Joseph Needham
, a historian of China, speculated that some
star charts
of the Chinese
Song Dynasty
may have been based on the Mercator projection;
[2]
however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used an
equirectangular projection
instead.
[3]
In the 13th century, the earliest extant
portolan charts
of the Mediterranean sea, which are generally not believed to be based on any particular map projection, but which have a closer fit to the Mercator projection than to alternatives, included
windrose networks
of criss-crossing lines which could be used to help set a ship's
bearing
in sailing between locations on the chart; the region of the Earth covered by such charts was small enough that a course of constant bearing would be approximately straight on the chart.
[4]
The charts have startling accuracy not found in the maps constructed by contemporary European or Arab scholars, and their construction remains enigmatic; based on cartometric analysis which seems to contradict the scholarly consensus, they have been speculated to have originated in some unknown pre-medieval cartographic tradition, possibly evidence of some ancient understanding of the Mercator projection.
[5]
German
polymath
Erhard Etzlaub
engraved miniature "compass maps" (about 10×8?cm) of Europe and parts of Africa that spanned latitudes 0°?67° to allow adjustment of his portable pocket-size
sundials
. The projection found on these maps, dating to 1511, was stated by
John Snyder
in 1987 to be the same projection as Mercator's.
However, given the geometry of a sundial, these maps may well have been based on the similar
central cylindrical projection
, a limiting case of the
gnomonic projection
, which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993.
Portuguese mathematician and cosmographer
Pedro Nunes
first described the mathematical principle of the
rhumb line
or loxodrome, a path with constant bearing as measured relative to true north, which can be used in
marine navigation
to pick which compass bearing to follow. In 1537, he proposed constructing a nautical atlas composed of several large-scale sheets in the equirectangular projection as a way to minimize distortion of directions. If these sheets were brought to the same scale and assembled, they would approximate the Mercator projection.
In 1541, Flemish geographer and cartographer
Gerardus Mercator
included a network of rhumb lines on a terrestrial
globe
he made for
Nicolas Perrenot
.
[8]
In 1569, Mercator announced a new projection by publishing a large world map measuring 202 by 124?cm (80 by 49?in) and printed in eighteen separate sheets. Mercator titled the map
Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata
: "A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained the method of construction or how he arrived at it. Various hypotheses have been tendered over the years, but in any case Mercator's friendship with
Pedro Nunes
and his access to the loxodromic tables Nunes created likely aided his efforts.
English mathematician
Edward Wright
published the first accurate tables for constructing the projection in 1599 and, in more detail, in 1610, calling his treatise "Certaine Errors in Navigation". The first mathematical formulation was publicized around 1645 by a mathematician named Henry Bond (
c.
?1600
?1678). However, the mathematics involved were developed but never published by mathematician
Thomas Harriot
starting around 1589.
The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact that
magnetic directions, instead of geographical directions
, were used in navigation. Only in the middle of the 18th century, after the
marine chronometer
was invented and the spatial distribution of
magnetic declination
was known, could the Mercator projection be fully adopted by navigators.
Despite those position-finding limitations, the Mercator projection can be found in many world maps in the centuries following Mercator's first publication. However, it did not begin to dominate world maps until the 19th century, when the problem of position determination had been largely solved. Once the Mercator became the usual projection for commercial and educational maps, it came under persistent criticism from cartographers for its unbalanced representation of landmasses and its inability to usefully show the polar regions.
The criticisms leveled against inappropriate use of the Mercator projection resulted in a flurry of new inventions in the late 19th and early 20th century, often directly touted as alternatives to the Mercator. Due to these pressures, publishers gradually reduced their use of the projection over the course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in the form of the
Web Mercator projection
.
Today, the Mercator can be found in marine charts, occasional world maps, and Web mapping services, but commercial atlases have largely abandoned it, and wall maps of the world can be found in many alternative projections.
Google Maps
, which relied on it since 2005, still uses it for local-area maps but dropped the projection from desktop platforms in 2017 for maps that are zoomed out of local areas. Many other online mapping services still exclusively use the Web Mercator.
