Are Originless Coordinates Possible for Spherical Surfaces and Planes?

[AverageJoe]

This is something that has always been in my mind, yet everywhere I look I can't find an answer.

Is there any type of coordinate system that has no origin?
As in, everything is found by relation to other elements within the model?

 

[Mentallic]

Do you mean as in a coordinate system that cannot have an origin, or doesn't necessarily need it? I can't imagine any system where an origin isn't used for convenience.

Take the surface of the Earth. Everything can be described in relation to something else so that no origin is required, but without this origin it would be impossible to give a point on Earth a set of coordinates.

 

[arildno]

This is something that has always been in my mind, yet everywhere I look I can't find an answer.

Is there any type of coordinate system that has no origin?
As in, everything is found by relation to other elements within the model?

Yes, that does exist.

Unfortunately, I have forgotten what we call it.

 

[Mentallic]

Yes, that does exist.

Unfortunately, I have forgotten what we call it.

Can you please explain how this system works?

 

[arildno]

Can you please explain how this system works?

I only vaguely remember it, so I can't say anything further.

It is an abstract geometry without using coordinates at all, basically.

I think..

 

[arildno]

Okay, I have muddled a lot here!

What I was thinking of is called the "affine plane", and the reason why it doesn't have an origin is that it doesn't have coordinates.

Once you impose coordinates you have, of course, an implied origin.

Sorry about this.

 

[rochfor1]

Any vector space will have an "origin."

 

[AverageJoe]

Hmm..Affine Space isn't exactly what I was thinking of.

 

[JSuarez]

Could it be possible that what you have in mind is a metric space (ir doesn't have to be a vector space, but the more interesting ones are), where you can calculate the distance between any two points, but none of these is a distinguished origin, but where the distance doesn't come from a norm (a norm is a special kind of distance, defined only on vector spaces, that measures distances from the origin, which allows us to define lenghts).

 

[HallsofIvy]

Could it be possible that what you have in mind is a metric space (ir doesn't have to be a vector space, but the more interesting ones are), where you can calculate the distance between any two points, but none of these is a distinguished origin, but where the distance doesn't come from a norm (a norm is a special kind of distance, defined only on vector spaces, that measures distances from the origin, which allows us to define lenghts).

But AverageJoe asked for a space that has a coordinate system but no origin. A metric space does not have a coordinate system. Although I suspect that this is the kind of thing he is thinking of.

(By the way, a metric space does not necessarily not have a norm. It may or may not. Given a norm, we can immediately define a metric but not necessarily the other way.)

 

[JSuarez]

When I stated:

but where the distance doesn't come from a norm

I meant a metric space with a distance that cannot be defined by a norm (for example, it's not translation invariant and/or homogeneous), I didn't meant it doesn't necessarily have a norm; many interesting metrics came from norms.

A metric space does not have a coordinate system

I don't completely agree (strictly speaking, the definition doesn't require it), but metric spaces that are also vector spaces have coordinate systems; we may forget the vector space structure and retain only the metric one.

But if we want to remain in the vector space category, we have two options: forget the origin and fall in affine space (already mentioned), or the Möbius homogeneous coordinate system.

 

[AverageJoe]

Maybe it's more of an Omni-Origin coordinate system. Every element is it's own origin, and their dimensions are ratios of their sister elements.

 

[JSuarez]

Then check the barycentric coordinate system (or Möbius coordinates):

http://mathworld.wolfram.com/BarycentricCoordinates.html"

http://en.wikipedia.org/wiki/Homogeneous_coordinates"

 

[HallsofIvy]

Here's a couple of ideas I had. For the surface of a sphere of radius R, identify each point by spherical coordinates. That is, each point is identified with $(R, \theta, \phi)$. Since R is not 0, there is no point in that surface with coordinates (0, 0, 0).

Or- given a plane, think of it as embedded in three dimensional space and give that space a coordinate system such that its origin was not on the plane. for example, if we take our three dimensional coordinates so that the plane contains (1, 0, 0), (0, 1, 0), and (0, 0, 1), then every point in the plane is given coordinates (x, y, z) where x+ y+ z= 1. Again, there is no point in the plane with coordinates (0, 0, 0). This would be similar to "barycentric coordinates".