[NetBehaviour] Cartographical Map Projections & Polyhedral Maps...

marc marc.garrett at furtherfield.org
Mon Sep 11 15:36:36 CEST 2006

Cartographical Map Projections & Polyhedral Maps.

Cartographical Map Projections.

Cartography is the science of map-making.

It comprises many problems and techniques, including:

* measuring Earth's shape and features
* collecting and storing information about terrain, places and people
* adapting three-dimensional features to flat models (my main concern)
* devising and designing conventions for graphical representation of data
* printing and publishing information.

There is an endless variety of geographical maps for every kind of 
purpose.  When looking at two different world maps one can wonder why 
the differences: do we draw the world as a rectangle, or an oval?  
Shouldn't it be a circle?  Should grid lines be parallel, straight or 
curved?  Does South America's "tail" bend eastwards or westwards?  
What's the "right" way (or, more properly, is there one?) to draw our 
unique planet?

One important concern of cartography is solving how to project, i.e. 
transform or map points from an almost spherical lump of rock (our 
Earth) onto either flat sheets of paper or not-so flat phosphorus-coated 



Polyhedral Maps.

Several approaches were presented for reducing distortion when 
transforming a spherical surface into a flat map, including:

* first mapping the sphere into an intermediate zero-Gaussian curvature 
surface like a cylinder or a cone, then converting the surface into a plane
* partially cutting the sphere and separately projecting each division 
in an interrupted map

Both techniques are combined in polyhedral maps:

1. inscribe the sphere in a polyhedron, then separately project regions 
of the sphere onto each polyhedral face
2. optionally, cut and disassemble the polyhedron into a flat map, 
called a "net" or fold-out

Intuitively, distortion in polyhedral maps is greater near vertices and 
edges, where the polyedron is farther from the inscribed sphere; also, 
increasing the number of faces is likely to reduce distortion (after 
all, a sphere is equivalent to a polyhedron with infinitely many faces). 
However, too many faces create additional gaps and direction changes in 
the unfolded map, greatly reducing its usefulness.


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