NAME¶
mapproj - Map projection routines
SYNOPSIS¶
package require
Tcl ?8.4?
package require
math::interpolate ?1.0?
package require
math::special ?0.2.1?
package require
mapproj ?1.0?
::mapproj::toPlateCarree lambda_0 phi_0 lambda
phi
::mapproj::fromPlateCarree lambda_0 phi_0 x y
::mapproj::toCylindricalEqualArea lambda_0 phi_0
lambda phi
::mapproj::fromCylindricalEqualArea lambda_0 phi_0 x
y
::mapproj::toMercator lambda_0 phi_0 lambda
phi
::mapproj::fromMercator lambda_0 phi_0 x y
::mapproj::toMillerCylindrical lambda_0 lambda phi
::mapproj::fromMillerCylindrical lambda_0 x y
::mapproj::toSinusoidal lambda_0 phi_0 lambda
phi
::mapproj::fromSinusoidal lambda_0 phi_0 x y
::mapproj::toMollweide lambda_0 lambda phi
::mapproj::fromMollweide lambda_0 x y
::mapproj::toEckertIV lambda_0 lambda phi
::mapproj::fromEckertIV lambda_0 x y
::mapproj::toEckertVI lambda_0 lambda phi
::mapproj::fromEckertVI lambda_0 x y
::mapproj::toRobinson lambda_0 lambda phi
::mapproj::fromRobinson lambda_0 x y
::mapproj::toCassini lambda_0 phi_0 lambda
phi
::mapproj::fromCassini lambda_0 phi_0 x y
::mapproj::toPeirceQuincuncial lambda_0 lambda phi
::mapproj::fromPeirceQuincuncial lambda_0 x y
::mapproj::toOrthographic lambda_0 phi_0 lambda
phi
::mapproj::fromOrthographic lambda_0 phi_0 x
y
::mapproj::toStereographic lambda_0 phi_0 lambda
phi
::mapproj::fromStereographic lambda_0 phi_0 x
y
::mapproj::toGnomonic lambda_0 phi_0 lambda
phi
::mapproj::fromGnomonic lambda_0 phi_0 x y
::mapproj::toAzimuthalEquidistant lambda_0 phi_0
lambda phi
::mapproj::fromAzimuthalEquidistant lambda_0 phi_0 x
y
::mapproj::toLambertAzimuthalEqualArea lambda_0 phi_0
lambda phi
::mapproj::fromLambertAzimuthalEqualArea lambda_0 phi_0
x y
::mapproj::toHammer lambda_0 lambda phi
::mapproj::fromHammer lambda_0 x y
::mapproj::toConicEquidistant lambda_0 phi_0 phi_1
phi_2 lambda phi
::mapproj::fromConicEquidistant lambda_0 phi_0 phi_1
phi_2 x y
::mapproj::toAlbersEqualAreaConic lambda_0 phi_0
phi_1 phi_2 lambda phi
::mapproj::fromAlbersEqualAreaConic lambda_0 phi_0
phi_1 phi_2 x y
::mapproj::toLambertConformalConic lambda_0 phi_0
phi_1 phi_2 lambda phi
::mapproj::fromLambertConformalConic lambda_0 phi_0
phi_1 phi_2 x y
::mapproj::toLambertCylindricalEqualArea lambda_0 phi_0
lambda phi
::mapproj::fromLambertCylindricalEqualArea lambda_0 phi_0
x y
::mapproj::toBehrmann lambda_0 phi_0 lambda
phi
::mapproj::fromBehrmann lambda_0 phi_0 x y
::mapproj::toTrystanEdwards lambda_0 phi_0 lambda
phi
::mapproj::fromTrystanEdwards lambda_0 phi_0 x
y
::mapproj::toHoboDyer lambda_0 phi_0 lambda
phi
::mapproj::fromHoboDyer lambda_0 phi_0 x y
::mapproj::toGallPeters lambda_0 phi_0 lambda
phi
::mapproj::fromGallPeters lambda_0 phi_0 x y
::mapproj::toBalthasart lambda_0 phi_0 lambda
phi
::mapproj::fromBalthasart lambda_0 phi_0 x y
DESCRIPTION¶
The
mapproj package provides a set of procedures for converting between
world co-ordinates (latitude and longitude) and map co-ordinates on a number
of different map projections.
COMMANDS¶
The following commands convert between world co-ordinates and map co-ordinates:
- ::mapproj::toPlateCarree lambda_0
phi_0 lambda phi
- Converts to the plate carr['e]e (cylindrical
equidistant) projection.
