Anyone familiar with map projections knows that the Mercator projection maps common to many of our elementary school classrooms grossly overemphasize the size of the northern hemisphere at the expense of the equatorial regions. Looking at a globe will show you that the “important” North American and European areas are really rather small compared to the vast tracts of equatorial expanse. What I was wondering was, where would one draw lines on a globe to seperate “polar” regions from “equatorial” regions, so that each had half the surface area of the Earth. (Assume a spherical Earth – I’m not that nerdy.)
45 degrees north and south latitude are halfway between the equator and the poles, but obviously the area on the equatorial side of this line is much bigger than that on the polar side. How far towards the equtor must we pull the lines in order to make the areas equal?
(wlog radius=1) Integrating to get the area with angle less than phi to the pole gives me 2.pi.(1 - cos phi), and area above the equator will then be 2.pi.
So the angle desired is (1 - cos phi)=1/2 or phi = cos[sup]-1[/sup]phi = 60 degrees.
Anyone want to check, correct, or explain why this answer is obvious for a reason I missed?
Area of a section of the earth is the integral of (cos(T) dT dP) where T is the latitude and P is the longitude. We can ignore the dP for this problem since we’re not concerned about longitude limits. Integrate cos(T) dT from T=0 to T1 and you get sin(T1), so it reaches half the maximum value at T1=pi/6=30 deg.
Actually, there is a great interest in figuring out the best way to present the features of a three-dimensional globe (or any significant portion thereof) on a two-dimensional surface. The calculations go far beyond attempting to simply move a central line.