General Relativity: An Introduction for Physicists · Problem 2.6

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · map-projections-conformal-equal-area

Problem

For the general map projection discussed in Exercise 2.5, show that the angle between two directions at some point on the Earth's surface will equal the angle between the corresponding directions on the map, provided that the functions

x

and

y

are chosen such that

\Omega(x,y)(dx^2 + dy^2) = a^2(d\lambda^2 + \cos^2\lambda\,d\phi^2),

for some function

\Omega(x,y)

. Show that the Mercator projection satisfies this condition. Write down the general requirement on

x

and

y

for an equal-area projection, in which the area of any region of the map is proportional to the corresponding area on the Earth's surface. Find such a projection. Is it possible to obtain a projection that simultaneously is equal-area and preserves angles?

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