# Reference ellipsoid

A reference ellipsoid is an ellipsoid flattened at the poles , usually an ellipsoid of revolution , which is used as a reference system for calculating surveying networks or for directly specifying geographic coordinates. As a mathematical figure of the earth, it should approximate the area of ​​constant height (see geoid ), with the historical development going from regional degree measurement to global adjustment of the gravitational field .

## history

The globe has been a scientifically recognized model of the earth since Greek natural philosophy . The first doubts about the exact spherical shape emerged in the 17th century; Around 1680 Isaac Newton was able to theoretically prove in a dispute with Giovanni Domenico Cassini and the Paris Academy that the earth's rotation must cause a flattening at the poles and not at the equator (see elongated ellipsoid ). The survey of France by Philippe de La Hire and Jacques Cassini (1683–1718) initially suggested the opposite. The empirical proof was only achieved in the middle of the 18th century by Pierre Bouguer and Alexis-Claude Clairaut , when the measurements of the expeditions to Peru (today's Ecuador ) and Lapland (1735–1741) were unequivocally evaluated. This first precise degree measurement also led to the definition of the meter as the 10 millionth part of the earth's quadrant , which, however, was “too short” by 0.022% due to inevitable small measurement errors .

In the 19th century, numerous mathematicians and geodesists began to deal with the determination of the ellipsoid dimensions. The determined values ​​of the equatorial radius varied between 6376.9 km ( Jean-Baptiste Joseph Delambre 1810) and 6378.3 km ( Clarke 1880 ), while the widely accepted Bessel ellipsoid resulted in 6377.397 km (the modern reference value is 6378.137 km). The fact that the differences exceeded the measurement accuracy of that time by five times is due to the location of the individual surveying networks on differently curved regions of the earth's surface (see vertical deviation ).

The flattening values , however, varied less - between 1: 294 and 1: 308, which means ± 0.5 km in the polar axis. Here Bessel's value (1: 299.15) was by far the best. Due to ever larger surveying networks, the result "leveled off" in the 20th century at around 1: 298.3 ( Friedrich Robert Helmert 1906, Feodossi Krassowski 1940), which corresponds to a 21.4 km difference between the equatorial and polar axes , while the Hayford Ellipsoid with 1: 297.0 was clearly out of line due to the type of geophysical reduction. Due to the great US influence after World War II, it was nevertheless chosen as the basis of the ED50 reference system, while the " Eastern Bloc " took the Krassowski values ​​as the norm. The latter were confirmed as the better ones in the 1970s by the global satellite network and global multilateralation (time of flight measurements on signals from quasars and geodetic satellites).

## Reference ellipsoids in practice

Reference ellipsoids are used by geodesists for calculations on the earth's surface and are also the most common reference system for other geosciences . Every regional administration and surveying of a state needs such a reference ellipsoid in order to

## Reference ellipsoids in theory

Since the physical earth figure, the geoid , has slight waves due to the irregularities of the earth's surface and gravitational field , calculations on a geometrically defined earth figure are much easier. The objects to be measured are projected vertically onto the ellipsoid and can then even be viewed on a small scale as in a plane . For this z. B. a Gauss-Krüger coordinate system is used.

The height indicates the distance to the ellipsoid, perpendicular to its surface. However, this perpendicular differs by the so-called. Perpendicular error of the real plumb line as a plumb line would pose. For measurements that are supposed to be more precise than a few decimeters per kilometer, this effect must be calculated and the measurements reduced by it. Depending on the terrain and geology, the vertical deviation in Central Europe can be 10–50 ″ and causes a difference between astronomical and ellipsoidal longitude and latitude ( or ). ${\ displaystyle h}$${\ displaystyle \ lambda}$${\ displaystyle \ varphi}$

### Conversion into geocentric Cartesian coordinates

In a geocentric, right-angled reference system, the origin of which is in the center of the ellipsoid of revolution and is aligned in the direction of the axis of rotation ( ) and the prime meridian ( ), then applies ${\ displaystyle Z}$${\ displaystyle X}$

{\ displaystyle {\ begin {aligned} X & = (N _ {\ varphi} + h) \ cos \ varphi \ cdot \ cos \ lambda \\ Y & = (N _ {\ varphi} + h) \ cos \ varphi \ cdot \ sin \ lambda \\ Z & = (N _ {\ varphi} (1- \ varepsilon ^ {2}) + h) \ sin \ varphi \ end {aligned}}}

With

• ${\ displaystyle a}$ - major semi-axis (parameters of the reference ellipsoid)
• ${\ displaystyle b}$ - small semi-axis (parameters of the reference ellipsoid)
• ${\ displaystyle \ varepsilon = {\ frac {\ sqrt {a ^ {2} -b ^ {2}}} {a}}}$- numerical eccentricity
• ${\ displaystyle N _ {\ varphi} = {\ frac {a} {\ sqrt {1- \ varepsilon ^ {2} \ sin ^ {2} \ varphi}}} \, \!}$- radius of curvature of the first vertical , d. H. the distance of the plumb line from the intersection of the extended plumb line with the Z axis.

