Angle measurement

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When measuring angles, technical equipment is used to determine the angle at which two straight lines or two other directions are to one another.

This article provides an overview of the operating principles. Detailed descriptions can be found in the linked articles, while under measuring device and direction measurement several precise measuring instruments and necessary reference directions are dealt with.

Rotation angle measurement

Reading on a scale

Often, as with a conventional protractor , angles are read after manual adjustment. The larger the radius of the scale , the more accurately the angle can be read.

Opto-electronic scanning of a code disk

Markings that can be read optically are applied to a pane of glass. This enables very high levels of accuracy to be achieved.

  • The absolute encoder immediately shows the angle.
  • The incremental encoder measures the difference between two angles. If a zero pulse is available, it can show the absolute angle after one revolution.

Detecting the position of a rotating magnetic field

Magnetic methods are less sensitive to dirt, but not as precise as opto-electronic scanning:

Voltage divider

A potentiometer easily converts an angle into an analog measurement signal. The disadvantage is the wear caused by the friction of the grinder.

Calculation by integrating the angular velocity

Fiber optics use the principle of the Sagnac interferometer . The device must be zeroed beforehand. The angular velocity is measured by shifting the phase of laser beams guided in a circle . The angle is calculated through mathematical integration .

Calculation over time

The time that a rotating cam needs for the angle to be measured is compared with the time for one revolution. The angle must not change during the measurement.

Measure fixed angles

In addition to the above-mentioned operating principles, there are also the following options for fixed angles:

Calculation from the measured position

If the position of a line is measured at two points, its angle can be calculated. This principle uses:

to teach

These devices are used to control and transfer angles:

Inclination measurement

Compass with inclinometer

Angle measuring devices with reference to the earth's surface, inclinometers measure the deviation from the horizontal or plumb direction . They often only have a small measuring range. The reference angle is always the earth's gravity field .

Manual reading

Necli hand slope knife
  • The plumb line measures the vertical.
  • The merchet is an ancient Egyptian inclinometer.
  • A slope knife (name variations: slope knife, slope knife, hand slope knife, hand slope knife, hand slope knife ) is a small optical instrument that essentially consists of a sight (aiming device) and a fluid-damped pendulum body with protractor.
  • With the historic plumb level and the horizontal was determined with the solder.
  • The scoliometer measures the angle of inclination of the back surface with a ball in a curved tube.
  • The spirit level uses the principle of the horizontal surface of a liquid. An air bubble in the tubular vial shows the angle of inclination in the longitudinal direction, a circular vial in both directions
High-precision dragonflies based on this principle are hanging and riding dragonflies .
  • The mercury horizon uses the principle of the horizontal surface of a liquid to reflect the measuring beam.
  • Devices from geodesy and astronomy see below .

Electronic inclinometer

The surface of a liquid always aligns horizontally. The position of the liquid is scanned without contact. The following operating principles are possible:

Liquid horizon

With the height compensator : A beam of light is directed at an angle onto the surface of a transparent liquid. The position of the reflected light beam is recorded with optical sensors.

Capacitive liquid inclination sensors

A dielectric liquid is enclosed in a semicircular vessel between two perpendicular plates. If the vessel is rotated, the capacity changes linearly with the angle. This change in capacitance is converted electronically into an output signal.

Electrolyte level

The level filled with electrolyte has one electrode in the middle and two electrodes on the edge. When the vessel is tilted, the conductivity changes . This is evaluated with an alternating voltage. A circular level with five electrodes is used for two-axis sensors. The electrodes are arranged like the five points on a dice .

Resistance measurement (magnetoresistive)
Magnetoresistive inclinometer
1: weight, 2: transducer magnet, 3: MR resistors, 4: connections

A magnet attached to a weight acts on a voltage divider made of magnetoresistive resistors . The incline can be measured directly based on the change in the resistance ratios.

Resistance measurement (mercury)

A resistance track is mounted in a vessel filled with a conductive liquid such as mercury . If the container is inclined, the resistance track is immersed in the conductive liquid to different degrees and bridges its part in the resistance. The measurable resistance changes depending on the unwetted length.

Thermal principle

The detuning of a sensor bridge by a rising heated gas in a miniature measuring chamber is used.

Direction measurement

If the measurement of a horizontal angle refers to geographic or magnetic north, it is called a direction measurement . It results in oriented measuring beams or courses in a defined reference system . Together with a distance measurement , it is used to locate vehicles and other objects, in geodesy and geometry to define a point in a coordinate system.

The angle encoders find the reference direction north in the following way:

Geographically north

In the magnetic field of the earth

Other reference directions

Instruments of geodesy and astronomy

Historical devices such as the astrolabe and Jacob's staff or the quadrant were still used for free-eyed measurements (without a telescopic sight). It was possible to achieve accuracies of 0.1 ° to a maximum of 0.02 ° ( Tycho Brahe ). From around 1660 onwards, the telescope , which had been invented 50 years earlier, was increasingly being built into measuring instruments, making it possible to achieve 100 to 1000 times greater accuracy.

  • The sextant is mainly used to measure elevation angles . It is held freely in the hand and is therefore used in astronomical navigation on ships. The double reflection stabilizes the reflection of the stars and the horizon even when the ground is unsteady.
    • The sextant is held horizontally to measure horizontal angles (terrestrial navigation) . The next three instruments must be leveled (aligned) with spirit levels and must be stationary during the measurement.
  • The theodolite measures elevation angle and horizontal direction
  • The total station measures the elevation angle, horizontal direction and distance to the target point
  • The passage instrument is aligned to the meridian and is used for the precise measurement of star words .

The heliometer measures small angular differences by tilting a split lens so that two images are aligned.

The vertical direction can be determined with the circumzenital or a prism astrolab by measuring star passages through the field of view of a telescope. The portable Ni2 astrolabe achieves an accuracy of ± 0.2 ", the less portable Danjon astrolabe up to ± 0.05".


  • Franz Löffler u. a .: Handbook of Engineering Geodesy. Mechanical and plant engineering. 2nd Edition. Herbert Wichmann Verlag, Heidelberg 2001, ISBN 978-3879072996