Maps and Projections
Mercator Projection
• Mercator projection was developed by Gerardus Mercator (1512-1594).
• A point on the sphere at latitude
projected onto
φ is the cylinder at height ln tan φ2 + π4 .
— θ is the longitude and φ is the latitude
• Why is the complicated expression ln tan
necessary?
φ
2
+
π
4
• Advantages of the Mercator projection
— Lines of latitude and lines of longitude (meridans)
are parallel
— Meridians and lines of latitude intersect at right
angles.
— Compass bearings can be found by drawing straight
lines
• Disadvantages of the Mercator projection
— Area distortion
Reason for the stretching factor ln
tan
φ
2
+
• As we move north, all horizontal distances must be
stretched by some factor
• The angle preserving condition requiresvertical distances be stretched by the same factor.
• It can be shown that if D (φ) is the distance on the
map from the equator to a parallel at latitude φ, then
dD
= sec φ
dφ
• Integration yields
D (φ) =
φ
0
sec (x) dx
π 4 • It can be shown by differentiation that the following
equation is correct
φ
0
φ
π
sec (t) dt = ln tan
+
2
4
φ
π
• This is the reason for the stretching factor = ln tan 2 + 4 .
• Mercator was secretive about
at his
how
he arrived
projection (i.e., the term ln tan φ2 + π4 ).
• Part of the reason for the secrecy was that if a country could out navigate its competitors , then that
country would have an advantage in trade.
• The lack of good maps for navigation was profound
in those days (17th century).
• Edward Wright (1561-1615) finally found the method
for producing an accurate Mercator map.
• From “Certaine Errors in Navigation, Arising either
of the ordinaire erroneous making or vsing of the sea
chart, compasse, crosse staffe and tables of declination of the sunne, and fixed starres detected and corrected, Valentine, Sims, London, 1599” by Edward
Wright:
the parts of the meridian at every poynt of
latitude must needs increase with the same proportion wherewith the Secantes or hypotenusae
of the arke, intercepted betweene those pointes
of latitude and the aequinoctial do increase ....
For.. by perpetuall addition of the Secantes∗
answerable to the latitudes of each point or parallel vnto the summe compounded of all former
secantes ... we may make a table which shall
shew the sections and points of latitude in the
meridians of the nautical planisphere: by which
sections, the parallels are to be drawne.
∗ This
perpetuall addition of the Secantes
is the process of taking
Riemann sums leading to the integral sec (x) dx described above.
• Wright was able to do this a few years before the
invention of the calculus, since his process of ”perpetuall addition of the Secantes” was essentially a
computation of the integral
φ
sec (x) dx.
0
• An outstanding mathematical problem of the midseventeenth
century:
prove rigorously that D (φ) =
ln tan φ2 + π4 is actually true.
— There was even a wager offered via the Royal Society of London for the solution of this problem.
• First to prove the conjecture, James Gregory in 1668
— But his proof, according to Edmund Halley was
“not without a long train of consequences and
complication of proportions, whereby the evidence of the demonstration is in a great measure
lost, and the Reader wearied before he attain it.”
• — First intelligible proof given by Isaac Barrow (16301677).
— Note: calculus had already been developed by
the time of Barrow’s proof.
— Barrow’s result in modern notation is
sec (x) dx = ln |sec x + tan x| + C
Rhumb lines and loxodromes
• A path of constant compass bearing is called a rhumb
line or loxodrome.
• A major advantage of the Mercator projection is that
rhumb lines are actually straight lines.
• But on the globe, a rhumb line is a spherical helix.
Here are some pictures of rhumb lines on the sphere.
• — ∗
Loxodrome, or rhumb line, of constant 10o bearing
An actual map (not a Mercator map) with a
loxodrome (at bearing of 20 degrees originating in the southern US):
• Mercator projection of the globe from -85 degrees to
85 degrees.
— The corresponding loxodromes would be straight
lines on this map.