Introduction
to Map Projections
A map projection is the systematic arrangement of the earth’s (or generating
globe’s) parallels and meridians onto a plane surface. These meridians
and parallels become the projection graticule. The graticule takes on different
forms depending on the type of projection plane surface (or projection
family), the point or line of tangency, the aspect, and the direction of
an imaginary projection light source. The projection process also involves
the transformation of earth features such as coastlines and land boundaries.
All map projections have some type of distortion or deformation. Depending
on the projection properties, the distortion may be of area, shape, size,
distance, direction, or scale. No projection is free from all distortions,
but each contains only some distortion. The cartographer or mapmaker must
select a projection which will result in a minimum of distortion in relation
to the map theme or purpose, the amount of land area shown, and the portion
of the earth’s surface being represented on the map.
The reader is also referred to Peter Dana's Introduction
to Map Projections (The Geographers Craft, Virtual Geography Department,
University of Texas at Austin) for additional information on map projections.
Note that several of Dana's map projection examples are linked to relevant
portions of this page.
Map projections are grouped into three families: Cylindrical, Conic
and Azimuthal, with Pseudocylindrical projections forming
a variation on the Cylindrical Family. These families are based on the
configuration of the plane onto which the globe (sphere) is projected.
Each family is suitable for representing select areas of the globe. Each
family produces a different appearance of the grid (or graticule) on the
projected planar surface. And each family allows for tangent and secant
case or tangency with the globe. These and other projection concepts and
principles are explained below.
For graphic examples of projection families, Figure 2.16 on page 45
of the Dent Textbook (Cartography, Thematic Map Design, Fourth
Edition) illustrates examples of the three families of projection, their
grid appearance with the normal aspect, and patterns of deformation given
the tangent (simple) and secant tangency or case. The U.S.G.S. Map Projections
Poster provides examples of numerous projection types with varied properties
and suggested uses. And as mentioned above, several of Dana's projections
are linked to explanations below.
Cylindrical projections are formed by wrapping a large, flat plane
(e.g., a large sheet of paper) around the globe to form a cylinder. The
points on the spherical grid are transferred to the cylinder which is then
unfolded into a flat plane. The equator is the "normal aspect" or viewpoint
for these projections. This family of projections are typically used to
represent the entire world. When projected from the center of the globe
with the normal aspect, the typical grid appearance for cylindrical projections
shows parallels and meridians forming straight perpendicular lines. The
spacing varies depending on the type of cylindrical projection. Conformal
cylindrical projections are also used for large scale topographic mapping
since they enable measurements of angles and distance (see Conformality
below).
Click
here for an example of the Projection of a Sphere onto a Cylinder
with the Normal (Equatorial) Aspect and Tangent Case.
Within the cylindrical family are pseudocylindrical projections.
These "cylinders" are not rectangular, but rather, curve inwards at the
poles. The resulting grid thus shows straight line parallels and central
meridian (the meridian in the center of the projected map), and all other
meridians form arcs which are concave from the perspective of the central
meridian. Pseudocylindrical projections are also often used for world maps.
The Robinson
,
Mollweide,
Eckert,
and
Sinusoidal
Projections are examples of Pseudocylindrical Projections.
With Conic projections, points from the globe graticule are transferred
to a cone which has been enveloped around the sphere. The cone is then
unrolled into a flat plane. The normal aspect is the north or south pole
where the axis of the cone (the point) coincides with the pole. Conic projections
can only represent one hemisphere, or a portion of one hemisphere, for
the cone does not extend far beyond the center of the sphere. Conic projections
are often used to project areas that have a greater east-west extent than
north-south, e.g., the United States. When projected from the center of
the globe, the typical grid appearance for Conic projections shows parallels
forming arcs of circles facing up in the Northern Hemisphere and down in
the Southern Hemisphere; and meridians are either straight or curved and
radiate outwards from the direction of the point of the cone.
Click
here for an example of the Projection of a Sphere onto a Cone with
the Normal (Polar) Aspect and Tangent Case.
With Azimuthal projections, the spherical (globe) grid is projected
onto a flat plane, thus it is also called a plane projection. The poles
are the "normal aspect" (the viewpoint or perspective) which results in
the simplest projected grid for this family of projections. That is, the
plane is normally placed above the north or south pole. Normally only one
hemisphere, or a portion of it, is represented on Azimuthal projections.
When projected from the center of the globe with the normal aspect, the
typical grid appearance for Azimuthal projections shows parallels forming
concentric circles, while meridians radiate out from the center.
Click
here for an example of the Projection of a Sphere onto a Plane
with the Equatorial Aspect and Tangent Case.