Cylindrical projections
edit
Although the surface of Earth is best modelled by an
oblate ellipsoid of revolution
, for
small scale
maps the ellipsoid is approximated by a sphere of radius
a
, where
a
is approximately 6,371?km. This spherical approximation of Earth can be modelled by a smaller sphere of radius
R
, called the
globe
in this section. The globe determines the scale of the map. The various
cylindrical projections
specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map.
[
page?needed
]
The fraction
R
/
a
is called the
representative fraction
(RF) or the
principal scale
of the projection. For example, a Mercator map printed in a book might have an equatorial width of 13.4?cm corresponding to a globe radius of 2.13?cm and an RF of approximately
1
/
300M
(M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198?cm corresponding to a globe radius of 31.5?cm and an RF of about
1
/
20M
.
A cylindrical map projection is specified by formulae linking the geographic coordinates of latitude?
φ
and longitude?
λ
to Cartesian coordinates on the map with origin on the equator and
x
-axis along the equator. By construction, all points on the same meridian lie on the same
generator
[a]
of the cylinder at a constant value of
x
, but the distance
y
along the generator (measured from the equator) is an arbitrary
[b]
function of latitude,
y
(
φ
). In general this function does not describe the geometrical projection (as of light rays onto a screen) from the centre of the globe to the cylinder, which is only one of an unlimited number of ways to conceptually project a cylindrical map.
Since the cylinder is tangential to the globe at the equator, the
scale factor
between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude, is
R
?cos?
φ
, the corresponding parallel on the map must have been stretched by a factor of
1
/
cos
φ
= sec
φ
. This scale factor on the parallel is conventionally denoted by
k
and the corresponding scale factor on the meridian is denoted by?
h
.
[28]
Scale factor
edit
The Mercator projection is
conformal
. One implication of that is the "isotropy of scale factors", which means that the point scale factor is independent of direction, so that small shapes are preserved by the projection. This implies that the vertical scale factor,
h
, equals the horizontal scale factor,
k
. Since
k
=
sec
φ
, so must
h
.
The graph shows the variation of this scale factor with latitude. Some numerical values are listed below.
- at latitude 30° the scale factor is??
k
?= sec?30°?=?1.15,
- at latitude 45° the scale factor is??
k
?= sec?45°?=?1.41,
- at latitude 60° the scale factor is??
k
?= sec?60°?=?2,
- at latitude 80° the scale factor is??
k
?= sec?80°?=?5.76,
- at latitude 85° the scale factor is??
k
?= sec?85°?=?11.5
The area scale factor is the product of the parallel and meridian scales
hk
= sec
2
φ
. For Greenland, taking 73° as a median latitude,
hk
= 11.7. For Australia, taking 25° as a median latitude,
hk
= 1.2. For Great Britain, taking 55° as a median latitude,
hk
= 3.04.
The variation with latitude is sometimes indicated by multiple
bar scales
as shown below.
The classic way of showing the distortion inherent in a projection is to use
Tissot's indicatrix
.
Nicolas Tissot
noted that the scale factors at a point on a map projection, specified by the numbers
h
and
k
, define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels.
[c]
For the Mercator projection,
h
?=?
k
, so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations.
Mercator projection transformations
edit
Derivation
edit
As discussed above, the isotropy condition implies that
h
=
k
=
sec
φ
. Consider a point on the globe of radius
R
with longitude
λ
and latitude
φ
. If
φ
is increased by an infinitesimal amount,
dφ
, the point moves
R
dφ
along a meridian of the globe of radius
R
, so the corresponding change in
y
,
dy
, must be
hR
dφ
=
R
?sec?
φ
dφ
. Therefore
y′
(
φ
)?=?
R
?sec?
φ
. Similarly, increasing
λ
by
dλ
moves the point
R
cos
φ
dλ
along a parallel of the globe, so
dx
=
kR
cos
φ
dλ
=
R
dλ
. That is,
x′
(
λ
)?=?
R
. Integrating the equations
-
with
x
(
λ
0
)?=?0 and
y
(0)?=?0, gives
x(λ)
and
y(φ)
. The value
λ
0
is the longitude of an arbitrary central meridian that is usually, but not always,
that of Greenwich
(i.e., zero). The angles
λ
and
φ
are expressed in radians. By the
integral of the secant function
,
[30]
[31]
-
The function
y
(
φ
) is plotted alongside
φ
for the case
R
?=?1: it tends to infinity at the poles. The linear
y
-axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels.
