﻿ The Earth as an Ellipsoid

# The Earth as an Ellipsoid

For many maps, including nearly all maps in commercial atlases, it may be assumed that the Earth is a sphere. Actually, it is more nearly a slightly flattened sphere - an oblate ellipsoid of revolution, also called an oblate spheroid. This is an ellipse rotated about its shorter axis. The flattening of the ellipse for the Earth is only about one part in three hundred; but it is sufficient to become a necessary part of calculations in plotting accurate maps at a scale of 1:100,000 or larger, and is significant even for 1:5,000,000-scale maps of the United States, affecting plotted shapes by up to 2/3 percent. On small-scale maps, including single-sheet world maps, the oblateness is negligible.

The bad news is that the Earth is not an exact ellipsoid. In fact, because the Earth is such a "lumpy" ellipsoid no single smooth ellipsoid will provide a perfect reference surface for the entire Earth.

One way to get around the irregularity of the ellipsoid is to measure the Earth's shape in different areas and to then create different reference ellipsoids used for mapping different regions on Earth.

For example, the ellipsoid shown in a yellow grid above is a fair match to the Earth's surface (shown in solid blue) in some areas but not in others. In some areas the Earth's surface protrudes above the even ellipsoid shape and in other areas the Earth's surface is lower than the ellipsoid's surface. We can use the yellow ellipsoid for precision mapping in areas where the Earth's surface is a close match.

We can use a different ellipsoid (shown in magenta) to map other areas where the magenta ellipsoid is a better match to the Earth's surface. We can specify many different standard ellipsoids to map different areas of the Earth.

##### Offset Ellipsoids

A refinement to the idea of using different ellipsoids lets us use a standard collection of ellipsoids without having to create hundreds of different reference ellipsoids to fit all the different lumpy regions of the Earth. The refinement is to use the same ellipsoid in different areas, but to offset the ellipsoid slightly to make it a better match. For example, in the illustration above we could use the magenta ellipsoid as is for mapping northern regions and we could also use it in southern regions if we moved it up slightly.

To see how this works, consider a translucent Earth in blue color with the center of the Earth marked by the intersection of three green axes.

For any ellipsoid we choose to use we can mark the center of the ellipsoid as well. The ellipsoid is shown in dotted outline with yellow axes marking the center.

To achieve a better fit between a given ellipsoid and a particular region of the Earth we can offset the standard ellipsoid from the center of the Earth.

If we illustrate the situation with a grid ellipsoid and a solid Earth we can see the ellipsoid is above the Earth's surface in some areas and below it in others. In those regions where the ellipsoid closely follows the surface of the Earth we can use it for higher-accuracy mapping.

To get it to fit better in other areas we can move the ellipsoid relative to the Earth.

A datum is a reference ellipsoid together with an offset from the center of the Earth, and is often referred to as a base coordinate system or simply as a base.  By specifying different offsets, we can use the same standard ellipsoids in many different regions of the Earth. Different countries will often use the same ellipsoid but with different offsets for standard government maps in those countries. In Manifold, each such unique combination is listed as a different datum or base in Manifold projection dialogs. There are very many choices.

There are over a dozen principal ellipsoids that are frequently used by one or more countries. The choice box in Manifold projections dialogs will call up a table of many datums, most of which use one of the most common ellipsoids. There are over a dozen principal ellipsoids with slightly different dimensions as a result from varying accuracy in geodetic measurements, that is , measurements of locations on the Earth.

Differences also arise because the curvature of the Earth’s surface is not uniform due to irregularities in the gravity field and even regional phenomena such as the rebound of crust in regions where very thick and heavy glaciers pushed down the crust during the last Ice Age from which the world is still emerging. Therefore the ellipsoid one calculates to be a best fit for the surface of the Earth depends not only on how the measurements are made but also where they are made.

Until recently, ellipsoids were only fitted to the Earth’s shape over a particular country or continent so that in effect every datum used in various countries employed an offset ellipsoid as illustrated above. The discrepancy between centers is usually a few hundred meters at most. In more recent years satellite-determined coordinate systems, such as the WGS series, have resulted in geocentric ellipsoids.

The center of a geocentric ellipsoid is the same as the center of the Earth. Satellite-computed geocentric ellipsoids represent the entire Earth more accurately on an overall, average basis than ellipsoids determined from ground measurements but they do not generally give the "best fit" for a particular region.

The many choices for the datum or base in Manifold projection dialogs arise because many different countries over the years have used hundreds of different combinations of "standard" ellipsoids with different offsets. When creating new maps, pick the datum that is the one most used by other maps with which you will work. For general-purpose mapping, the safest choice is the Manifold default, the WGS 84 World Geodetic datum used for almost all GPS work.

##### Grids

In some regions the Earth is so lumpy and irregular that even a well-chosen ellipsoid suitably nudged off center does not provide a close enough match to the median surface of the Earth for high-accuracy mapping.    In such locations instead of using smooth, geometric surfaces such as the surface of an ellipsoid the coordinates of maps will be projected and transformed using grids, which are arrays of numbers that show the distortion of the median surface in the region covered by the grid.

The use of grids is generally driven by governments since usually only governments have the funds to create grids for various countries and regions within those countries and usually only governments can set the legal standards requiring the use of such grids for cartographic work of interest to those governments.   Governments typically also set the standard for how such grids are used to transform coordinates from one projection into the government-required one by publishing formulae for using the grids or software that implements those formulae.

