**This ****applet****
demonstrates different ****map
projections.**** They can be scaled to different
sizes, or deliberately or randomly rotated. Experiment using the
mouse.** Or look at a topographic
Earth, Earth at night, Tissot's Indicatrix,
the Moon, Mars, or Jupiter.

**Some thoughts**(Return to top)

There is an earlier version here.

Good introductions to map projections include: the University of Colorado and the United States Geological Service. Hans Havlicek has an impressive gallery. There are useful collections of formulae at the Escola Universitària Politècnica Barcelona en español (e català) and at MathWorld. For links there is Oddens' bookmarks, aussi Liens Utiles en français, e Geogr@afia on line em português. Some other pages link here.

**Lines on the image:** In an attempt to
illustrate how different maps distort areas differently, the sea
in the image above shows a division of the earth into roughly
equal area "rectangles" (and "triangles" near
the poles, thought some projections show these as rectangles).
Lines of longditude (meridians) are shown roughtly about every 15
degrees (so at 0 and +/- 15, 30, 45, 60, 75, 90, 105, 120, 135,
150, 165, and 180 degrees) while lines of latitude (parallels)
are shown less evenly (at about 0 and +/- 14.6, 30, 48.6, and 90
degrees, i.e. the angles whose sines are 0 and +/- 1/4, 1/2, 3/4
and 1). Each rectangle (including those obscured by land)
therefore covers about 1/192 of the earth or just over 0.5%.

**Scaling:** Each projection shown in the applet
has its own particular properties. In some cases these can be
affected by scaling the X and Y axes on the map just by clicking
on the map to give a new right hand corner.

As an example, the Mercator projection is conformal and can be used for navigational purposes since angles on the map represent bearings on a compass (and so straight lines are loxodromes); any scaling needs to preserve this property - but since a complete Mercator projection is of infinite height, different rectangles can include different amounts of northern and southern extremities.

Similarly the polar azimuthal projections keep lines of lattitude as concentric circles, and this needs to be preserved in scaling; but for the infinite gnomonic and stereographic maps, different amounts of the sphere can be included at different scales (the gnomonic projection - which is conformal and projects great circles to straight lines - cannot even cover a hemisphere, while the streographic projection can almost cover the whole sphere but only at the cost of losing much of the interesting detail. When centred on the North Pole, the azimuthal equal distance projection is essentially the same, minus Antartica, as the symbol of the United Nations and, more curiously, as the prefered map of the Flat Earth Society.

For the plain Longitude/Latitude, Equal Area Cylindrical, Mollweide and Triangular projections, scaling can be done almost freely, enabling different lines of latitude to have distances consistent with those on the prime meridian near that line of latitude.

A scaled Equal Area Cylindrical projection is known as a "secant" rather than the basic "tangent" projection, but the difference is in fact minor. Despite this, the various names for this projection includes the Lambert Cylindrical Equal Area, Behrmann Cylindrical Equal Area, Tristran Edwards, Peters, Gall Orthographic, and Balthasart projections. The claims made for the originality and political importance of the Peters projection are particularly odd - there were already many equal area projections.

The Baar equal area projection produces various results in between the Cylindrical equal area and the Sinusoidal equal area projections. Specific examples include the Adams and Kavraisky V projections. This idea of finding intermediate projections is common, with the aim of reducing the worst distortions of the extremes.

Many conical projections can be seen as azimuthal projections which have been split and partially unwound.

There is a heart shaped equal area projection which starts with the Werner Projection, but can be unwound to produce a Sinusoidal projection. It is an alternative to the Bonne projection which also lies between the Werner and Sinusoidal projections.

The triangular projection is simply there to show that virtually anything is possible (its properties are that it preserves areas, it has lines of latitude as horizontal lines, and it fits inside a triangle - it is slightly similar to a sinusoidal projection at the top, and to the cylindrical projection at the bottom). Its use is not recomended for any purpose.

**Changing the centre of the map** can for
example enable Pacific islanders to feel as if they are at the
centre of the earth, rather than at the edge. Most maps of the
world tend to put the centre close to the west coast of Africa (where
the Greenwich meridian meets the equator), but this leads to
major distortions near the poles, and so it is common to produce
additional azimuthal projections with polar centres.

**Changing the direction.** A transverse
projection is at right angles to a standard projection. The
Transverse Mercator is currently popular for navigation (away
from the poles). A reverse projection is at two right angles (i.e.
upside down), perhaps so the penguins can feel superior to the
polar bears. An oblique projection is at any other angle, and is
harder to calculate. This version uses a different method to that
used in the earlier
applet and is probably marginally more accurate and faster
when producing oblique projections, but it is certainly slower
when rescaling.

Copyright 2002 Henry Bottomley. Any comments?

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