A
map projection is any method of representing the
surface of a sphere or other shape on a
plane. Map projections are
necessary for creating
maps. All map projections
distort the surface in some fashion. Depending on the purpose of
the map, some distortions are acceptable and others are not;
therefore different map projections exist in order to preserve some
properties of the sphere-like body at the expense of other
properties. There is no limit to the number of possible map
projections.
Background
For simplicity, this article usually assumes that the surface to be
mapped is the surface of a sphere. However, the Earth and other
sufficiently large celestial bodies are generally better modeled as
oblate spheroids, and small objects
such as asteroids may have irregular shapes. These other surfaces
can be mapped as well. Therefore, more generally, a map projection
is any method of "flattening" into a plane a continuous surface
having curvature in all three spatial dimensions.
Projection as used here is not limited to
perspective projections, such as
those resulting from casting a shadow on a screen. Rather, any
mathematical function transforming coordinates from the curved
surface to the plane is a projection.
Carl Friedrich Gauss's
Theorema Egregium proved that a sphere
cannot be represented on a plane without distortion. Since any
method of representing a sphere's surface on a plane is a map
projection, all map projections distort. Every distinct map
projection distorts in a distinct way. The study of map projections
is the characterization of these distortions.
A
map of the earth is a representation of a
curved surface on a plane. Therefore a map projection must have
been used to create the map, and, conversely, maps could not exist
without map projections. Maps can be more useful than
globes in many situations: they are more compact and
easier to store; they readily accommodate an enormous range of
scales; they are viewed easily on computer displays; they can
facilitate measuring properties of the terrain being mapped; they
can show larger portions of the Earth's surface at once; and they
are cheaper to produce and transport. These useful traits of maps
motivate the development of map projections.
Metric properties of maps
Many properties can be measured on the Earth's surface
independently of its geography. Some of these properties are:
Map projections can be constructed to preserve one or more of these
properties, though not all of them simultaneously. Each projection
preserves or compromises or approximates basic metric properties in
different ways. The purpose of the map determines which projection
should form the base for the map. Because many purposes exist for
maps, many projections have been created to suit those
purposes.
Another major concern that drives the choice of a projection is the
compatibility of data sets. Data sets are geographic information.
As such, their collection depends on the chosen model of the Earth.
Different models assign slightly different coordinates to the same
location, so it is important that the model be known and that the
chosen projection be compatible with that model. On small areas
(large scale) data compatibility issues are more important since
metric distortions are minimal at this level. In very large areas
(small scale), on the other hand, distortion is a more important
factor to consider.
Construction of a map projection
The creation of a map projection involves three steps:
- Selection of a model for the shape of the Earth or planetary
body (usually choosing between a sphere or
ellipsoid). Because the Earth's actual
shape is irregular, information is lost in this step.
- Transformation of geographic coordinates (longitude and latitude) to
Cartesian (x,y) or
polar plane coordinates.
Cartesian coordinates normally have a simple relation to eastings and northings defined on a
grid superimposed on the projection.
Some of the simplest map projections are literally projections, as
obtained by placinga light source at some definite point relative
to the globe and projecting its features onto a specified surface.
This is
not the case for most projections which
are defined
only in terms of mathematical formulae
that have no direct physical interpretation.
Choosing a projection surface
A surface that can be unfolded or unrolled into a plane or sheet
without stretching, tearing or shrinking is called a
developable surface. The
cylinder,
cone and of course the plane are all
developable surfaces. The sphere and ellipsoid are not developable
surfaces. As noted in the introduction, any projection of a sphere
(or an ellipsoid) onto a plane will have to distort the image. (To
compare, you cannot flatten an orange peel without tearing or
warping it.)
One way of describing a projection is first to project from the
Earth's surface to a developable surface such as a cylinder or
cone, and then to unroll the surface into a plane. While the first
step inevitably distorts some properties of the globe, the
developable surface can then be unfolded without further
distortion.
Aspects of the projection
Once a choice is made between projecting onto a cylinder, cone, or
plane, the
aspect of the shape must be specified.
