Map Projections

## Basic Definitions and Concepts

### The Shape of the Earth

Since a map is a representation, the original shape of the represented subject must first be defined. An important branch of cartography, geodesy studies the Earth shape and how it is related to its surface's features.

#### Spheres, spheroids and geoids

Geoid, from the Greek for "Earth-shaped", is the common definition of our world's shape. This recursive description is necessary because no simple geometric shape matches the Earth:

• like in all space bodies above a certain mass, the Earth's materials aggregated in a spherical shape, which minimizes gravity and potential energy
• however, quick rotation around its axis caused a bulging at the middle (Equator) and a flattening at the poles; the resulting shape is called an spheroid or oblate ellipsoid. The equatorial diameter is nearly 1/300 longer than polar diameter
• on average, the surface rests perpendicular to the gravitational force at each point, which also influences land level. However, mass concentration is not uniform, due to irregular crust density and land distribution. In addition to the rotation bulge, some researchers concluded that the southern hemisphere is expanded and its pole depressed, while the other half is compressed with a raised pole (the resulting shape resembles a pear, but average distortion of curvature does not exceed 50 meters). Others have the opinion that the Equator itself is elliptical
• finally, the surface is not smooth, further complicating the shape

Taken in account, those factors greatly complicate the cartographer's job but, depending on the task, some irregularities can be ignored. For instance, although important locally, terrain levels are minuscule in planetary scale: the tallest land peak stands less than 9km above sea level, or nearly 1/1440 of Earth diameter; the depth of the most profound sea abyss is roughly 1/1150 diameter.

For maps covering very large areas, especially worldwide, the Earth may be assumed perfectly spherical, since any shape imprecision is dwarfed by unavoidable errors in data and media resolution. This assumption holds for most of this document. Conversely, for very small areas terrain features dominate and measurements can be based on a flat Earth.

#### The Datum

For highly precise maps of smaller regions, the basic ellipsoidal shape can not be ignored. A geodetic datum is a set of parameters (including axis lengths and offset from true center of the Earth) defining a reference ellipsoid. For each mapped region, a different datum can be carefully chosen so that it best matches average sea level, therefore terrain features. Thus, data acquisition for a map involves surveying, or measuring heights and distances of reference points as deviations from a specific datum (a delicate task: due to mentioned irregularities, gravity - and therefore plumb bobs and levers - is not always aligned towards the center of the Earth).

Several standard datums were adopted for regional or national maps. International datums do exist, but may not fit any particular area as well as a local one.

### Coordinate Systems

#### Latitudes and Longitudes

Although the Earth is a three-dimensional object, when supposed spherical its surface has a constant radius, so any point on it is uniquely identified using a polar two-coordinate system.

 Wooden sphere with an octant removed for clarity; copper arrows define the coordinate system's origins. The white "point" is located by two angles or coordinates: its latitude and longitude. Every point has a counterpart directly on the opposite side, called its antipode (not shown here). Selected parallels (in red) and meridians (in blue), here spaced 15° apart, comprise a spherical graticule. The number of possible parallels and meridians if infinite; how many should be presented depend on the globe's (or map's) purpose and size.

Given a polar axis (around which the planet rotates daily), an orthogonal plane which divides the globe in halves (i.e., an equatorial plane) and an arbitrary reference axis on it, any surface point determines a latitude, or the smallest angle, measured from the center of the Earth, from it towards the equatorial plane, and a longitude, or the smallest angle from the arbitrary axis to the projection of the point on the Equator determined by the latitude.

A graticule is a spherical grid of coordinate lines over the planetary surface, comprising circles on planes normal - i.e., perpendicular to the north-south axis - called parallels, and semicircular arcs with that axis as chord, called meridians. True to their name, no parallels ever cross one another, while all meridians meet at each geographic pole. Every parallel crosses every meridian at an angle of 90°. This and other properties help assessing map distortion.

