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Pentasplat Xplanation | by lloydb
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Pentasplat Xplanation

 

The shape is built by cascading several transforms, like the sprocket.

 

 

 

 

 

 

 

 

1. Make a conformal cube projection using math from Lee, L. P. (1976): Conformal Projections Based on Elliptic Functions. Cartographica. Monograph 16, supplement no. 1 to Canadian Cartographer, v. 13., 1976.

 

 

 

 

2. Rearrange the net just enough to make it confusing.

 

 

 

 

3. Spin it like this; position so the three big squares meet at (0,0).

 

 

 

 

4-5. Close up the gap using z = z^(3/4).

 

 

 

 

6. Tumble the sphere so the north pole moves to the center of the 2D shape.

 

 

 

 

7. Increase lobes from 3 to 5 using z = z^(5/3).

 

 

 

 

8. Adjust the contents to show 1 sphere instead of 5/3 of a sphere. Briefly,

 

 

 

 

- stereographically map the sphere to the plane

- z = z^(3/5)

- map the plane back to the sphere

- all the while evading icky numerical trouble at the south pole by handling the north and south hemispheres separately

- fool with the azimuth so that sectors are not misplaced.

 

 

 

 

 

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Uploaded on August 29, 2012