The aspect describes how the developable surface is placed relative
to the globe: it may be
normal (such that the surface's
axis of symmetry coincides with the Earth's axis),
transverse (at right angles to the Earth's axis) or
oblique (any angle in between). The developable surface
may also be either
tangent or
secant to the sphere or
ellipsoid. Tangent means the surface touches but does not slice
through the globe; secant means the surface does slice through the
globe. Insofar as preserving metric properties goes, it is never
advantageous to move the developable surface away from contact with
the globe, so that possibility is not discussed here.
Scale
A
globe is the only way to represent the earth
with constant
scale throughout the
entire map in all directions. A map cannot achieve that property
for any area, no matter how small. It can, however, achieve
constant scale along specific lines.
Some possible properties are:
- The scale depends on location, but not on direction. This is
equivalent to preservation of angles, the defining characteristic
of a conformal map.
- Scale is constant along any parallel in the direction of the
parallel. This applies for any cylindrical or pseudocylindrical
projection in normal aspect.
- Combination of the above: the scale depends on latitude only,
not on longitude or direction. This applies for the Mercator projection in normal
aspect.
- Scale is constant along all straight lines radiating from a
particular geographic location. This is the defining characteristic
of an equidistant projection such as the Azimuthal equidistant
projection. There are also projections (Maurer, Close) where
true distances from two points are preserved.
Choosing a model for the shape of the Earth
Projection construction is also affected by how the shape of the
Earth is approximated. In the following discussion on projection
categories, a
sphere is assumed. However, the
Earth is not exactly spherical but is closer in shape to an oblate
ellipsoid, a shape which bulges around the
equator. Selecting a model for a shape of
the Earth involves choosing between the advantages and
disadvantages of a sphere versus an ellipsoid. Spherical models are
useful for small-scale maps such as world atlases and globes, since
the error at that scale is not usually noticeable or important
enough to justify using the more complicated ellipsoid. The
ellipsoidal model is commonly used to construct
topographic maps and for other large and
medium scale maps that need to accurately depict the land
surface.
A third model of the shape of the Earth is called a
geoid, which is a complex and more or less accurate
representation of the global mean sea level surface that is
obtained through a combination of terrestrial and satellite gravity
measurements. This model is not used for mapping due to its
complexity but is instead used for control purposes in the
construction of
geographic datum.
(In geodesy, plural of "datum" is "datums" rather than "data".) A
geoid is used to construct a datum by adding irregularities to the
ellipsoid in order to better match the Earth's actual shape (it
takes into account the large scale features in the Earth's gravity
field associated with
mantle
convection patterns, as well as the gravity signatures of very
large geomorphic features such as mountain ranges, plateaus and
plains). Historically, datums have been based on ellipsoids that
best represent the geoid within the region the datum is intended to
map. Each ellipsoid has a distinct major and minor axis. Different
controls (modifications) are added to the ellipsoid in order to
construct the datum, which is specialized for a specific geographic
regions (such as the
North American
Datum). A few modern datums, such as
WGS84
(the one used in the Global Positioning System
GPS), are optimized to represent the entire earth as
well as possible with a single ellipsoid, at the expense of some
accuracy in smaller regions.
Classification
A fundamental projection classification is based on the type of
projection surface onto which the globe is conceptually projected.
The projections are described in terms of placing a gigantic
surface in contact with the earth, followed by an implied scaling
operation. These surfaces are cylindrical (e.g.
Mercator), conic (e.g.,
Albers), or azimuthal or plane (e.g.
stereographic). Many
mathematical projections, however, do not neatly fit into any of
these three conceptual projection methods. Hence other peer
categories have been described in the literature, such as
pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal,
and
polyconic.
Another way to classify projections is according to properties of
the model they preserve. Some of the more common categories
are:
- Preserving direction (azimuthal), a trait possible
only from one or two points to every other point
- Preserving shape locally (conformal or
orthomorphic)
- Preserving area (equal-area or equiareal or
equivalent or authalic)
- Preserving distance (equidistant), a trait possible
only between one or two points and every other point
- Preserving shortest route, a trait preserved only by the
gnomonic projection
NOTE: Because the sphere is not a developable surface, it is
impossible to construct a map projection that is both equal-area
and conformal.
Projections by surface
Cylindrical
The Mercator projection
The term "normal cylindrical projection" is used to refer to any
projection in which
meridians
are mapped to equally spaced vertical lines and
circles of latitude (parallels) are
mapped to horizontal lines.