Inverse transformations
edit
-
The expression on the right of the second equation defines the
Gudermannian function
; i.e.,
φ
?=?gd(
y
/
R
): the direct equation may therefore be written as
y
?=?
R
·gd
?1
(
φ
).
[30]
Alternative expressions
edit
There are many alternative expressions for
y
(
φ
), all derived by elementary manipulations.
[31]
-
Corresponding inverses are:
-
For angles expressed in degrees:
-
The above formulae are written in terms of the globe radius
R
. It is often convenient to work directly with the map width
W
?=?2
π
R
. For example, the basic transformation equations become
-
Truncation and aspect ratio
edit
The ordinate
y
of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with the result that European countries were moved toward the centre of the map. The
aspect ratio
of his map is
198
/
120
= 1.65. Even more extreme truncations have been used: a
Finnish school atlas
was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97.
Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case the maximum latitude attained must correspond to
y
?=?±
W
/
2
, or equivalently
y
/
R
?=?
π
. Any of the inverse transformation formulae may be used to calculate the corresponding latitudes:
-
Small element geometry
edit
The relations between
y
(
φ
) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding
small
elements on the globe and map. The figure below shows a point P at latitude?
φ
and longitude?
λ
on the globe and a nearby point Q at latitude
φ
?+?
δφ
and longitude
λ
?+?
δλ
. The vertical lines PK and MQ are arcs of meridians of length
Rδφ
.
[d]
The horizontal lines PM and KQ are arcs of parallels of length
R
(cos?
φ
)
δλ
. The corresponding points on the projection define a rectangle of width?
δx
and height?
δy
.
For small elements, the angle PKQ is approximately a right angle and therefore
-
The previously mentioned scaling factors from globe to cylinder are given by
- parallel scale factor
- meridian scale factor
Since the meridians are mapped to lines of constant
x
, we must have
x
=
R
(
λ
?
λ
0
)
and
δx
?=?
Rδλ
, (
λ
in radians). Therefore, in the limit of infinitesimally small elements
-
In the case of the Mercator projection,
y'
(
φ
) =
R
sec
φ
, so this gives us
h
=
k
and
α
=
β
. The fact that
h
=
k
is the isotropy of scale factors discussed above. The fact that
α
=
β
reflects another implication of the mapping being conformal, namely the fact that a sailing course of constant azimuth on the globe is mapped into the same constant grid bearing on the map.
Formulae for distance
edit
Converting ruler distance on the Mercator map into true (
great circle
) distance on the sphere is straightforward along the equator but nowhere else. One problem is the variation of scale with latitude, and another is that straight lines on the map (
rhumb lines
), other than the meridians or the equator, do not correspond to great circles.
The distinction between rhumb (sailing) distance and great circle (true) distance was clearly understood by Mercator. (See
Legend 12
on the 1569 map.) He stressed that the rhumb line distance is an acceptable approximation for true great circle distance for courses of short or moderate distance, particularly at lower latitudes. He even quantifies his statement: "When the great circle distances which are to be measured in the vicinity of the equator do not exceed 20 degrees of a great circle, or 15 degrees near Spain and France, or 8 and even 10 degrees in northern parts it is convenient to use rhumb line distances".
For a ruler measurement of a
short
line, with midpoint at latitude?
φ
, where the scale factor is
k
?=?sec?
φ
?=?
1
/
cos?
φ
:
- True distance = rhumb distance ? ruler distance × cos?
φ
/ RF.???(short lines)
With radius and great circle circumference equal to 6,371?km and 40,030?km respectively an RF of
1
/
300M
, for which
R
?=?2.12?cm and
W
?=?13.34?cm, implies that a ruler measurement of 3?mm. in any direction from a point on the equator corresponds to approximately 900?km. The corresponding distances for latitudes 20°, 40°, 60° and 80° are 846?km, 689?km, 450?km and 156?km respectively.
Longer distances require various approaches.
