Can a point on any 2d surface be uniquely identified by only 2 coordinates?

[Werg22]

This is obviously true for perfectly "flat" surfaces - and, intuitively, it seems to be true for all other sorts of 2d surfaces. Is this a proven result? If so, can we generalize?; can we say a n-elements system of coordinates uniquely identifies a point on any n dimensional object? It carries on to 3 dimensions, I think this is more than obvious, but otherwise?

 

[Vid]

To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide coordinates on it — except at the International Date Line and the poles, where longitude is undefined. This example illustrates that in general it is not possible to extend anyone coordinate patch to the entire surface; surfaces, like manifolds of all dimensions, are usually constructed by patching together multiple coordinate systems.

http://en.wikipedia.org/wiki/Surface

 

[Werg22]

Hummm, interesting. Thanks, I should research before coming here to ask questions.

 

[daniel_i_l]

Isn't the fact that any point on the surface can be identified by only 2 coordinates the definition of a 2d surface?

 

[HallsofIvy]

This is obviously true for perfectly "flat" surfaces - and, intuitively, it seems to be true for all other sorts of 2d surfaces. Is this a proven result? If so, can we generalize?; can we say a n-elements system of coordinates uniquely identifies a point on any n dimensional object? It carries on to 3 dimensions, I think this is more than obvious, but otherwise?

What, exactly, is your definition of "2d surface"? Most definitions of "dimension" are precisely the number of coordinates required to specify a point.

 

[daniel_i_l]

What, exactly, is your definition of "2d surface"? Most definitions of "dimension" are precisely the number of coordinates required to specify a point.

Well from what I understood from the wikipedia site, a surface is 2d if for every point there exists a neighborhood so that all the points in the neighborhood can be described using 2 coordinates in some coordinate system. Interestingly, the coordinate system doesn't have to be universal.

 

[mathwonk]

the way you phrased the question, the coordinates do not have to be unique, but they do have to specify a unique point. thus your question is equivalent to asking whether for any 2 dimensional surface (supply definition), there is a map from a subset of R^2 onto the surface.

(you said nothing about continuity of coordinates so the answer is trivially yes. now if you want continuity of the parameter map, it is more interesting, but seems likely true, unless you allow really huge, say not paracompact, surfaces.)

 

[mathwonk]

in fact since there is a continuous map from an interval onto a square, every point of say the square, hence most other surfaces, can be uniquely specified by one coordinate.

 

[dans595]

I'm a little worried about the uniqueness comment, mathwonk.

Aren't space filling curves necessarily not injective?

EDIT: I guess these sets have the same cardinality. But a bijective map therebetween is necessarily discontinuous.

 

[mathwonk]

we seem to disagree over the meaning of the question.

what he asked was that the coordinates uniquely specify the point, not thaT THE POINT UNIQUELY specify the coordinates..

"Can a point on any 2d surface be uniquely identified by only 2 coordinates?"

that has nothing to do with injectivity of the map from coordinates to points.

that has to do only with well - definedness of the map.perhaps what he meant waS DIFFERENT FROM WHAT HE ACTUALLY ASKED, I CANNOT TELL THAT.

 

[dans595]

Yes, I see what you mean.

I read the question sort of like the unique representation of vectors as coordinates of a basis.