Parametrizing Paths: Learn How to Do It for Functions of Multiple Variables

[nivekious]

I'm trying to figure out what is meant by parametrizing a path, and how it would be done for a function of multiple variables. Can someone help me?

 

[Pere Callahan]

If you mean by path a curve C in, say R^n, then "parametrization" usually refers to a function

$$\gamma:[a,b]\to\mathbb{R}^n$$

such that $\gamma([a,b])=C$.
This is just the rough idea, does it help?

 

[ice109]

not meant to derail but what are some techniques for parameterizing arbitrary paths? probably interpolation?

 

[HallsofIvy]

Since "interpolation" is an approximating method, no it is not at all appropriate for a problem like this.

The whole point of parameterization is that a path, whether in 2 dimensional space or 3 dimensional space, is a one dimensional object. We should be able to specify any point on it by one number.

Another important thing to remember is that there is no single "correct" parameterization for any path- there always exist an infinite number of such parameters.

One fairly useful "method" is this: if y= f(x) is a function of x then we can use x itself as parameter: x= x and y= f(x) or, if you prefer to call your parameter "t", x= t, y= f(t).

It is always possible to use "arclength" as parameter: select some point on the path as t= 0 and one direction as t> 0 (both of those can be done arbitrarily). The (x,y) corresponding to t> 0 is the point at distance t from the point at t=0 in that direction, and the (x,y) corresponding to t< 0 is the point at distance t from the point at t=0 in the opposite direction. Of course, determining formulas for x and y, as functions of t involves doing a (typically) difficult integral to find the arclength itself- I suspect you haven't done that yet.

You can think "physically": imagine an object moving along the path with a given speed. Then (x(t), y(t)) is the point your object is at at time t. That's often done, of course, in Physics.

Finally, you can use some kind of geometric property. I know that, if I measure angle $\theta$ from the positive x axis, the point on a circle of radius R, center at the origin, at angle $\theta$ is given by (Rcos($\theta$), Rsin($\theta$) so I can use x= Rcos($\theta$), y= Rsin($\theta$) as parametric equations for that circle.

The important thing to remember is that there is no single "correct" parameterization: every path has an infinite number of possible parameterizations.