Isothermal network

Isothermal networks are required in order to realize conformal, orthogonal maps in geodesy .

If one looks at the network of lengths and widths on the globe and spans areas of z. B. 1 ° × 1 °, these are still almost square at the equator (they are always right-angled), but the further north you go, the more this area is compressed in a west-east direction. The former square merges into the shape of a piece of cake, although the area is still z. B. 1 ° × 1 °!

In geodesy, one needs images that are as conformal as possible and that span an orthogonal network. At the equator, the almost square area described has an edge length of around 111 km (= 40,000/360). If you want to keep the north or south direction of this figure true to length, you have to increase the coordinates in the east and west direction by a small factor the further you go north or south, because the former squares are getting narrower and narrower. If you consider the density of the longitude and latitude on the sphere, then you can say that this density of the longitude at the North Pole - compared to the equator - is higher than that of the latitude. This compensation factor is called the mesh density . Networks that are formed according to this procedure are called isothermal networks (such networks also play an important role in heat theory, hence the name). In order to mathematically represent the complex situation for the sphere, one can write (easting = y; northing = x): ${\displaystyle \lambda }$

(1)    ${\displaystyle \mathrm {d} s^{2}=\lambda ^{2}\cdot \mathrm {d} y^{2}+\mathrm {d} x^{2}.}$

You write

(2)    ${\displaystyle \mathrm {d} s^{2}=\lambda ^{2}({\bar {u}},{\bar {v}})\cdot (\mathrm {d} {\bar {u}}^{2}+\mathrm {d} {\bar {v}}^{2}).}$

then one receives the basic requirement of isothermal networks. The following applies:

(3)    ${\displaystyle E=G=g_{11}=g_{22}=\lambda ^{2}({\bar {u}},{\bar {v}});\quad F=g_{12}=0.}$

The network formed from these parameters is conformal and orthogonal. The designations , and represent the classic designations of the fundamental quantities of the first order , while , and are designated as components of the first fundamental tensor. There is , and . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle g_{11}}$ ${\displaystyle g_{12}}$ ${\displaystyle g_{22}}$ ${\displaystyle E=g_{11}}$ ${\displaystyle F=g_{12}}$ ${\displaystyle G=g_{22}}$

literature

• Walter Großmann: Geodetic calculations and images in the national survey . Stuttgart 1976.
• Bernhard Heck: Calculation methods and evaluation models for national surveying . Karlsruhe 1987.