## The Conformal Mercator Projection
Map projections are attempts to portray the surface of the earth or a portion
of the earth on a flat surface. Some distortions of conformity, distance,
direction, scale, and area always result from this process. Some projections
minimize distortions in some of these properties at the expense of maximizing
errors in others. Other projection are attempts to only moderately distort
all of these properties. ## History
The "Mercator Projection" is named after the Flemish cartographer Gerard
Kremer (1512-1594) , who later adopted the latin name Gerardus Mercator.
He published the first maps using his new developed projection in 1569,
but only in the beginning of the 17th century, after 1599 when Edward Wright
published a detailed explanation of the technique Mercator used, the conformal
Mercator Projection became popular among nautical cartographers. ## The conformal Projection
In the 16th century, trading over the new discovered seaways had developed
to an important economical factor in western Europe. The compass was the
principal navigation instrument and navigation itself was based on dead
reckoning. Therefore, it was most convenient to sail tracks of constant
heading or sailing along a loxodrome.
The cylindrical projection and the Mercator projection have in common
that meridians lines and lines of parallels are straight and intersect
orthogonally. Also since the meridian lines don't converge towards the
poles (as they do on a globe), the scale of the chart must increase as
the latitude increases. The difference to the Mercator projection is that
a cylindrical projection has different horizontal and vertical scaling
factors and thus it is not conformal. ## The Mathematical Framework
In mathematics, a projection is a function which takes the coordinates
of a point on one surface and transforms them into the coordinates of a
point on another surface. More specific in cartography the projection function
takes a point from the surface of the globe and transforms it to a point
on a plane. Lat : Latitude in degrees Lon : Longitude in degrees X : east -west distance on the chart (e.g. in mm) Y : north-south distance on the chart (e.g. in mm) c : scale of the chart (e.g. in mm/degree) ## The conformal Mercator projectionAn infinitesimal small square with sides (dLon, dLat) located on the globe at longitude "Lon" and latitude "Lat" has a size on the globe of dX = c' * dLon * cos(Lat) dY = c' * dLat
with c' the scaling factor between the units of Lon/Lat (°) and the
units of X/Y (e.g. km). c = c' * cos(Lat)To obtain conformity, the scaling has to be applied for both directions X and Y: dX = c * dLon dY = c * dLat / cos(Lat) The transformation function is obtained by integrating these differential equations. The chart origin (X,Y) = (0,0) is mapped to location (Lon,Lat) = (0°,0°) to resolve the "integration constant": X = c * Lon Y = c * (ln ( tan ( Lat/2 + 45°))
This is the Mercator conformal projection function, transforming an arbitrary
location on the surface of the Earth (Lon,Lat) to a point on a two dimensional
chart (X,Y). Longitudes east of the Prime Meridian have positive sign and
map to positive X values; longitudes west of the Prime Meridian map to
negative values of X. In a similar way, latitudes north of the equator
map to positive Y values; latitudes south of the equator have a negative
sign and map to negative Y values. ## The cylindrical projectionThe cylindrical projection, which is often used as geometrical illustration of the Mercator projection has the following projection function: X = c * Lon Y = c * tan(Lat) This cylindrical projection does not map a loxodrome to a straight line in the chart. In a conformal Mercator projection the scaling factor c is the same for both X and Y dimensions. The cylindrical projection has different scaling factor for X and Y directions and thus is not conformal. This is illustrated in the pictures below for a NW loxodrome:
The cylindrical projection is nevertheless useful as an approximation
of the Mercator projection in a small plotting range (e.g. a couple of
degrees of latitude around the location of a vessel). ## The gnomonic Projection
The conformal Mercator projection has the great advantage that rhumb lines
or loxodromes are mapped to straight lines in the chart. The gnomonic projection
produces charts in which great-circle segments are mapped as straight lines.
Since a great-circle segment is the shortest track between two points
on the globe, gnomonic charts can be advantageously used for nautical navigation
to find the shortest route between two ports. X = c * ( sin(Lon - Lon_sp) * cos(Lat) ) / cos(a) Y = c * ( cos(Lat_sp) * sin(Lat) - sin(Lat_sp) * cos(Lat) * cos(Lon - Lon_sp) ) / cos(a) with "a" the angular distance of the point (Lat,Lon) to the center of the projection (Lon_sp, Lat_sp): cos(a) = sin(Lat_sp) * sin(Lat) + cos(Lat_sp) * cos(Lat) * cos(Lon - Lon_sp) If the location (Lon_sp,Lat_sp) = (0°,0°) is chosen as point of tangency (standard point) the projection function simplifies to: X = c * tan(Lon) Y = c * tan(Lat) / cos(Lon) The principle of this gnomonic projection and the resulting chart grid of meridian lines and parallels of constant latitude is illustrated in the pictures below:
The example of the loxodrome in the picture on the right also illustrates
that - even on this scale - there is no real significant difference between a loxodrome
and a great-circle segment (the loxodrome is almost a straight line). - distance and bearings for great-circle track between two locations
- distance and courses for loxodrome track between two locations
As an example the great-circle distance across the Atlantic Ocean between Cape Town (South Africa; 36°S 17°E) and Montevideo (Uruguay; 37°S 56°W) is 3428 nautical miles. The loxodrome distance is 3521 nautical miles. So the great-circle track is about 100 nautical miles (less than 3%) shorter than the loxodrome track. However, for ocean sailing, taking advantage of favourable winds and currents is usualy of much more importance than sailing the shortest track. |

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