theodolite n : a surveying instrument for measuring horizontal and vertical angles, consisting of a small telescope mounted on a tripod [syn: transit]

English

Etymology

New Latin theodolitus, of unknown origin

Noun

1. a surveying instrument, consisting of a small mounted telescope, used to measure horizontal and vertical angles

See also

• tribrach

A theodolite () is an instrument for measuring both horizontal and vertical angles, as used in triangulation networks. It is a key tool in surveying and engineering work, but theodolites have been adapted for other specialized purposes in fields like meteorology and rocket launch technology. A modern theodolite consists of a telescope mounted movably within two perpendicular axes, the horizontal or trunnion axis, and the vertical axis. When the telescope is pointed at a desired object, the angle of each of these axes can be measured with great precision, typically on the scale of arcseconds.

The transit refers to a specialized type of theodolite that was developed in the early 19th century. It featured a telescope that could "flop over" ("transit the scope") to allow easy back-sighting and doubling of angles for error reduction. Some transit instruments were capable of reading angles directly to thirty arc-seconds. In the middle of the 20th century, transits came to be known as a simple form of theodolite with less precision, lacking features such as scale magnification and mechanical meters. The importance of transits is waning since compact, accurate electronic theodolites have become widespread tools, but transits still find use as a lightweight tool for construction sites. Some transits do not measure vertical angles.

The builder's level is often mistaken for a transit, but is actually a type of inclinometer. It measures neither horizontal nor vertical angles. It simply combines a spirit level and telescope to allow the user to visually establish a line of sight along a level plane.

Concept of operation

Both axes of a theodolite are equipped with graduated circles that can be read out through magnifying lenses. The vertical circle (the one associated with the horizontal axis) should read 90° or 100 grad when the sight axis is horizontal (or 270°, 300 grad, when the instrument is in its second position, "turned over" or "plunged"). If not, we call half of the difference with 300 grad index error.

The horizontal and vertical axes of a theodolite must be mutually perpendicular. The condition where they deviate from perpendicularity (and the amount by which) is referred to as horizontal axis error. The optical axis of the telescope, called sight axis and defined by the optical center of the objective and the center of the crosshairs in its focal plane, must similarly be perpendicular to the horizontal axis. If not, we call the deviation from perpendicularity collimation error.

Horizontal axis error, collimation error and index error are regularly determined by calibration, and removed by mechanical adjustment at the factory in case they grow overly large. Their existence is taken into account in the choice of measurement procedure in order to eliminate their effect on the measurement results.

A theodolite is mounted on the tripod head by means of a forced centering plate or tribrach, containing four thumbscrews (or in some modern theodolites three thumbscrews) for rapid levelling. Before use, a theodolite must be placed precisely and vertically over the point to be measured — centering — and its vertical axis aligned with local gravity — leveling. The former is done using a plumb bob, laser plummet or optical plummet, the latter using a spirit level. Fast and accurate procedures for doing both have been developed.

History

In old texts, one might find the term diopter used as a synonym for theodolite. This usage would derive from an older instrument called a dioptra.

Prior to the theodolite, instruments such as the geometric square and various graduated circles (see circumferentor) and semi-circles (see graphometer) were used to obtain either vertical or horizontal angle measurements. It was only a matter of time before someone put two measuring devices into a single instrument that could measure both angles simultaneously. Gregorius Reisch showed such an instrument in the appendix of his book, Margarita Philosophica, which he published in Strasburg in 1512. It was described in the appendix by Martin Waldseemüller, a Rhineland topographer and cartographer, who made the device in the same year.

The first occurrence of the name theodolite, or 'theodelitus', is found in the surveying textbook A geometric practice named Pantometria (1571) by Leonard Digges. This was published posthumously by his son, Thomas Digges. Digges senior invented the name, but its origin is unclear. Thus the name originally applies only to the azimuth instrument and only later became associated with the altazimuth instrument. The 1728 Cyclopaedia compares graphometer to "half-theodolite". Even as late as the 19th century, the instrument for measuring horizontal angles only was called a simple theodolite and the altazimuth instrument, the plain theodolite.

The earliest altazimuth instruments consisted of a base graduated with a full circle at the limb and a vertical angle measuring device, most often a semi-circle. An alidade on the base was used to sight an object for horizontal angle measurement and a second alidade was mounted on the vertical semi-circle. Later instruments had a single alidade on the vertical semi-circle and the entire semi-circle was mounted so as to be used to indicate horizontal angles directly. Eventually, the simple, open-sight alidade was replaced with a sighting telescope. This was first done by Jonathan Sisson in 1725. As technology progressed, in the 1840s, the vertical partial circle was replaced with a full circle and both vertical and horizontal circles were finely graduated. This was the transit theodolite. This, with continuing refinements, evolved into the modern theodolite used by surveyors today.

Using theodolites in surveying

Triangulation, as invented by Gemma Frisius around 1533, consists of making such direction plots of the surrounding landscape from two separate standpoints. After that, the two graphing papers are superimposed, providing a scale model of the landscape, or rather the targets in it. The true scale can be obtained by just measuring one distance both in the real terrain and in the graphical representation.

Modern triangulation as, e.g., practiced by Snellius, is the same procedure executed by numerical means. Photogrammetric block adjustment of stereo pairs of aerial photographs is a modern, three-dimensional variant.

In the late 1780s Jesse Ramsden, a Yorkshireman from Halifax, England who had developed the dividing engine for dividing angular scales accurately to within a second of arc, was commissioned to build a new instrument for the British Ordnance Survey. The Ramsden theodolite was used over the next few years to map the whole of southern Britain by triangulation.

