Understanding Map Projections

Unless explicitly stated, it’s impossible to tell from
the parameters alone which convention is being
used. If you use the wrong method, your results can
return inaccurate coordinates. The only way to
determine how the parameters are defined is by
checking a control point whose coordinates are
known in the two systems.
Molodensky method
The Molodensky method converts directly between
two geographic coordinate systems without actually
converting to an X,Y,Z system. The Molodensky
method requires three shifts (∆X,∆Y,∆Z) and the
differences between the semimajor axes (∆a) and the
flattenings (∆f) of the two spheroids. The Projection
Engine automatically calculates the spheroid
differences according to the datums involved.
( M + h)
∆ϕ
= − sin ϕ cos λ∆X
− sin ϕ sin λ∆Y
2
e sin ϕ cosϕ
+ cosϕ∆Z
+
2 2 1/
(1 − e sin ϕ)
+ sin ϕ cosϕ(
M
a b
+ N ) ∆f
b a
( N + h) cosϕ
∆λ
= − sin λ∆X
+ cos λ∆Y
2
∆a
M and N are the meridional and prime vertical radii
of curvature, respectively, at a given latitude. The
equations for M and N are:
a(1
− e )
M =
2 2
(1 − e sin ϕ)
a
N =
2 2
( 1 − e sin ϕ)
2
3/ 2
1/ 2
You solve for ∆λ and ∆ϕ. The amounts are added
automatically by the Projection Engine.
Abridged Molodensky method
The Abridged Molodensky method is a simplified
version of the Molodensky method. The equations
are:
M ∆ϕ
= − sin ϕ cos λ∆X
− sin ϕ sin λ∆Y
+ cosϕ∆Z
+ ( a∆f
+ f∆a)
⋅ 2sin ϕ cosϕ
N cosϕ
∆λ
= − sin λ∆X
+ cos λ∆Y
∆h
= cosϕ
cos λ∆X
+ cosϕ
sin λ ∆Y
+ sin ϕ∆Z
+ ( a∆f
2
+ f∆a) sin
ϕ −∆a
∆h
= cosϕ
cos λ ∆X
+ cosϕ
sin λ ∆Y
+ sin ϕ ∆Z
− (1 − e
a(1
− f )
+
2 2
(1 − e sin ϕ)
1/ 2
2
2
sin ϕ)
2
1/ 2
sin ϕ ∆f
∆a
h
ϕ
λ
a
b
f
e
ellipsoid height (meters)
latitude
longitude
semimajor axis of the spheroid (meters)
semiminor axis of the spheroid (meters)
flattening of the spheroid
eccentricity of the spheroid
26 • Understanding Map Projections