PATHS, PARAMETERIZED CURVES, AND PARAMETERIZATION

What are Paths?

Definition: A path is a curve in the plane or three dimensional space -- or in whatever dimension we are working -- with an orientation, i.e. a direction of travel specified.

For example, "the unit circle, traversed counter--clockwise" specifies a path. Also, "the unit circle, traversed clockwise" specifies another. They are not the same because their orientations differ.

A path is called simple in case it never crosses itself. The two paths above are simple; any sort of "figure--eight" path would not be. But clearly all reasonable paths can be "broken up" into a collection of simple pieces. Now, the kind of thing that we are going to do with paths is to compute the work done on moving some particle along a given path in the presence of a given force field. Clearly this work is the sum of the amoints of work done in moving the particle along each of the pieces. So it we develop methods for dealing with simple paths, we can then deal with non-simple paths as well -- just add up the results for the pieces.

So mostly we will think about simple paths. There are two key types of these: Simple paths can be either open or closed. The two paths in the first example are closed. Here is an open path:

"The part of the parabola y= x2 starting at (0,0) and ending at (1,1)"

A closed, simple path is one that encloses a region, and has no endpoints. An open, simple path encloses no region and has two end points: a starting point and a finishing point, and they are different.

What are Parameterized Curves?

To actually compute things associated with paths (like the amount of work done moving something along one of them...), one needs formulas, not just a verbal description like "the unit circle, traversed counter--clockwise" or even "The part of the parabola y= x2 starting at (0,0) and ending at (1,1)".

Definition: A parameterized curve in the plane is a function that assigns to each t in some given interval [a,b] a point

(x(t),y(t))

in the plane. In three dimensional space, t would be assigned to a point

(x(t),y(t),z(t))

in three dimension space, and so on.

What is parameterization?

Clearly, any parameterized curve determines a single path. The curve in question is the set of points "traced out" as t varies, and there is a start, namely (x(a),y(a)) and an end, namely (x(a),y(a)) so there is an orientation. To go from a parameterized curve to a path, essentially you just plot it, and draw in an arrow to record the orientation.

Parameterization is the process of doing the reverse: finding a parameterized curve that traces out a given path.

For example, here are several parameterizations of the unit circle, traversed counter clockwise:

(1) ((x(t),y(t)) = (cos(2*pi*t),sin(2*pi*t)), [0,1]

(2) ((x(t),y(t)) = (cos(t2),sin(t2)), [0,2*pi]

(3) ((x(t),y(t)) = (cos(t),sin(t)), [0,2*pi]

Here are two for "the part of the parabola y= x2 starting at (0,0) and ending at tt>(1,1)"

(1) ((x(t),y(t)) = (t,t2), [0,1]

(2) ((x(t),y(t)) = (t2,t4), [0,1]

(3) ((x(t),y(t)) = (t1/2,t), [0,1]

Note that we always give the interval over which t ranges right after we specify ((x(t),y(t)). It is needed in the specification, or you wouldn't know where to start and stop drawing.

Make sure you understand why all of these are in fact parameterizations of the paths they claim to parameterize...

As you can see, there are many ways to parameterize a given path. Infinitely many in fact. Some are simpler than others.Our problem is to develop mwthods for finding one, preferably as simple as possible.

How do I parameterize a given path?

Basically, there are three important cases:
  • The path is a straight line segment from a point (a,b) to a point (c,d).
  • The path is a circle, or an arc of a circle.
  • The path is given using an equation of the form y = f(x).
    • Of course, you may have to do some work to arrive at one of these cases. For example, you may be given the curve in implicit form, such as

    x4 + 2x2y + y4 = 0

    and then you would have to solve for y as a function of x, or, what's better here, for x as a function of y. Then you would proceed as in the thrid case above with the roles of y and x reversed.

    Examples

    Here are some examples: