, \sin\theta, z]$$0<\theta<2\pi, 0.(Smaurer1 No, this is not cylindrical coordinates; it is the translation of cylindrical coordinates $(r,\theta,z)$ into rectilinear coordinates! You make the same mistake with spherical coordinates later. In any event, the meaning of different coordinate systems would make a great topic for this wiki, with excellent visuals, but it would need to be much expanded to make sense to the reader.).To imagine what it looks like, it is useful to imagine cutting the cylinder at random z values and the cross section will be a unit circle. Stacking them all together will give you a cylindrical surface.

Spherical Coordinates

A unit spherical surface is also very familiar to us. Using spherical coordinates, the parameterization is$[\sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta),\cos(\phi)]$$0<\phi<\frac{\pi}{2}, 0<\theta<2\pi$. Using the same technique of cutting the sphere at different height, we could imagine that at each level is a circle, but difference is that they have different radius at each level.

Plane

In addition, if you decide to fix you vertical position (your z coordinate) and move anywhere you want in the horizontal x-y plane then you will trace out the xy plane. However, that plane could be translated and rotated by any angle in any direction to generate a general plane and it could look like the graph to the left.

Let $\vec r$ be the point on a plane $\pi$, and let $\vec a$ be any point on the line. In addition, let $\vec u$ and $\vec v$ be two vectors lying on the plane. Then

$\pi: \vec r = \vec a + s \vec u + t \vec v \qquad s,t\in \mathbb{R}$

In the diagram to the right, the blue arrow is $\vec a$, the green arrow is $\vec u$ and the purple arrow is $\vec v$.

Why It's Interesting

An interesting thing about the surface plot of $z=x^2-y^2$, also called a saddle, is that it has a saddle point. Depending on the direction you are going, the saddle point could be both a maximum or minimum. For example, if you approach the surface in the direction of $y=0$, the saddle point is a maximum. On the other hand, if you approach the surface in the direction of $x=0$, the saddle point is a minimum.

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Future Directions for this Page

This page was my first page and it was not well conceived. I had no idea what a page should look like nor any idea about research. Should anyone chooses to carry on. The first thing you should do it to find primary and secondary literature for ideas and backup. Also, you could really use a lot of animation and graphs here. Or, you can tear this apart and only use part of it. For example, if you are doing page on maxwell equations, you can use part of this page.

If you are able, please consider adding to or editing this page!

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