Projecting the Earth: A Diamond or Tetragon?

[k_squared]

I was thinking, I know that the Earth has to be distorted when projected on a flat map because the space nearer the poles of the Earth does not have a much circumference but is expending the same amount of flat space. Could you not do an accurate flat projection by making the Earth a diamond, or a diamond with curved sides? Or some possible combination of 'rotated' tetragon?

I hope this is the appropriate forum. Either way, thanks a lot!

 

[espen180]

There are many ways to project the Earth to a map.

A few are discussed here: http://en.wikipedia.org/wiki/Map_projection

 

[HallsofIvy]

However, there is no way to project a sphere onto a plane without distorting some portion. It is shown in Differential Geometry that a surface can be mapped onto a plane without distortion if and only if, through every point in the surface, there exist a least one straight line that is completely contained in the surface (a "ruled surface").

 

[espen180]

It comes from the fact that a sphere has curvature (4π if I am not mistaken) while a sphere missing a point is flat, doesn't it?

 

[wofsy]

It comes from the fact that a sphere has curvature (4π if I am not mistaken) while a sphere missing a point is flat, doesn't it?

A sphere missing a point is not flat. Topologically it is the same as a plane but not geometrically.

The best you can do is map the sphere- point conformally onto the plane. Stereographic and Mercator projections are examples.