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# mapproj(n) 1.1 tcllib "Tcl Library"

## Name

mapproj - Map projection routines

## Synopsis

- package require Tcl ?8.5 9?
- package require math::interpolate ?1.1?
- package require math::special ?0.2.2?
- package require mapproj ?1.1?

- ::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 carrée (cylindrical equidistant) projection.

::mapproj::fromPlateCarree lambda_0 phi_0 x y

  Converts from the plate carrée (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 carrée, 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 carrée 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 © 2007 Kevin B. Kenny <kennykb@acm.org>