-
Tangency or Case
Tangency or Case refers to the location or locations that
a projection surface touches or cuts through the globe. There are two types
of tangency: the Tangent Case and the Secant Case. Case is very important
because while all projections contain distortions, scale deformation is
virtually lacking at the point or line(s) of tangency. Distortion increases
away from tangency. It is important, therefore, when projecting the spherical
surface onto a map projection, to locate tangency on or near the area of
central focus or greatest interest.
With the tangent (simple) case, the projection surface (azimuthal
plane, cylindrical or conic surface) touches the globe at one point or
along one line as demonstrated in Fig. 2.16 on page 45 of the Dent textbook,
where the Azimuthal projection with normal (polar) aspect touches the north
or south pole, the cylindrical with normal (equatorial) aspect touches
the equator, and the cone with normal (polar) aspect touches a mid-latitude
parallel.
Additional graphic examples of the tangent case include the following
versions of the Normal
Cylindrical, Transverse
Cylindrical, Oblique
Cylindrical, Conic,
and Azimuthal
projections.
In secant case, the projection surface cuts through the globe to
touch the surface at two lines as demonstrated also in Fig. 2.16 where
the Azimuthal projection cuts the globe at a high-latitude parallel, the
Cylinder cuts at two mid-latitude parallels, and the Cone cuts at a high-latitude
parallel, and one just above the equator. Secant tangency is useful for
reducing distortion of larger land areas, e.g., large regions (such as
the United States), whole continents or the world.
Additional graphic examples of the secant case include the following
versions of the Cylindrical,
Conic,
and Azimuthal
projections.
-
Aspect
Each projection surface (family) can be positioned over the globe from
one of four aspects (also referred to as perspectives or viewpoints):
Polar,
Equatorial, Transverse or Oblique. The aspect may sometimes
be indicated in the name or description of a projection, e.g.,
Transverse
Mercator, Oblique
Orthographic, Space
Oblique Mercator, etc. The appearance of the graticule will vary
depending on the projection aspect. Aspect should be selected so that the
area of greatest interest takes central focus on the projected map. If
the North Pole is of greatest interest, for example, a polar aspect would
be chosen. Aspect, together with case, can help reduce distortion on map
projections.
With a Polar Aspect the projection surface is placed over the north
or south pole and the point or line of tangency is at or near that pole.
The North
Polar Stereographic projection illustrates the polar aspect.
The Equatorial Aspect places the projection surface over the equator.
With the tangent case the surface touches at the equator. The secant case
cuts through the globe above and below the equator but the perspective
is as if the globe were viewed from the equator. The Miller
Cylindrical projection is an example of an equatorial aspect. World
maps are most often projected from an equatorial aspect.
The Transverse Aspect places the projection surface 90 degrees from
the normal position, e.g., for a Polar Azimuthal projection the equator
would be the Transverse Aspect, while for an Equatorial Cylindrical projection
the poles would be the Transverse Aspect. A transverse
projection of a sphere onto a cylinder illustrates the transverse
aspect.
The Oblique Aspect of a projection surface is placed above or on
any position between, but not including, the equator and the poles. It
may be centered on a parallel or on a meridian. Oblique aspects are useful
for centering smaller regions on a map projection, thus reducing the map
distortion. A land area such as India, for example, would be distorted
more when projected from either an equatorial or a polar aspect than from
an oblique aspect directly above the Indian subcontinent. Examples of an
Oblique Aspect are shown in the Oblique
Mercator projection of Alaska, an Oblique
Aspect Orthographic projection, and in the Lambert
Conformal Conic projection of Texas.
-
Central Meridian
The Central Meridian is the meridian that passes through the center
of a projection. Like the case, distortion is minimized along the central
meridian. Thus, it is important to select a central meridian that runs
through the center of the area of interest on a map. For a map of Africa,
for example, 20 degrees East forms a suitable central meridian. Likewise
for a map of the world, the central meridian determines which continents
are placed in the center of the projection. Note, for example, that North
and South America are at the center of the world maps shown on the U.S.G.S.
Map Projections poster, while many wall maps of the world place Europe
and Africa in the center. Some examples of varied central meridians include:
a. Behrmanns
Cylindrical Equal Area projection of the world, with 0 degrees
as the Central Meridian;
b. Albers
Equal Area Conic projection of the United States with 96 degrees
West as the Central Meridian;
c. Lambert
Azimuthal Equal Area projection from a Polar Aspect with 0 and
180 degrees forming the Central Meridian.