The US uses what are called NADCON formulae (from the NADCON software package published by the US government) while many countries outside the US tend to use the Canadian equivalent NTv2 formulae.  Manifold supports both to enable use of all grids worldwide.

Many grids have been published by governments for use in various countries, regions, provinces and even for very local settings such as counties and even individual cities.  Most are openly published on Internet but a handful require payment of a fee or special licensing.    Some, such as some of the grids for Canadian provinces, were once published openly but since then are available only upon agreeing to a license.   In such cases if we can get a copy of the grid from someone who obtained it when it was published with an open license we can use it without agreeing to the later license.

Manifold includes all grids that are openly published now and also those which at some time in the past were published without restrictions.   All grids within Manifold that once were openly published but which now can be only obtained with restrictions were obtained from sources which obtained those grids legally at a time when no restrictions applied.   The resulting collection of grids in Manifold is as of this writing much larger than the built-in collection of grids in any other general-purpose GIS or DBMS package.

Of over 100 grids used in various localities throughout the world only a handful, perhaps three or four, cannot be obtained without paying a fee and agreeing to a license.  Those are all for local work in small regions, such as Bavaria or Brandenburg, and thus not having them built-in is not a big deal for most users.   If you have such grids Manifold can work with them.

##### Historical Notes

A frequently-used official Earth ellipsoid was defined in 1924, when the International Union of Geodesy and Geophysics (IUGG) adopted a flattening of exactly 1 part in 297 and a semi-major axis (or Equatorial radius) of exactly 6,378,388 m. The radius of the Earth along the polar axis is then 1/297 less than 6,378,388 or approximately 6,356,911.9 m. This is called the International ellipsoid and is based on John Fillmore Hayford’s calculations in 1909 from U.S. Coast and Geodetic Survey measurements made entirely within the United States. Although this datum is most accurate in the US it is amusing to note that this ellipsoid was adopted for international use and not adopted for use in North America. The datums still used in many countries employ the International ellipsoid together with a particular offset that best aligns the International ellipsoid with the Earth's surface in the region of that particular country.

##### The North American Datum of 1927 / NAD 27

The first official geodetic datum in the United States was the New England Datum, adopted in 1879. It was based on surveys in the Eastern and Northeastern states and referenced to the Clarke ellipsoid of 1866, with triangulation station Principio, in Maryland, as the origin. The first transcontinental arc of triangulation was completed in 1899, connecting independent surveys along the Pacific Coast of the U.S. In the intervening years, other surveys were extended to the Gulf of Mexico. The New England Datum was thus extended to the south and west without major readjustment of the surveys in the east. In 1901, this expanded network was officially designated the United States Standard Datum, and triangulation station Meades Ranch, in Kansas, was the origin. In 1913, after the geodetic organizations of Canada and Mexico formally agreed to base their triangulation networks on the United States network, the datum was renamed the North American Datum.

By the mid-1920’s, the problems of adjusting new surveys to fit into the existing network were acute. Therefore, during the 5-year period 1927-1932 all available primary data were adjusted into a system now known as the North American 1927 Datum. The coordinates of station Meades Ranch were not changed but the revised coordinates of the network comprised the North American 1927 Datum.

##### Recent Ellipsoids

The ellipsoid adopted for use in North America is the result of the 1866 evaluation by the British geodesist Alexander Ross Clarke using measurements made by others of meridian arcs in western Europe, Russia, India, South Africa, and Peru. This resulted in an adopted Equatorial radius of 6,378,206.4 m and a polar radius of 6,356,583.8 m, or an approximate flattening of 1/294.9787. Since Clarke is also known for an 1880 revision used in Africa, the Clarke 1866 ellipsoid is named with the year. Once again it is amusing to note that the ellipsoid used for the North American Datum is based on data compiled outside of North America and thus is a less accurate choice than the International ellipsoid.

Satellite tracking data have provided geodesists with new measurements to define the best Earth-fitting ellipsoid and for relating existing coordinate systems to the Earth’s center of mass. The Defense Mapping Agency’s efforts produce the World Geodetic System 1966 (WGS 66) followed by more recent evaluations (WGS 72, WGS 84).

The North American 1927 Datum has been replaced with a new datum, the North American Datum 1983 (NAD 83) that is Earth-centered based on satellite tracking data, using the Geodetic Reference System 1980 (GRS 80) ellipsoid, an ellipsoid very similar to that for the WGS 72.

##### Eccentricity

In most map projection formulas, some form of the eccentricity e is used rather than the flattening f ratio.

##### Other Planets and Satellites

For the mapping of other planets and natural satellites, only Mars and Earth among the inner planets is treated as an ellipsoid. The Moon, Mercury, Venus, and the satellites of Jupiter and Saturn are taken as spheres.

Numbers for radius (Major axis and minor axis are the same for a sphere) for many planets, minor planets and satellites may be found from a variety of Internet sites. Recent planetary missions have greatly improved accuracy.

## Notes

The illustrations for this topic are greatly exaggerated to illustrate the concepts. The actual differences between ellipsoids and the Earth's surface are rarely more than a few hundred meters.

When we talk about the Earth being "lumpy" we do not mean variations in height such as mountain ranges; instead, we are referring to what would be an irregular average sea level in that region.

It is difficult for humans to realize just how smooth the surface of the Earth really is on a large scale. Although we are impressed by the dramatic rise of mountains on a human scale, if the Earth were reduced to the size of a billiard ball the Earth would be much smoother than the billiard ball. The difference in height between the median surface and even the Himalayan Mountains is much less than the variations in smoothness of billiard balls.