# Projecting the Earth

1. Jan 12, 2010

### 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!

2. Jan 12, 2010

### espen180

3. Jan 12, 2010

### 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").

4. Jan 12, 2010

### 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?

5. Jan 12, 2010

### wofsy

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.