Angular difference

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Angular differences are the subject of special measurements and calculation methods in a wide variety of specialist areas. Such methods can be found in natural sciences such as astronomy , physics, geodesy or geology, but also in technical applications such as navigation and instrument science .

In many cases, the angular differences measured or computationally treated in such a method cannot be obtained by direct angular measurement , but rather as an object

In the following, some applications are listed as examples and their special features are briefly treated.

Geodesy, navigation

For direction determinations on earth or in the sky, the angles themselves are usually the subject of the measurement, but in special cases also their differences or changes in location or time. Many of these differential measurements are made with special optics or high-precision micrometers .


The celestial navigation and nautical science knows several measurement and evaluation methods using angle differences (see above). Two of these methods have been further developed for astrometry and earth measurement :

  • The same height method is used for the highly precise determination of the true (astronomical) plumb line or for terrestrial time and length determination . Actually, the stars are observed at a constant zenith distance (astronomically mostly 10 °, geodetically 30 °), but it actually comes down to measuring small angle differences using special micrometers or small time differences ("target minus actual"). During the evaluation (as in the nautical method, but with additional reduction variables ) the difference between the (indirectly) observed “mean elevation angle ” of all stars and the ellipsoidally calculated elevation angles of the individual stars is formed.
  • With some special methods, angle differences are measured directly in the instrument, e.g. B. with the heliometer invented by Fraunhofer , which works with a two-part telescope lens . With two other special instruments, the Danjon astrolabe and the circumzenital , the calculation of the angular difference between the "calculated" and the observed star resulting from the plumb line disturbance or the clock error is carried out in a similar way to the above method for the same heights .
  • With the Horrebow-Talcott method for determining the polar movement and latitude, culminations of successive pairs of stars in the south and north branches of the meridian are observed by pivoting the telescope through 180 ° and measuring the difference in elevation angles directly in the field of view with a micrometer.


Astronomy knows numerous methods and calculation models that are based on small angle differences. Some examples are:

Larger angle differences also play a role:

Physics, construction, technology

Special methods with angle differences that go beyond mere angle measurement can be found in several areas - among others

See also


  • Thomas Westermann: Mathematics for Engineers . 6th edition. Springer, ISBN 978-3-642-12760-1 .
  • Heribert Kahmen : Applied Geodesy: Surveying . 20th edition. Gruyter, Walter de GmbH, ISBN 3-11-018464-8 , Berlin ≈2005.
  • Franz Ackerl: Geodesy and Photogrammetry Part I (Instrumentology), Georg Fromme Verlag, Vienna 1950
  • Albert Schödlbauer : Geodetic Astronomy - Basics and Concepts . 634 p., Verlag de Gruyter, Berlin 2000
  • J.Bennett, M.Donahue, N.Schneider, M.Voith: Astronomy (Chapters 3, 5, 13 and 15) . Editor Harald Lesch, 5th edition (1170 pages), Pearson-Studienverlag, Munich-Boston-Harlow-Sydney-Madrid 2010
  • HÜTTE, Des Ingenieurs Taschenbuch , 26th edition (anthology) or 28th edition, Volume II, Berlin 1955.

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