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

In summary, the conversation discusses the definition of a two-dimensional surface and whether a point on such a surface can be uniquely identified by only two coordinates. The concept of coordinate patches and the possibility of extending a single coordinate system to cover the entire surface is also mentioned. There is some disagreement over the interpretation of the question and the meaning of "uniquely identifying" a point, but it is generally agreed that a continuous map from a subset of R^2 to the surface can achieve this.
  • #1
Werg22
1,431
1
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?
 
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  • #2
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
 
  • #3
Hummm, interesting. Thanks, I should research before coming here to ask questions.
 
  • #4
Isn't the fact that any point on the surface can be identified by only 2 coordinates the definition of a 2d surface?
 
  • #5
Werg22 said:
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.
 
  • #6
HallsofIvy said:
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.
 
  • #7
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.)
 
  • #8
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.
 
  • #9
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.
 
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  • #10
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.
 
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  • #11
Yes, I see what you mean.

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

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

Yes, a point on any 2d surface can be uniquely identified by only 2 coordinates. These coordinates are typically referred to as x and y coordinates, and they represent the horizontal and vertical position of the point on the surface.

2. What is the significance of identifying a point on a 2d surface with only 2 coordinates?

Identifying a point on a 2d surface with only 2 coordinates allows for a simple and efficient way to describe the location of the point. This is important in many fields such as mathematics, physics, and computer science where precise location information is necessary.

3. Are there any exceptions to being able to uniquely identify a point with 2 coordinates?

There are some exceptions where a point on a 2d surface cannot be uniquely identified with only 2 coordinates. This can occur in non-Euclidean geometries where traditional Cartesian coordinates may not accurately represent the shape of the surface.

4. How does the shape of the 2d surface affect the ability to identify a point with 2 coordinates?

The shape of the 2d surface can affect the ability to identify a point with 2 coordinates in cases where the surface is curved or non-planar. In these cases, a different coordinate system may be necessary to accurately describe the location of a point.

5. Is there a limit to the precision of 2 coordinates in identifying a point on a 2d surface?

Yes, there is a limit to the precision of 2 coordinates in identifying a point on a 2d surface. This is due to the concept of rounding errors, where the coordinates may not be able to represent the exact location of the point and can only provide an approximation.

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