Angular difference

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

specially developed measuring instruments ( e.g. heliometer or optical micrometer)
special evaluation methods (e.g. method of equal heights ) where the angle differences only result from the combination of several steps,

or computational modeling that also process angles measured in different planes.

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 distance measurement with scissors telescopes is based on angle measurement between the two telescopic sights of the measuring device, which is technically realized as the angle difference between two mirror positions. The optical distance measurement has the same principle, only that the measuring base is a horizontal or vertical measuring stick and the angle or its change is measured with a theodolite .
The geodetic backward section - the determination of one's own location from three directions to distant target points - corresponds to the geometric determination of peripheral angles . However, these are not determined directly, but from differences in direction . The primarily measured directions are actually angles on the pitch circle of the theodolite between its zero point and the target direction.
The accuracy of these point determinations - which is critical on circles - can easily be estimated from the angle differences that occur when the measured point is shifted by a few centimeters.
The baseline method of nautical science uses the difference between the observed elevation angle and the elevation angle calculated for the assumed location for each of the two or three observed stars .
Something similar takes place with astronomical location determination with a prism astrolab or a circumzenital .
In the adjustment calculation , geodesists work with distance or angle differences between each measurement and the initial geometric model of the surveying network . The squares of these differences are minimized according to the Gaussian method.
For precision measurements and in instrument science , the reduction in the measured directions due to instrumental errors or refraction is used as an angle difference, with differential refraction of different light colors even as a difference in angle differences.
Further difference methods can be found e.g. B. for measurements with angular prisms , when aligning in a straight line ( alignment ), with the flat plate micrometer , with the deck bearing and with the sails in the navigation.

Astrogeodesy

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

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

the daily and annual parallax of stars as a result of the earth's rotation and the earth's orbit around the sun

variable telescope bending with large telescopes (see also reduction (measurement) )

the measurement of double stars with a micrometer or heliometer ; the motion of the stars around their center of mass is in turn the difference in angle differences

the changes in the parallax of the sun or the moon due to the ellipticity of the orbit of the earth and the moon

small fluctuations in the solar radius , see also helioseismology

some effects of the not quite constant rotation of the earth - e.g. B. on the time difference dUT1 or on the pole coordinates .

Larger angle differences also play a role:

Morning and evening distance of the sun
Elongation , the current distance of a planet from the sun
Mean anomaly , a time-proportional, fictitious angle for calculating planetary orbits ; the difference from the true anomaly is in the Kepler's equation , a
Historical: Measurement of the sun disk (mutual distance between the two sun edges ) with a suitable shadow projector or from the time difference that results from the apparent movement due to the rotation of the earth.

Physics, construction, technology

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

when measuring the torsion of components in a non-geometric way
with the torsion bar , the torsion balance and the gravimeter
for changes in angle due to shear stress
as a bend of bimetal strips z. B. for temperature measurement
in the angular position of damping flaps u. Ä.
as thermal effects and pillar rotation during deformation measurements.

literature

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.

Web links

Angular difference, math for engineers, page 452