Azimuthal projections are constructed from one of three perspectives
where for each it is as if a light source were shown upon the globe and
the arcs of the parallels and meridians were projected onto the flat, tangent,
straight line surface. The three projection perspectives are Gnomonic,
Orthographic, and Stereographic. The appearance of the graticule
differs for each perspective. The U.S.G.S. Map Projections Poster gives
good graphic examples of each.
The light source for the Gnomonic perspective is from the center
of the earth through to the spherical surface where it is projected onto
a plane.
In the Orthographic perspective, the spherical surface is transformed
to a projection plane from infinity, that is, as if a light source were
shown from an infinite distance through the globe and onto a planar surface.
An example of an orthographic projection is the Oblique
Aspect Orthographic Projection.
In the Stereographic projection the perspective is a point at the
opposite end of the globe. In other words, the light is a point source
shown from a point on the globe through to the other end of the globe (e.g.,
a South Pole point of projection would shine light through to the North
Pole). An example of a stereographic projection is the North
Polar Stereographic Projection.
-
Mathematical Properties
As noted in the overview, all map projections contain some types of distortion.
Such distortion may be of shape, area, distance, direction, or scale. Some
projections preserve shape and direction while distorting area. Others
maintain area but distort shape and scale. In many projections scale may
vary from place to place, and in all projections distortion will increase
away from the places of tangency. The types of distortions are a function
of the way the projection is constructed. Most projections have been derived
mathematically, thus the types of distortion are often a function of certain
mathematical relationships specific to a given projection. The most commonly
described mathematical relationships or properties are conformality,
equivalence (equal area) and equidistance. Such properties are
often indicated in the names of projections, e.g., Behrmann
Cylindrical Equal Area, Lambert
Conformal Conic, Albers
Equal Area Conic, and Lambert
Azimuthal Equal Area projection. Note that the name of each of
these projections contains the projection creator (Behrmann, etc.), the
projection family (cylindrical, etc.) and the mathematical relationship
(conformal, etc.).
The main mathematical relationships are described qualitatively below.
On a conformal projection scale is the same in every direction from
any point on the map, thus deformation of scale increases regularly in
all directions. Parallels and meridians intersect at right angles and the
shapes of very small areas ("orthomorphic"), and angles with very short
sides are preserved. As there is no angular deformation, and true angles
are maintained, angular measurements can be made from conformal projections.
These projections are useful for large scale mapping, especially for military
and other navigational uses where angular measurements are needed. Topographic
maps and navigational charts use conformal projections. These projections
are also commonly used for world general reference maps. Some examples
of Comformal projections include the Mercator,Lambert
Conformal Conic, and Space
Oblique Mercator projections.
Equal area or equivalent maps maintain true relationships
of areas. That is, at a given scale, for every part, as well as the whole,
map area is proportional to the corresponding area on the Earth. Deformation
occurs in elliptical fashion away from tangency thus shapes are distorted.
Because they maintain true areas, however, they are useful for comparing
regional distributions of geographic phenomenon, e.g., world maps of population
density, per capita income, literacy and various other human-oriented statistics
that can be cartographically portrayed using symbols per unit area ( e.g.,
choropleth and dot mapping techniques). Examples of Equal Area projections
include the Behrmann
Cylindrical Equal Area, Peters,Mollweide,
Albers
Conic Equal Area, and Lambert
Azimuthal Equal Area projections.
In the equidistant projections scale is preserved (not distorted)
in the direction perpendicular to the line of zero distortion or radially
outwards from a point of zero distortion. The name arises from the fact
that in the normal aspect of Cylindrical, Conic and Azimuthal projections
the principal scale is preserved along the meridians and therefore all
parallels on the map are equidistantly spaced. Uses of these projections
include measuring bearings and distances to other places in the world,
(thus useful for airline networks), and for representing very small areas,
such as a portion of a city, without scale distortion. The projection has
comparatively small amounts of angular deformation and the area scale does
not become excessively large. They are therefore a good compromise between
conformality and equivalence and are often used in atlases as the base
for general reference maps of countries or continents. The Equidistant
Conic and Azimuthal
Equidistant projections are examples of equidistance.
Certain projections offer a compromise between conformality, equivalence
and equidistance. In these projections there is some distortion of shape,
area, distance, direction and scale, but each is only distorted in moderate
amounts. The Robinson
Projection is an example of a compromise between all types of distortion.
This projection was developed by graphical rather than mathematical means.
References
Go to Introduction to
Map Projections above
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Created 6/5/97 by Laurie A. B. Garo. Last updated 9/20/00 by
lg.
The URL for this page is http://www.uncc.edu/lagaro/cwg/mapproj/intro_mp.html