### Calculation of φ , λ and h from Cartesian coordinates

The ellipsoidal length can be determined exactly as ${\ displaystyle \ lambda}$

${\ displaystyle \ lambda = \ arctan \ left [{\ frac {Y} {X}} \ right]}$

For a given , the height is as ${\ displaystyle \ varphi}$${\ displaystyle h}$

${\ displaystyle h = {\ frac {\ sqrt {X ^ {2} + Y ^ {2}}} {\ cos \ varphi}} - N _ {\ varphi}}$

Although this relationship is exact, the formula offers itself

${\ displaystyle h \ approx {\ sqrt {X ^ {2} + Y ^ {2}}} \ cdot \ cos \ varphi + Z \ sin \ varphi -a ^ {2} / N _ {\ varphi}}$

rather for practical calculations because of the error

${\ displaystyle \ Delta h \ approx {\ frac {1} {2}} (h + {\ sqrt {X ^ {2} + Y ^ {2}}}) (\ Delta \ varphi) ^ {2}}$

depends only on the square of the error in . The result is therefore several orders of magnitude more accurate. ${\ displaystyle \ varphi}$

For the calculation of recourse to approximation methods must. Due to the rotational symmetry, the problem is moved to the XZ plane ( ). In the general case, is then replaced by . ${\ displaystyle \ varphi}$${\ displaystyle \ lambda = 0}$${\ displaystyle X}$${\ displaystyle {\ sqrt {X ^ {2} + Y ^ {2}}}}$

Ellipse with a circle of curvature. The length of the green route is .${\ displaystyle N _ {\ varphi}}$

The perpendicular of the searched point on the ellipse has the slope . The extended perpendicular goes through the center M of the circle of curvature , which touches the ellipse at the base of the perpendicular. The coordinates of the center are ${\ displaystyle (X, Z)}$${\ displaystyle \ tan \ varphi}$

${\ displaystyle {\ begin {pmatrix} X_ {M} \\ Z_ {M} \ end {pmatrix}} = {\ begin {pmatrix} \ varepsilon ^ {2} a \ cos ^ {3} t \\ - { \ tilde {\ varepsilon}} ^ {2} b \ sin ^ {3} t \ end {pmatrix}}}$

With

• ${\ displaystyle t}$- parametric latitude , d. i.e., points on the ellipse are described by${\ displaystyle \ left ({\ begin {smallmatrix} a \ cos t \\ b \ sin t \ end {smallmatrix}} \ right)}$
• ${\ displaystyle {\ tilde {\ varepsilon}} = {\ frac {\ sqrt {a ^ {2} -b ^ {2}}} {b}}}$

This applies

${\ displaystyle \ tan \ varphi = {\ frac {Z + {\ tilde {\ varepsilon}} ^ {2} b \ sin ^ {3} t} {X- \ varepsilon ^ {2} a \ cos ^ {3} t}}}$

This is an iterative solution because there and over are related. An obvious starting value would be ${\ displaystyle \ varphi}$${\ displaystyle t}$${\ displaystyle \ tan \ varphi = {\ frac {a} {b}} \ tan t}$

${\ displaystyle \ tan t_ {0} = {\ frac {a} {b}} {\ frac {Z} {X}}}$.

With this choice, an accuracy of is achieved after one iteration step

${\ displaystyle \ Delta \ varphi \ approx {\ frac {3} {2}} \ varepsilon ^ {3} {\ frac {ah ^ {2}} {(a + h) ^ {3}}} \ sin ^ {3} \ varphi \ cos ^ {3} \ varphi}$.

This means that on the earth's surface the maximum error is 0.00000003 ″ and the global maximum error (at ) is 0.0018 ″. ${\ displaystyle \ varphi}$${\ displaystyle h = 2a}$

With a good choice of , the maximum error for points in space can be reduced even further. With ${\ displaystyle t_ {0}}$

${\ displaystyle \ tan t_ {0} = {\ frac {b} {a}} {\ frac {Z} {X}} \ left (1 + {\ tilde {\ varepsilon}} {\ frac {b} { \ sqrt {X ^ {2} + Z ^ {2}}}} \ right)}$

the angle (for the parameters of the earth) is determined with an accuracy of 0.0000001 ″ by inserting it once into the iteration formula (regardless of the value of ). ${\ displaystyle \ varphi}$${\ displaystyle h}$

## Important reference ellipsoids

The ellipsoids used in different regions of the shape and size are generally produced by its semi-major axis and the flattening (Engl. Flattening ) set. Furthermore, that centrally located “ fundamental point ” has to be defined on which the reference ellipsoid touches the geoid and thus gives it an unambiguous altitude . Both definitions together are called " geodetic datum ". ${\ displaystyle a}$ ${\ displaystyle f}$

Even if two countries use the same ellipsoid (e.g. Germany and Austria the Bessel ellipsoid ), they differ in this central point or fundamental point . Therefore, the coordinates of the common border points can differ by up to one kilometer.

