Mapping a Sphere to a Plane

Date: 11/28/2001 at 12:15:18
From: Candice 
Subject: Maps of the world

Maps of the world are always distorted in some way when put on a flat 
map instead of globe.  Why?

I have to answer this question on a college level and an elementary 
level. I would explain this to elementary students with an orange 
peel. We can see that an orange peel covers the whole orange before 
we peel it. Then after we peel it we can see that the peel would not 
lie flat on a table. I 'm not sure where to go from here.

I'm not sure which facts of geometry I should use to investigate this 
on a college level. I think that using the formula of the surface 
area of a sphere might work, but I'm not sure how to use this formula 
to answer the question.

Thanks,
Candice

Date: 11/28/2001 at 13:18:31
From: Doctor Tom
Subject: Re: Maps of the world

Hi Candice,

The proof that it's impossible uses differential geometry, but I can 
perhaps indicate the idea behind the proof.

Any smooth surface has a "curvature" at every point. In the case of 
the plane, the curvature happens to be zero everywhere. The curvature 
may not be exactly what you think it is. On a cone, away from the tip, 
the curvature is also zero. (There are different sorts of curvature, 
but the type I'm interested in has zero curvature.) And, in fact, a 
cone could be slit and flattened out on a plane.

You can locally measure this curvature if you "live" on such a surface 
as follows: Take a point and put a nail in the ground. Take a piece of 
string of length 1 tied to the nail, and draw a "circle" by connecting 
the other end to a pencil and dragging it around as far from the nail 
as possible. Now measure the length of the circle. If it's 2*pi, the 
curvature is zero. If it's bigger, the curvature is negative. If it's 
smaller, the curvature is positive.

A sphere obviously has positive curvature everywhere since the circle 
on the surface of the sphere is smaller than it would be on a plane.  
(If you have trouble imagining this, think of a 1-foot string on a 
2-foot sphere.) Things like saddles are the opposite - the circle on a 
saddle would be bigger than on a plane.

If you try to imagine mapping a sphere to a plane in a way that 
preserves all lengths, you're obviously out of luck. The the points on 
the plane will all have to be the same distance away from the image of 
the center, but then you will have changed the lengths around the 
circle, so the scaling is wrong there.

You can map a small enough piece of a sphere to the plane with an 
arbitrarily small error, but as you decrease the allowable error, the 
map gets smaller and smaller, and none of them are mathematically 
perfect.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/