The mapping of meridians to vertical lines can be visualized by
imagining a cylinder (of which the axis coincides with the Earth's
axis of rotation) wrapped around the Earth and then projecting onto
the cylinder, and subsequently unfolding the cylinder.
By the geometry of their construction, cylindrical projections
stretch distances east-west. The amount of stretch is the same at
any chosen latitude on all cylindrical projections, and is given by
the
secant of the
latitude as a multiple of the equator's scale. The
various cylindrical projections are distinguished from each other
solely by their north-south stretching (where latitude is given by
φ):
- North-south stretching is equal to the east-west stretching
(secant φ): The east-west scale matches the north-south scale:
conformal cylindrical or Mercator; this distorts areas
excessively in high latitudes (see also transverse Mercator).
- North-south stretching growing rapidly with latitude, even
faster than east-west stretching (secant² φ: The cylindric
perspective (= central cylindrical) projection; unsuitable because
distortion is even worse than in the Mercator projection.
- North-south stretching grows with latitude, but less quickly
than the east-west stretching: such as the Miller cylindrical projection
(secant[4φ/5]).
- North-south distances neither stretched nor compressed (1):
equidistant cylindrical or plate carrée.
- North-south compression precisely the reciprocal of east-west
stretching (cosine φ): equal-area cylindrical (with many named
specializations such as Gall–Peters or Gall
orthographic, Behrmann, and
Lambert
cylindrical equal-area). This divides north-south distances by
a factor equal to the secant of the latitude, preserving area but
heavily distorting shapes.
In the first case (Mercator), the east-west scale always equals the
north-south scale. In the second case (central cylindrical), the
north-south scale exceeds the east-west scale everywhere away from
the equator. Each remaining case has a pair of identical latitudes
of opposite sign (or else the equator) at which the east-west scale
matches the north-south-scale.
Normal cylindrical projections map the whole Earth as a finite
rectangle, except in the first two cases, where the rectangle
stretches infinitely tall while retaining constant width.
Pseudocylindrical
A sinusoidal projection shows relative sizes accurately, but
grossly distorts shapes.
Distortion can be reduced by "interrupting" the map.
Pseudocylindrical projections represent the
central
meridian and each
parallel as a single straight line
segment, but not the other meridians. Each pseudocylindrical
projection represents a point on the Earth along the straight line
representing its parallel, at a distance which is a function of its
difference in longitude from the central meridian.
- Sinusoidal: the
north-south scale and the east-west scale are the same throughout
the map, creating an equal-area map. On the map, as in reality, the
length of each parallel is proportional to the cosine of the
latitude. Thus the shape of the map for the whole earth is the
region between two symmetric rotated cosine curves.
The true distance between two points on the same meridian
corresponds to the distance on the map between the two parallels,
which is smaller than the distance between the two points on the
map. The true distance between two points on the same parallel –
and the true area of shapes on the map – are not distorted. The
meridians drawn on the map help the user to realize the shape
distortion and mentally compensate for it.
Hybrid
The
HEALPix projection combines an
equal-area cylindrical projection in equatorial regions with the
Collignon projection in polar
areas.
Conical
Pseudoconical
- Bonne
- Werner cordiform
designates a pole and a meridian; distances from the pole are
preserved, as are distances from the meridian (which is straight)
along the parallels
- Continuous American
polyconic
Azimuthal (projections onto a plane)
Azimuthal projections have the property that
directions from a central point are preserved (and hence, great
circles through the central point are represented by straight lines
on the map). Usually these projections also have radial symmetry in
the scales and hence in the distortions: map distances from the
central point are computed by a function
r(
d) of
the true distance
d, independent of the angle;
correspondingly, circles with the central point as center are
mapped into circles which have as center the central point on the
map.
The mapping of radial lines can be visualized by imagining a
plane tangent to the
Earth, with the central point as tangent point.
The radial scale is
r'(
d) and the transverse
scale
r(
d)/(
R sin(
d/
R))
where
R is the radius of the Earth.
Some azimuthal projections are true
perspective projections; that is,
they can be constructed mechanically, projecting the surface of the
Earth by extending lines from a
points of perspective (along an
infinite line through the tangent point and the tangent point's
antipode) onto the plane:
- The gnomonic projection
displays great circles as straight
lines. Can be constructed by using a point of perspective at the
center of the Earth. r(d) =
c tan(d/R); a hemisphere already
requires an infinite map,
- The General
Perspective Projection can be constructed by using a point of
perspective outside the earth. Photographs of Earth (such as those
from the International Space
Station) give this perspective.