- ::mapproj::fromPlateCarree lambda_0
phi_0 x y
- Converts from the plate carr['e]e (cylindrical
equidistant) projection.
- ::mapproj::toCylindricalEqualArea lambda_0
phi_0 lambda phi
- Converts to the cylindrical equal-area projection.
- ::mapproj::fromCylindricalEqualArea lambda_0
phi_0 x y
- Converts from the cylindrical equal-area projection.
- ::mapproj::toMercator lambda_0 phi_0
lambda phi
- Converts to the Mercator (cylindrical conformal)
projection.
- ::mapproj::fromMercator lambda_0 phi_0
x y
- Converts from the Mercator (cylindrical conformal)
projection.
- ::mapproj::toMillerCylindrical lambda_0
lambda phi
- Converts to the Miller Cylindrical projection.
- ::mapproj::fromMillerCylindrical lambda_0
x y
- Converts from the Miller Cylindrical projection.
- ::mapproj::toSinusoidal lambda_0 phi_0
lambda phi
- Converts to the sinusoidal (Sanson-Flamsteed) projection.
projection.
- ::mapproj::fromSinusoidal lambda_0
phi_0 x y
- Converts from the sinusoidal (Sanson-Flamsteed) projection.
projection.
- ::mapproj::toMollweide lambda_0 lambda
phi
- Converts to the Mollweide projection.
- ::mapproj::fromMollweide lambda_0 x
y
- Converts from the Mollweide projection.
- ::mapproj::toEckertIV lambda_0 lambda
phi
- Converts to the Eckert IV projection.
- ::mapproj::fromEckertIV lambda_0 x
y
- Converts from the Eckert IV projection.
- ::mapproj::toEckertVI lambda_0 lambda
phi
- Converts to the Eckert VI projection.
- ::mapproj::fromEckertVI lambda_0 x
y
- Converts from the Eckert VI projection.
- ::mapproj::toRobinson lambda_0 lambda
phi
- Converts to the Robinson projection.
- ::mapproj::fromRobinson lambda_0 x
y
- Converts from the Robinson projection.
- ::mapproj::toCassini lambda_0 phi_0
lambda phi
- Converts to the Cassini (transverse cylindrical
equidistant) projection.
- ::mapproj::fromCassini lambda_0 phi_0
x y
- Converts from the Cassini (transverse cylindrical
equidistant) projection.
- ::mapproj::toPeirceQuincuncial lambda_0
lambda phi
- Converts to the Peirce Quincuncial Projection.
- ::mapproj::fromPeirceQuincuncial lambda_0
x y
- Converts from the Peirce Quincuncial Projection.
- ::mapproj::toOrthographic lambda_0
phi_0 lambda phi
- Converts to the orthographic projection.
- ::mapproj::fromOrthographic lambda_0
phi_0 x y
- Converts from the orthographic projection.
- ::mapproj::toStereographic lambda_0
phi_0 lambda phi
- Converts to the stereographic (azimuthal conformal)
projection.
- ::mapproj::fromStereographic lambda_0
phi_0 x y
- Converts from the stereographic (azimuthal conformal)
projection.
- ::mapproj::toGnomonic lambda_0 phi_0
lambda phi
- Converts to the gnomonic projection.
- ::mapproj::fromGnomonic lambda_0 phi_0
x y
- Converts from the gnomonic projection.
- ::mapproj::toAzimuthalEquidistant lambda_0
phi_0 lambda phi
- Converts to the azimuthal equidistant projection.
- ::mapproj::fromAzimuthalEquidistant lambda_0
phi_0 x y
- Converts from the azimuthal equidistant projection.
- ::mapproj::toLambertAzimuthalEqualArea
lambda_0 phi_0 lambda phi
- Converts to the Lambert azimuthal equal-area
projection.
- ::mapproj::fromLambertAzimuthalEqualArea
lambda_0 phi_0 x y
- Converts from the Lambert azimuthal equal-area
projection.
- ::mapproj::toHammer lambda_0 lambda
phi
- Converts to the Hammer projection.
- ::mapproj::fromHammer lambda_0 x
y
- Converts from the Hammer projection.
- ::mapproj::toConicEquidistant lambda_0
phi_0 phi_1 phi_2 lambda phi
- Converts to the conic equidistant projection.
- ::mapproj::fromConicEquidistant lambda_0
phi_0 phi_1 phi_2 x y
- Converts from the conic equidistant projection.