Both sets of parallels and meridians are infinite, but of course only a subset can be included in any map. A point's latitude and longitude, both usually measured in degrees, define the crossing of a parallel and a meridian, respectively. So, latitudes mean north-to-south angles from the equatorial plane, while longitudes express west-to-east angles from a particular meridian defined by the reference axis. Latitudes conventionally range from 90° South to 90° North, while longitudes range from 180° West to 180° East. reference axis.

#### Parallels and Their Properties

A natural reference, the longest parallel divides the Earth in two equal hemispheres, north and south; thus its name, Equator. Four other important parallels are defined by astronomical constraints. The geographical north-south axis is actually tilted slightly less than 23.5° from the plane of the Earth's orbit around the sun. This accounts for the different seasons and different lengths of day and night periods throughout the year.

 Schematic cross section of Earth's orbit. AT is the axial tilt, about 23.5°

Every year about December 21st, the solar rays fall vertically upon a parallel near 23.5°S. That is the longest day in the southern hemisphere (notice how most of it is exposed to the sun, so that date is called the southern summer solstice), but the shortest day in the northern hemisphere (therefore winter solstice); not only shorter daylight periods but a shallower angle of incidence of solar rays explain the lower temperatures north of Equator.

Near June 21st, a similar phenomenon happens along the parallel opposite North. By definition, these two parallels encircle the torrid or tropical zone; they are named after the zodiacal constellations where the sun is at those dates, thus Tropic of Capricorn (south) and Tropic of Cancer (north). In regions south of the Tropic of Capricorn the sun around noon appears to run always north of the observer; at the same hour, in places north of the Tropic of Cancer the sun runs always south of the observer, while in tropical regions the sun appears sometimes south, sometimes north, depending on the season.

Subtracting the axial tilt from 90° we get the latitudes of the Arctic (about 66.5°N) and Antarctic (about 66.5°S) polar circles. Around December 21 the sun does not set at the Antarctic circle for a full day. Going south, we get even longer consecutive daylight periods, up to six months at the pole. There are correspondingly long nights at the Antarctic winter. Of course, the same occurs at the northern latitudes, with a six-month offset.

Points on the same parallel suffer similar rates of exposure to the sun, therefore are prone to similar climates (disregarding other factors like altitude, wind/sea conditions and terrain).

A point's latitude can be inferred from the sun's angle above the horizon at noon (the moment when the sun appears highest at the sky and a vertical stake projects its shortest shadow); Sailors use instruments like the sextant for measuring it.

 East-west distances between points separated by one minute of solar time at different latitudes. Distance is zero at poles, where one "sees" every moment of the day at any time.

#### Meridians and Their Properties

All points on a meridian have the same solar, or local, time.  Due to different day lengths throughout the year, correction formulas are applied to convert it to a local mean time.  Since it would be impractical having nearby regions with different time reckonings (one nautical mile, approximately 1853 meters, corresponds to an angle of 0°1' along the Equator, or a temporal difference of 4 seconds), the world is divided in 24 fuses, or time zones, each 15° wide.  For everyday purposes, every point inside a zone is considered having the same standard time (actually, a few countries still use solar time).  In practice, the time-jumping boundaries seldom follow the meridians, bending (usually at national or regional borderlines) to keep related places conveniently synchronized.

Unlike the Equator, there's no easily defined prime or "main" meridian, which was fixed (mainly by political consensus) in 1884 over the Royal Observatory in Greenwich, near London, UK.  This choice's only obvious advantage is setting the opposite meridian (near or at the left or right edges of many world maps) away from most inhabited areas. That opposite meridian is the base  of the international date line, which separates world halves in two different days. Again, this line is somewhat  irregular in order to keep national territories (mostly Pacific islands) in a single timezone.
Compared to finding a point's latitude, getting its longitude is a much more involved procedure, usually comparing the time separating the noon at the reference meridian and at the point in question.

 www.progonos.com/furuti    December 19, 2012