Projections - A Primer of GIS: Fundamental Geographic and Cartographic Concepts - page 88

countries or regions of the world. Polar areas have strongly distorted shapes,
but most continents evidence only minor distortion.
Equations for calculating a Lambert cylindrical equal-area projection

x = R λ

y = R sin φ

Where φ is the latitude and λ is the longitude, R is the radius of the earth

measured at the scale of map.

Mercator Projection
The very common Mercator projection uses only slightly more complicated
equations.
Equations for a Mercator Projection (Snyder, 1993)

x = R λ

y = R ln tan ( π /4 + φ /2)

Where is the longitude (- if in the western hemisphere) for determining

values of the y axis and

λ

is the latitude (+ if north, - if south of the equator).
R is the radius of the earth measured at the scale of the map. The term ln
refers to the natural logarithm to the base e . All angles again are measured
in radians.

φ

Distortions

Distortions arising through projections are unavoidable. They have signifi-
cant consequences for accuracy, so it helps to know more about distortions
in order to choose the best projection for different purposes and to be able
to take distortions into account.
As a general place to start out, we can categorize distortions in terms we
have already seen: the four projection properties of angles, areas, distances,
and direction. Many projections distort one or two of these projection prop-
erties. Distortion of angles (including shapes) is sometimes easy to detect,
especially for large areas when familiar shapes of states, continents, or even
provinces are distorted; but projections of small areas may lack readily visi-
ble evidence of distortions and require the use of special graphics or statisti-
cal measures to determine the distortions. The same applies to areas. The
distortions arising related to distance can be significant because, as you
know now, no projection for large areas accurately shows distances for all
points, but can only be accurate for a few points. Small areas are another
matter, but you still should check to see what distortion a projection creates.
Direction can likewise be distorted in a subtle fashion that is not visually
noticeable, but is of significance should the map be used for navigation pur-
poses.