- ::mapproj::toAlbersEqualAreaConic lambda_0
phi_0 phi_1 phi_2 lambda phi
- Converts to the Albers equal-area conic projection.
- ::mapproj::fromAlbersEqualAreaConic lambda_0
phi_0 phi_1 phi_2 x y
- Converts from the Albers equal-area conic projection.
- ::mapproj::toLambertConformalConic lambda_0
phi_0 phi_1 phi_2 lambda phi
- Converts to the Lambert conformal conic projection.
- ::mapproj::fromLambertConformalConic lambda_0
phi_0 phi_1 phi_2 x y
- Converts from the Lambert conformal conic projection.
Among the cylindrical equal-area projections, there are a number of choices of
standard parallels that have names:
- ::mapproj::toLambertCylindricalEqualArea
lambda_0 phi_0 lambda phi
- Converts to the Lambert cylindrical equal area projection.
(standard parallel is the Equator.)
- ::mapproj::fromLambertCylindricalEqualArea
lambda_0 phi_0 x y
- Converts from the Lambert cylindrical equal area
projection. (standard parallel is the Equator.)
- ::mapproj::toBehrmann lambda_0 phi_0
lambda phi
- Converts to the Behrmann cylindrical equal area projection.
(standard parallels are 30 degrees North and South)
- ::mapproj::fromBehrmann lambda_0 phi_0
x y
- Converts from the Behrmann cylindrical equal area
projection. (standard parallels are 30 degrees North and South.)
- ::mapproj::toTrystanEdwards lambda_0
phi_0 lambda phi
- Converts to the Trystan Edwards cylindrical equal area
projection. (standard parallels are 37.4 degrees North and South)
- ::mapproj::fromTrystanEdwards lambda_0
phi_0 x y
- Converts from the Trystan Edwards cylindrical equal area
projection. (standard parallels are 37.4 degrees North and South.)
- ::mapproj::toHoboDyer lambda_0 phi_0
lambda phi
- Converts to the Hobo-Dyer cylindrical equal area
projection. (standard parallels are 37.5 degrees North and South)
- ::mapproj::fromHoboDyer lambda_0 phi_0
x y
- Converts from the Hobo-Dyer cylindrical equal area
projection. (standard parallels are 37.5 degrees North and South.)
- ::mapproj::toGallPeters lambda_0 phi_0
lambda phi
- Converts to the Gall-Peters cylindrical equal area
projection. (standard parallels are 45 degrees North and South)
- ::mapproj::fromGallPeters lambda_0
phi_0 x y
- Converts from the Gall-Peters cylindrical equal area
projection. (standard parallels are 45 degrees North and South.)
- ::mapproj::toBalthasart lambda_0 phi_0
lambda phi
- Converts to the Balthasart cylindrical equal area
projection. (standard parallels are 50 degrees North and South)
- ::mapproj::fromBalthasart lambda_0
phi_0 x y
- Converts from the Balthasart cylindrical equal area
projection. (standard parallels are 50 degrees North and South.)
ARGUMENTS¶
The following arguments are accepted by the projection commands:
- lambda
- Longitude of the point to be projected, in degrees.
- phi
- Latitude of the point to be projected, in degrees.
- lambda_0
- Longitude of the center of the sheet, in degrees. For many
projections, this figure is also the reference meridian of the
projection.
- phi_0
- Latitude of the center of the sheet, in degrees. For the
azimuthal projections, this figure is also the latitude of the center of
the projection.
- phi_1
- Latitude of the first reference parallel, for projections
that use reference parallels.
- phi_2
- Latitude of the second reference parallel, for projections
that use reference parallels.
- x
- X co-ordinate of a point on the map, in units of Earth
radii.
- y
- Y co-ordinate of a point on the map, in units of Earth
radii.
RESULTS¶
For all of the procedures whose names begin with 'to', the return value is a
list comprising an
x co-ordinate and a
y co-ordinate. The
co-ordinates are relative to the center of the map sheet to be drawn, measured
in Earth radii at the reference location on the map. For all of the functions
whose names begin with 'from', the return value is a list comprising the
longitude and latitude, in degrees.
CHOOSING A PROJECTION¶
This package offers a great many projections, because no single projection is
appropriate to all maps. This section attempts to provide guidance on how to
choose a projection.