On the equator
edit
Scale is unity on the equator (for a non-secant projection). Therefore, interpreting ruler measurements on the equator is simple:
- True distance = ruler distance / RF ??? (equator)
For the above model, with RF?=?
1
/
300M
, 1?cm corresponds to 3,000?km.
On other parallels
edit
On any other parallel the scale factor is sec
φ
so that
- Parallel distance = ruler distance × cos?
φ
/ RF ??? (parallel).
For the above model 1?cm corresponds to 1,500?km at a latitude of 60°.
This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. The difference is small for short distances but increases as
λ
, the longitudinal separation, increases. For two points, A and B, separated by 10° of longitude on the parallel at 60° the distance along the parallel is approximately 0.5?km greater than the great circle distance. (The distance AB along the parallel is (
a
?cos?
φ
)?
λ
. The length of the chord AB is 2(
a
?cos?
φ
)?sin?
λ
/
2
. This chord subtends an angle at the centre equal to 2arcsin(cos?
φ
?sin?
λ
/
2
) and the great circle distance between A and B is 2
a
?arcsin(cos?
φ
?sin?
λ
/
2
).) In the extreme case where the longitudinal separation is 180°, the distance along the parallel is one half of the circumference of that parallel; i.e., 10,007.5?km. On the other hand, the
geodesic
between these points is a great circle arc through the pole subtending an angle of 60° at the center: the length of this arc is one sixth of the great circle circumference, about 6,672?km. The difference is 3,338?km so the ruler distance measured from the map is quite misleading even after correcting for the latitude variation of the scale factor.
On a meridian
edit
A meridian of the map is a great circle on the globe but the continuous scale variation means ruler measurement alone cannot yield the true distance between distant points on the meridian. However, if the map is marked with an accurate and finely spaced latitude scale from which the latitude may be read directly?as is the case for the
Mercator 1569 world map
(sheets 3, 9, 15) and all subsequent nautical charts?the meridian distance between two latitudes
φ
1
and
φ
2
is simply
-
If the latitudes of the end points cannot be determined with confidence then they can be found instead by calculation on the ruler distance. Calling the ruler distances of the end points on the map meridian as measured from the equator
y
1
and
y
2
, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulæ:
-
where
R
may be calculated from the width
W
of the map by
R
?=?
W
/
2
π
. For example, on a map with
R
?=?1 the values of
y
?=?0, 1, 2, 3 correspond to latitudes of
φ
?=?0°, 50°, 75°, 84° and therefore the successive intervals of 1?cm on the map correspond to latitude intervals on the globe of 50°, 25°, 9° and distances of 5,560?km, 2,780?km, and 1,000?km on the Earth.
On a rhumb
edit
A straight line on the Mercator map at angle
α
to the meridians is a
rhumb line
. When
α
?=?
π
/
2
or
3
π
/
2
the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When
α
?=?0 or
π
it corresponds to a meridian great circle (if continued around the Earth). For all other values it is a spiral from pole to pole on the globe intersecting all meridians at the same angle, and is thus not a great circle.
[31]
This section discusses only the last of these cases.
If
α
is neither 0 nor
π
then the
above figure
of the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudes
φ
; and
φ
?+?
δφ
is
a
?sec?
α
δφ
. Since
α
is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth:
-
Once again, if Δ
φ
may be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudes
φ
1
and
φ
2
is given by the above. If there is no such scale then the ruler distances between the end points and the equator,
y
1
and
y
2
, give the result via an inverse formula:
-
These formulæ give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.
[e]
Generalization to the ellipsoid
edit
When the Earth is modelled by a
spheroid
(
ellipsoid
of revolution) the Mercator projection must be modified if it is to remain
conformal
. The transformation equations and scale factor for the non-secant version are
[32]
The scale factor is unity on the equator, as it must be since the cylinder is tangential to the ellipsoid at the equator. The ellipsoidal correction of the scale factor increases with latitude but it is never greater than
e
2
, a correction of less than 1%. (The value of
e
2
is about 0.006 for all reference ellipsoids.) This is much smaller than the scale inaccuracy, except very close to the equator. Only accurate Mercator projections of regions near the equator will necessitate the ellipsoidal corrections.
The inverse is solved iteratively, as the
isometric latitude
is involved.