The axes of the ellipsoids differ by up to 0.01% depending on the region from whose measurements they were determined. The increase in accuracy in determining the flattening (difference in the ellipsoidal axes around 21 km) is related to the launch of the first artificial satellites . These showed very clear path disturbances with regard to the paths that had been calculated in advance. On the basis of the errors one could calculate back and determine the flattening more precisely. ${\ displaystyle f = (from) / a}$

Regional ellipsoids 1810–1906 and globally determined earth ellipsoids 1924–1984
and development of knowledge of the mean equatorial radius and flattening of the earth
Ellipsoid year Major semi-axis a
[meter]
Small semi-axis b
[meters]
Number = 1 / flattening
( n = 1 / f = a / ( a - b ))
Remarks EPSG code
Delambre , France 1810 6 376 985 308.6465 Pioneering work
Schmidt 1828 6,376,804.37 302.02 Pioneering work
GB Airy 1830 6,377,563.4 6,356,256.91 299,3249646
Airy modified in 1830 1830 6,377,340,189 6,356,034,447 299,3249514 EPSG :: 7002
Everest (India) 1830 6,377,276,345 300,8017 EPSG :: 7015
Bessel 1841 1841 6,377,397.155 6,356,078.963 299.1528128 ideally adapted in Eurasia; often used in Central Europe EPSG :: 7004
Clarke 1866 6,378,206,400 294.9786982 ideally adapted in Asia EPSG :: 7008
Clarke 1880 / IGN 1880 6,378,249.17 6,356,514.99 293.4663 EPSG :: 7011
Friedrich Robert Helmert 1906 6,378,200,000 298.3 EPSG :: 7020
Australian Nat. 6,378,160,000 298.25 EPSG :: 7003
Modif. Fisherman 1960 6,378,155,000 298.3
Boarding school 1924 Hayford 1924 6,378,388,000 6,356 911,946 297.0 ideally adapted, published in America
as early as 1909
EPSG :: 7022
Krassowski 1940 6,378,245,000 6,356,863.019 298.3 EPSG :: 7024
Boarding school 1967 Lucerne 1967 6,378,165,000 298.25
SAD69 (South America) 1969 6,378,160,000 298.25
WGS72 (World Geodetic System 1972) 1972 6,378,135,000 ≈ 6,356,750.52 298.26 EPSG :: 7043
GRS 80 ( Geodetic Reference System 1980 ) 1980 6,378,137,000 ≈ 6 356 752.3141 298.257222101 EPSG :: 7019
WGS84 ( World Geodetic System 1984 ) 1984 6,378,137,000 ≈ 6 356 752.3142 298.257223563 for GPS - survey EPSG :: 7030

The Bessel ellipsoid is ideally adapted for Eurasia, so that its “800 m error” is favorable for the geodesy of Europe - similar to the opposite 200 m of the Hayford ellipsoid (after John Fillmore Hayford ) for America.

For many countries of Central Europe that is Bessel ellipsoid important also the ellipsoids of Hayford and Krasovsky (spelling inconsistent), and GPS - surveying the WGS84 .

The results of Delambre and von Schmidt are pioneering work and are based on only limited measurements. On the other hand, the big difference between Everest (Asia) and Hayford (America) arises from the geological geoid curvature of different continents. Hayford was able to eliminate part of this effect by mathematically reducing the isostasy , so that its values ​​were considered better than the European comparison values ​​at the time.

## literature

4. ${\ displaystyle b = - ({\ tfrac {a} {n}} - a) = - ({\ tfrac {6 \, 377 \, 397 {,} 155 \; \ mathrm {m}} {\ color { Green} 299 {,} 1528128}} - 6 \, 377 \, 397 {,} 155 \; \ mathrm {m}) = 6 \, 356 \, 078 {,} 96282 \; \ mathrm {m} \ simeq 6 \, 356 \, 078 {,} {\ color {Red} 963} \; \ mathrm {m}}$
5. Incorrectly too${\ displaystyle n = {\ tfrac {1} {f}} = {\ tfrac {a} {ab}} = {\ tfrac {6 \, 377 \, 397 {,} 155 \; \ mathrm {m}} {6 \, 377 \, 397 {,} 155 \; \ mathrm {m} -6 \, 356 \, 078 {,} {\ color {Red} 963} \; \ mathrm {m}}} = {\ color {Green} 299 {,} 15281} {\ color {Red} 53513205998}}$
6. georepository.com epsg.io EPSG: 7004 uses This value comes from the US Army Map Service Technical Manual ; 1943. “ Remarks: Original Bessel definition is a = 3272077.14 and b = 3261139.33 toise. This used a weighted mean of values ​​from several authors but did not account for differences in the length of the various toise: the "Bessel toise" is therefore of uncertain length. ${\ displaystyle n = {\ tfrac {1} {f}} = {\ color {Green} 299 {,} 1528128}}$