In network measurement, the use of forced centering speeds up operations while maintaining the highest precision. The theodolite or the target can be rapidly removed from, or socketed into, the forced centering plate with sub-mm precision. Nowadays GPS antennas used for geodetic positioning use a similar mounting system. The height of the reference point of the theodolite -- or the target -- above the ground bench mark must be measured precisely.

The American transit gained popularity during the 19th century with American railroad engineers pushing west. The transit replaced the railroad compass, sextant and octant and was distinguished by having a telescope shorter than the base arms, allowing the telescope to be vertically rotated past straight down. The transit had the ability to 'flop' over on its vertical circle and easily show the exact 180 degree sight to the user. This facilitated the viewing of long straight lines, such as when surveying the American Wild West. Previously the user rotated the telescope on its horizontal circle to 180 and had to carefully check his angle when turning 180 degree turns.

Modern theodolites

In today's theodolites, the reading out of the horizontal and vertical circles is usually done electronically. The readout is done by a rotary encoder, which can be absolute, e.g. using Gray codes, or incremental, using equidistant light and dark radial bands. In the latter case the circles spin rapidly, reducing angle measurement to electronic measurement of time differences. Additionally, lately CCD sensors have been added to the focal plane of the telescope allowing both auto-targeting and the automated measurement of residual target offset. All this is implemented in embedded software.

Also, many modern theodolites are equipped with integrated electro-optical distance measuring devices, generally infrared based, allowing the measurement in one go of complete three-dimensional vectors -- albeit in instrument-defined polar co-ordinates -- which can then be transformed to a pre-existing co-ordinate system in the area by means of a sufficient number of control points. This technique is called a resection solution or free station position surveying and is widely used in mapping surveying. The instruments, "intelligent" theodolites called self-registering tachometers or "total stations", perform the necessary operations, saving data into internal registering units, or into external data storage devices. Typically, ruggedized laptops or PDAs are used as data collectors for this purpose.

Gyrotheodolites

The gyrotheodolite is used when the north-south reference bearing of the meridian is required in the absence of astronomical star sights. This mainly occurs in the underground mining industry and in tunnel engineering. For example, where a conduit must pass under a river, a vertical shaft on each side of the river might be connected by a horizontal tunnel. A gyrotheodolite can be operated at the surface and then again at the foot of the shafts to identify the directions needed to tunnel between the base of the two shafts. Unlike an artificial horizon or inertial navigation system, a gyrotheodolite cannot be relocated while it is operating. It must be restarted again at each site.

General

A gyrotheodolite comprises a normal theodolite with an attachment that contains a gyroscope mounted so as to sense rotation of the Earth and from that the alignment of the meridian. The meridian is the plane that contains both the axis of the Earth’s rotation and the observer. The intersection of the meridian plane with the horizontal contains the true north-south geographic reference bearing required. The gyrotheodolite is usually refered to as being able to determine or find true north.

Construction

When not in operation the gyroscope assembly is anchored within the instrument. The electrically powered gyroscope is started while restrained and then released for operation. During operation the gyroscope is supported within the instrument assembly typically on a thin vertical tape that constrains the gyroscope spinner axis to remain horizontal. The alignment of the spin axis is however permitted to rotate in azimuth by only the small amount required during operation. An initial approximate estimate of the meridian is needed. This might be determined with a magnetic compass, from an existing survey network or by the use of the gyrotheodolite in an extended tracking mode.

How it works

When the spinner is released from restraint with its axis of rotation aligned close to the meridian, the gyroscopic reaction of spin and Earth’s rotation results in precession of the spin axis in the direction of alignment with the plane of the meridian. This is because the daily rotation of the Earth is in effect continuously tilting the east-west axis of the station. The spinner axis then accelerates towards and overshoots the meridian, it then slows to a halt at an extreme point before similarly swinging back towards the initial point of release. This oscillation in azimuth of the spinner axis about the meridian repeats with a period of a few minutes. In practice the amplitude of oscillation will only gradually reduce as energy is lost due to the minimal damping present. Gyrotheodolites employ an undamped oscillating system because a determination can be obtained in less than about 20 minutes, while the asymptotic settling of a damped gyrocompass would take many times that before any reasonable determination of meridian could possibly be made.

Operation

The attachment containing the gyroscope is mounted so as to rotate with the theodolite. A separate optical system within the attachment permits the operator to rotate the theodolite and thereby bring a zero mark on the attachment into conicidence with the gyroscope spin axis. By tracking the spin axis as it oscillates about the meridian, a record of the azimuth of a series of the extreme stationary points of that oscillation may be determined by reading the theodolite azimuth circle. A mid point can later be computed from these records that represents a refined estimate of the meridian. Careful setup and repeated observations can give an estimate that is within about 10 arc seconds of the true meridian. This estimate of the meridian contains errors due to the zero torque of the suspension not being aligned precisely with the true meridian and to measurement errors of the slightly damped extremes of oscillation. These errors can be moderated by refining the initial estimate of the meridian to within a few arc minutes and correctly aligning the zero torque of the suspension.

Limitations

A gyrotheodolite will function at the equator and in both the northern and southern hemispheres. The meridian is undefined at the geographic poles. A gyrotheodolite can not be used at the poles where the Earth’s axis is precisely perpendicular to the horizontal axis of the spinner, indeed it is not normally used within about 15 degrees of the pole because the east-west component of the Earth’s rotation is insufficient to obtain reliable results. When available, astronomical star sights are able to give the meridian bearing to better than one hundred times the accuracy of the gyrotheodolite. Where this extra precision is not required, the gyrotheodolite is able to produce a result quickly without the need for night observations.

References

See also

• Tacheometry
• clinometer
• Surveying
• Rankine's method
• Dumpy level