I understand that the key to parametrization is to realize that the

[glebovg]

I understand that the key to parametrization is to realize that the goal of this method is to describe the location of all points on a geometric object, a curve, a surface, or a region. However, I am looking for a general rule for parameterization. How would one know which parametrization to use except for the obvious cases?

 

[Erland]

If the curve, surface, or, more generally, manifold, is given by a set of equations, you can, locally, near a point, solve some of the variables för the remaining ones, so that the latter ones can be viewed as parameters. This is the content of the Implicit Function Theorem. http://en.wikipedia.org/wiki/Implicit_function_theorem

But this may only work locally, and parametrization will not always be the simplest or most useful one. For example, if the unit circle is given by the equation x^2 + y^2 = 1, then this method will give x as the parameter and y = +-sqrt(1-x^2), with different signs for the upper and the lower semicircle. The parametrization x=cos(t), y=sin(t), is much nicer but cannot be found with this method.

There is no general method for finding the "best" parametrization.

 

[glebovg]

What are the most common parameterization? If you are given a geometric object how would you decide which parametrization to use?

 

[Erland]

What are the most common parameterization? If you are given a geometric object how would you decide which parametrization to use?

There is no general rule. It depends upon the geometric object you want to parametrize. It must be decided in each case separately. Sorry, but there are no shortcuts.

 

[glebovg]

For example, if you wanted to describe a torus or the Möbius strip parametrically, where would you start?

 

[HallsofIvy]

You would start by thinking about it and describing its shape. A torus is easy- it's a set of circles whose center lie on a circle. So you would take one parameter a the angle, [itex]\theta[/tex] gives a specific point on the circle that all the the cross sections have their center on and take the other, $\phi$ as an angle in that cross section. Each of x, y, and z can be expressed as functions of those two parameters.

Another way to do that is to think of the torus as a rectangle of paper where "opposite sides" have been pasted together. If one side of the torus is much longer than the other, pasting together the long sides gives a cylinder. Pasting together the ends of the cylinder gives a torus. Each point on that rectangle can be given (x, y) coordinates and then the folding and pasting map those into (x, y, z) coordinates for the torus. That can also be used for the Klein bottle except that after pasting together the long sides we paste the short sides together in "reversed" order. That is, if our original rectangle had vertices at (2, 1), (-2, 1), (-2, -1), and (2,-1), pasting the long sides would paste (-2, 1) to (-2, -1) and (2, 1) to (2, -1). For the cylinder, you paste the short sides together so that (-2, 1) maps to (2, 1) and (-2, -1) to (-2, 1). For the Klein bottle, instead, you paste the shorts sides together so that (-2, 1) matches with (2, -1) and (-2,-1) to (2, 1) (impossible to do in Euclidean three dimensional space).