Map Projections
Map projections
This is not meant to be an exhaustive treatment of map
projections, but deals with the basic concepts and some examples
that are available in MapInfo. More thorough treatment can be
found in resources listed elsewhere in the page.
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A map projection is the systematic
arrangement of the earth’s (or generating
sphere’s) surface onto a plane surface. This
process of representing the curved surface of a sphere is
often described as analogous to peeling an orange and
flattening out the skin.
A better analogy is to think of the sphere as
translucent, with a light source at its centre. The
projection is the shadow cast by the surface features of
the sphere. Clearly, the shadow, and therefore the
configuration of the surface feature shadows, will vary
according to the shape of the plane and where it is
placed. These variations are what accounts for the
differences between projections. These variations also
mean that the surface features are to some extent
distorted or deformed.
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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 projection should be be
selected which will result in a minimum of distortion in relation
to the map purpose, the amount of land area shown, and the
portion of the earth’s surface being represented on the map
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The earth’s (or generating globe’s)
surface, and locations on them, are normally defined by
reference to meridians of longitude (lines which run
longitudinally, from pole to pole and converge there,
called lines of longitude) and parallels of latitude
(which run laterally across the globe and are parallel). The
central line of latitude is referred to as the equator (0
degrees latitude), and the meridian of longitude that
runs through the Greenwich Observatory in London is
designated the Prime Meridian (0 degrees longitude).
These two lines define two intersecting planes from which
all other positions are measured in degrees.
In a projection, the lines of longitude and latitude
become the projection grid or graticule. The graticule
takes on different forms depending on the type of
projection plane surface (or projection group), 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.
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Projection Groups
Map projections are grouped into three groups: Cylindrical (including
Pseudocylindrical), Conic and Azimuthal. These
groups are based on the configuration of the plane onto which the
globe (sphere) is projected. Each group is suitable for
representing select areas of the globe. Each group produces a
different appearance of the grid (or graticule) on the projected
planar surface. And each group allows for tangent
and secant case or tangency with the globe.
These and other projection concepts and principles are explained
below.
Cylindrical Projections
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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 group 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. Within the cylindrical group
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.
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Conic Projections
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In 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. |
Azimuthal Projections
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With Azimuthal or plane projections,
the spherical (globe) grid is projected onto a flat
plane. The poles are the 'normal aspect' (the viewpoint
or perspective) which results in the simplest projected
grid for this group 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. |
Tangency
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.
Tangent Case
With the tangent (simple) case, the projection
surface (azimuthal plane, cylindrical or conic surface) touches
the globe at one point or along one line, 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.
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Secant Case
In secant case, the projection surface cuts
through the globe to touch the surface at two lines 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 near the equator. Secant
tangency is useful for reducing distortion of larger land
areas, e.g., large regions (such as the Australia), whole
continents or the world.
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Aspect
Each projection surface (group) 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 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. World maps are most often projected from
an equatorial aspect.
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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. |
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.
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 Australia, for example,
132 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 most maps of the world place Europe and Africa in the
center, using the Prime (Greenwhich) Meriadian as the centre.
Perspective in Azmuthal Projections
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.
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).
Distortion
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.
The main mathematical relationships are:
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.
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).
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.
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.
Non Traditional Projections
The Fuller Projection
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Chris Rywalt's animation illustrating
Buckminster Fuller's Dymaxion Map Projection of the
Earth. Fuller started with the data for the spherical
Earth surface. He projected the data from the sphere onto
an icosahedron -- the twenty- sided Platonic solid -- and
then unfolded that icosahedron out flat. The
animation is available in a small (320x240 QuickTime) and
large (640x480 QuickTime) versions. Download them from here:
Small
version UNFOLDS.MOV
Large
version UNFOLD.MOV
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The advantages of this method are
many: for one thing, it's possible to align the surface
data with the icosahedron in such a way that, when
unfolded, no landmass is cut into, which allows us to see
the Earth's landmasses as one continent; also, this
method results in nearly no distortion of either size or
shape of the landmasses, unlike most other
projections. |
Last updated: 08 March 2001 18:13