First, consider the type of data that you intend to display on the map. If the
data are
directional (
e.g., winds, ocean currents, or magnetic
fields) then you need to use a projection that preserves angles; these are
known as
conformal projections. Conformal projections include the
Mercator, the Albers azimuthal equal-area, the stereographic, and the Peirce
Quincuncial projection. If the data are
thematic, describing properties
of land or water, such as temperature, population density, land use, or
demographics; then you need a projection that will show these data with the
areas on the map proportional to the areas in real life. These so-called
equal area projections include the various cylindrical equal area
projections, the sinusoidal projection, the Lambert azimuthal equal-area
projection, the Albers equal-area conic projection, and several of the
world-map projections (Miller Cylindrical, Mollweide, Eckert IV, Eckert VI,
Robinson, and Hammer). If the significant factor in your data is distance from
a central point or line (such as air routes), then you will do best with an
equidistant projection such as
plate carr['e]e, Cassini,
azimuthal equidistant, or conic equidistant. If direction from a central point
is a critical factor in your data (for instance, air routes, radio antenna
pointing), then you will almost surely want to use one of the azimuthal
projections. Appropriate choices are azimuthal equidistant, azimuthal
equal-area, stereographic, and perhaps orthographic.
Next, consider how much of the Earth your map will cover, and the general shape
of the area of interest. For maps of the entire Earth, the cylindrical equal
area, Eckert IV and VI, Mollweide, Robinson, and Hammer projections are good
overall choices. The Mercator projection is traditional, but the extreme
distortions of area at high latitudes make it a poor choice unless a conformal
projection is required. The Peirce projection is a better choice of conformal
projection, having less distortion of landforms. The Miller Cylindrical is a
compromise designed to give shapes similar to the traditional Mercator, but
with less polar stretching. The Peirce Quincuncial projection shows all the
continents with acceptable distortion if a reference meridian close to +20
degrees is chosen. The Robinson projection yields attractive maps for things
like political divisions, but should be avoided in presenting scientific data,
since other projections have moe useful geometric properties.
If the map will cover a hemisphere, then choose stereographic,
azimuthal-equidistant, Hammer, or Mollweide projections; these all project the
hemisphere into a circle.
If the map will cover a large area (at least a few hundred km on a side), but
less than a hemisphere, then you have several choices. Azimuthal projections
are usually good (choose stereographic, azimuthal equidistant, or Lambert
azimuthal equal-area according to whether shapes, distances from a central
point, or areas are important). Azimuthal projections (and possibly the
Cassini projection) are the only really good choices for mapping the polar
regions.
If the large area is in one of the temperate zones and is round or has a
primarily east-west extent, then the conic projections are good choices.
Choose the Lambert conformal conic, the conic equidistant, or the Albers
equal-area conic according to whether shape, distance, or area are the most
important parameters. For any of these, the reference parallels should be
chosen at approximately 1/6 and 5/6 of the range of latitudes to be displayed.
For instance, maps of the 48 coterminous United States are attractive with
reference parallels of 28.5 and 45.5 degrees.
If the large area is equatorial and is round or has a primarily east-west
extent, then the Mercator projection is a good choice for a conformal
projection; Lambert cylindrical equal-area and sinusoidal projections are good
equal-area projections; and the
plate carr['e]e is a good equidistant
projection.
Large areas having a primarily North-South aspect, particularly those spanning
the Equator, need some other choices. The Cassini projection is a good choice
for an equidistant projection (for instance, a Cassini projection with a
central meridian of 80 degrees West produces an attractive map of the
Americas). The cylindrical equal-area, Albers equal-area conic, sinusoidal,
Mollweide and Hammer projections are possible choices for equal-area
projections. A good conformal projection in this situation is the Transverse
Mercator, which alas, is not yet implemented.
Small areas begin to get into a realm where the ellipticity of the Earth affects
the map scale. This package does not attempt to handle accurate mapping for
large-scale topographic maps. If slight scale errors are acceptable in your
application, then any of the projections appropriate to large areas should
work for small ones as well.
There are a few projections that are included for their special properties. The
orthographic projection produces views of the Earth as seen from space. The
gnomonic projection produces a map on which all great circles (the shortest
distance between two points on the Earth's surface) are rendered as straight
lines. While this projection is useful for navigational planning, it has
extreme distortions of shape and area, and can display only a limited area of
the Earth (substantially less than a hemisphere).
KEYWORDS¶
geodesy, map, projection
COPYRIGHT¶
Copyright (c) 2007 Kevin B. Kenny <kennykb@acm.org>