Projecting the Earth: A Diamond or Tetragon?

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In summary, the conversation discusses the idea of accurately projecting the Earth onto a flat map. It is mentioned that there are several ways to project the Earth, but no method can completely eliminate distortion. The concept of a ruled surface and the difference between a sphere and a sphere missing a point are also briefly discussed. Ultimately, it is concluded that the best approach is to conformally map the Earth onto a flat surface, as seen in projections such as Stereographic and Mercator.
  • #1
k_squared
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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!
 
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  • #3
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
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
espen180 said:
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.
 

1. What is the purpose of "Projecting the Earth: A Diamond or Tetragon?"

The purpose of this project is to explore and compare the different ways in which the Earth can be represented on a flat surface, such as a map or globe. This includes examining the pros and cons of using a diamond or tetragon-shaped projection compared to the traditional rectangular projection.

2. What are the advantages of using a diamond or tetragon-shaped projection?

One advantage is that these projections can provide a more accurate representation of the Earth's surface, especially when it comes to areas near the poles. They also allow for less distortion in terms of distance and direction. Additionally, these projections can be visually appealing and provide a unique perspective of the Earth.

3. Are there any drawbacks to using a diamond or tetragon-shaped projection?

One drawback is that these projections may not be as familiar to people as the traditional rectangular projection, which could make it more difficult for some individuals to interpret and use the map. There may also be challenges in accurately representing the entire globe on a diamond or tetragon-shaped projection without some distortion in certain areas.

4. How do diamond and tetragon-shaped projections compare to each other?

Both projections have their own strengths and weaknesses. The diamond projection may be better at maintaining accurate shapes and distances near the poles, while the tetragon projection may be better at representing the entire globe with less distortion. It ultimately depends on the purpose and context of the map being used.

5. How can "Projecting the Earth: A Diamond or Tetragon?" impact our understanding of the world?

This project can help us to critically examine and question the ways in which we typically represent the Earth on a flat surface. By exploring alternative projections such as the diamond and tetragon, we can gain a deeper understanding of the complexities of our planet and how we choose to visualize it. It also allows for a more inclusive representation of the world, as different projections may highlight different regions or perspectives.

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