- The orthographic
projection maps each point on the earth to the closest point on
the plane. Can be constructed from a point of perspective an
infinite distance from the tangent point; r(d) =
c sin(d/R). Can display up to a
hemisphere on a finite circle. Photographs of Earth from far enough
away, such as the Moon, give this
perspective.
- The azimuthal conformal projection, also known as the stereographic projection, can be
constructed by using the tangent point's antipode as the point of perspective.
r(d) =
c tan(d/2R); the scale is
c/(2R cos²(d/2R)). Can
display nearly the entire sphere on a finite circle. The full
sphere requires an infinite map.
Other azimuthal projections are not true
perspective projections:
- Azimuthal
equidistant: r(d) = cd; it is used
by amateur radio operators to know the
direction to point their antennas toward a point and see the
distance to it. Distance from the tangent point on the map is
proportional to surface distance on the earth (; for the case where
the tangent point is the North Pole, see the flag of the United Nations)
- Lambert
azimuthal equal-area. Distance from the tangent point on the
map is proportional to straight-line distance through the earth:
r(d) =
c sin(d/2R)
- Logarithmic
azimuthal is constructed so that each point's distance from the
center of the map is the logarithm of its distance from the tangent
point on the Earth. Works well with cognitive maps . r(d) =
c ln(d/d_{0}); locations
closer than at a distance equal to the constant d_{0} are
not shown (, figure 6-5)
Projections by preservation of a metric property
Conformal
Conformal map projections preserve
angles locally:
Equal-area
These projections preserve area:
Equidistant
These preserve distance from some standard point or line:
Gnomonic
Great circles are displayed as straight
lines:
Retroazimuthal
Direction to a fixed location B (the bearing at the starting
location A of the shortest route) corresponds to the direction on
the map from A to B:
Compromise projections
Compromise projections give up the idea of perfectly preserving
metric properties, seeking instead to strike a balance between
distortions, or to simply make things "look right". Most of these
types of projections distort shape in the polar regions more than
at the equator:
Other noteworthy projections
See also
References
- Fran Evanisko, American River College, lectures for Geography
20: "Cartographic Design for GIS", Fall 2002
- Snyder, J.P., Album of Map Projections, United States
Geological Survey Professional Paper 1453, United States Government
Printing Office, 1989.
- This paper can be downloaded from USGS
pages
- Paul Andersons' Gallery of Map Projections - PDF
versions of numerous projections, created and released into the
Public Domain by Paul B. Anderson ... member of the International
Cartographic Association's Commission on Map Projections"]
External links
- A Cornucopia of Map Projections - A visualization of
distortion on a vast array of map projections in a single
image.
- G.Projector, free software by NASA GISS can render
many projections.
- Map Projections. The world we live in... HyperMaths.org: Sorted list
and descriptions
- RadicalCartography.net: Table of examples and
properties of all common projections
- UFF.br: An interactive JAVA applet to study deformations
(area, distance and angle) of map projections
- US Geological Survey overview
- USGS Map Projections: A Working Manual, freely
downloadable book by USGS with details on most
projections, including formulas and sample calculations.
- Map projections intro
- MathWorld's formulae
- Progonos.com: How Projections Work
- PDFs of projections
- Mapthematics: GIFs of projections
- U.S. WWII Newsmap, "Maps are Not True for All Purposes, These
are three of many projections", hosted by the UNT Libraries
Digital Collections
- BTInternet: Java applet for interactive
projections
- 3DSoftware: USGS info
- Geodesy, Cartography and Map Reading from Colorado
State University
- MapRef: A
collection of map projections and reference systems for
Europe
- KartoWeb: What is a map projection?
- NewMag: The World Turned Upside Down by Katy
Kramer
- PROJ.4
MapTools: Cartographic projections library
- GMT
(Generic Mapping Tools), for creating maps, processing data, and
learning first-hand about projections
- ESRI
publication.
- World Map Projections by Stephen Wolfram based on work by Yu-Sung
Chang, Wolfram
Demonstrations Project.
- B.J.S.Cahill Butterfly Map Resource Page:
Octahedral Map of the World