Curve Parametrization: Minimum Parameters for Unique Point Specification

[TrickyDicky]

What is the minimum number of parameters needed to uniquely specify a point in a curved line?

 

[Ben Niehoff]

One?

What are you trying to get at?

 

[TrickyDicky]

One?

What are you trying to get at?

I have some concepts muddled and I'm trying to clear them up.
I know that topologically is enough with one parameter since bending is ignored, but if we write the line element of say a circle I would say that since we must embed it in a plane it should require two parameters: one plus the radius of curvature but I'm not sure. Maybe there is an intuitive way to understand this.

 

[micromass]

A circle can be described as (cos(t),sin(t)). So it requires only one parameter, namely t...

 

[TrickyDicky]

A circle can be described as (cos(t),sin(t)). So it requires only one parameter, namely t...

Yes, maybe I'm conflating parameters with coordinates, you just used a pair of numbers to define a point in the circle.

 

[micromass]

Well, to describe a point in [itex]\mathbb{R}^2[/itex], you'll need two values. So to describe a curve you'll need 2 coordinates. If that's what you're getting at.

 

[TrickyDicky]

Well, to describe a point in [itex]\mathbb{R}^2[/itex], you'll need two values. So to describe a curve you'll need 2 coordinates. If that's what you're getting at.

Basically what I was thinking is that for every curved manifold of a given dimension (one in the case of the circle), one needs n+1 coordinates to describe it in terms of the Euclidean embedding space, right?
But since the curvature is intrinsic it shouldn't need to be embedded, (locally it would be Euclidean) so what is the line element of a circle with one coordinate?

 

[TrickyDicky]

anybody there? Did I ask wrong?

 

[micromass]

Basically what I was thinking is that for every curved manifold of a given dimension (one in the case of the circle), one needs n+1 coordinates to describe it in terms of the Euclidean embedding space, right?

This is certainly not true in general. It happens to be true for 1-dimensional manifolds though.
Take the Klein bottle for example, this has dimension 2, but one needs 4 coordinates to describe it, i.e. it can only be embedded in [itex]\mathbb{R}^4[/itex].

In general, take a look at the Whitney embedding theorem http://en.wikipedia.org/wiki/Whitney_embedding_theorem
which states that every smooth n-dimensional manifold can be embedded in [itex]\mathbb{R}^{2n}[/itex]. One cannot do better in general, although in some specific cases we can.

 

[TrickyDicky]

This is certainly not true in general. It happens to be true for 1-dimensional manifolds though.
Take the Klein bottle for example, this has dimension 2, but one needs 4 coordinates to describe it, i.e. it can only be embedded in [itex]\mathbb{R}^4[/itex].

In general, take a look at the Whitney embedding theorem http://en.wikipedia.org/wiki/Whitney_embedding_theorem
which states that every smooth n-dimensional manifold can be embedded in [itex]\mathbb{R}^{2n}[/itex]. One cannot do better in general, although in some specific cases we can.

You are right of course, I wasn't very precise, thanks for the reference.

My point was that if we have a curved manifold, say a two-sphere embedded in a 3-space, its line element may have 3 coordinates, but locally it will have a Euclidean form, a local chart with dimension 2, is this right?

 

[chiro]

What is the minimum number of parameters needed to uniquely specify a point in a curved line?

If a proper parametrization exists, then one if your object is a line, no matter how many dimensions that curve is embedded in.

For arbitrary objects, finding an analytical parametrization is pretty damn hard though.

 

[TrickyDicky]

If a proper parametrization exists, then one if your object is a line, no matter how many dimensions that curve is embedded in.

Sure, I was mixing there parameters, coordinates and dimensions.

 

[TrickyDicky]

If we had for instance an intrinsically curved surface, could we use the curvature parameter as a third coordinate in a 3-space embedding?
(Sorry if this is very basic stuff for you guys, I'm not sure where I should post this kind of questions.)