Very often, when conducting geodetic, design and construction work, it becomes necessary to determine various spatial quantities. It is far from always possible to do this with the help of such trivial tools as roulette.

Theodolite is a device designed for geodetic, design and construction work.

**Measurement of vertical angles by a theodolite is a very popular function.** Knowing the value of this parameter and using the formulas of trigonometry, one can determine the linear dimensions of the object of interest (height) or the distance to the object.

## Theodolite device itself

Theodolite device.

Theodolite is irreplaceable when carrying out verification of measurements on the ground of distances, horizontal and vertical angles. The main working elements are round graded scales. Conducting the entire possible range of research is provided by the presence of both horizontal and vertical measuring scale.

In the same plane with the measuring limb there is another rotating circle - alidade. On it is fixed a device for making counts on the limb.

In addition to the limbs and alidad, the measuring device includes the following essential units:

- Stand with built-in level. To ensure greater stability and setting relative to the horizontal level, the base has three legs, equipped with devices regulating their length.
- Optical telescope.
- Screws fine-tuning and fixing optics on the subject of measurement.
- Cylindrical level, reflecting the vertical angles of inclination of the device itself in the measurement process.
- Optical plummet.
- A readout microscope (bar or scale, depending on the design of the theodolite) for taking measurements.

## Proper verification

In the process of measuring with the help of a theodolite, the occurrence of errors is possible, which can later lead to serious errors in determining the magnitude of vertical angles. Obtaining accurate results is guaranteed with regular passage of a complex of calibration procedures. In case of occurrence of the deviation of the results of calibration from the regulated values, when it is possible, the theodolite is adjusted.

Types of verification of theodolite: a - installation by two screws; b - installation on the third screw; in - checking the correctness of the installation level.

There are four types of verification (adjustment):

- Checking the mutual perpendicularity of the axes of the cylindrical level and the rotation of the device in the vertical plane.

The operational position is determined by the air bubble inside the level capsule. It should be in the middle of the scale. The deviations of the position of the air bubble relative to the center of the scale at the level are compensated by the adjustment of the lifting screws.

Once the working position is established, the alidade and level are rotated through 180 °. In the new position, the whole procedure is repeated anew. It is necessary to achieve independence of the level reading from the angle of rotation of the alidade.

- Check the position of the threads of the measuring grid. The vertical thread in the optics of the device is combined with a pre-installed pole, which has a strictly vertical position. They must match. If deviations are detected, it is necessary to adjust the position of the eyepiece. The fixing fastener is loosened and the position of the eyepiece is shifted until the line of the thread and the plumb line completely coincide.

- Check the perpendicularity of the position of the theodolite sighting axis relative to the axis of rotation of the telescope. This verification is also called the collimation error verification. If the condition of perpendicularity is not fulfilled, the theodolite sighting axis during rotation will describe a conical surface instead of a plane, which will deduce the accuracy of measuring vertical angles beyond all permissible limits.

At the initial stage of verification, a remote reference point should be set, the line of sight of which is as close as possible to the horizontal plane.

When the measuring wheel is located to the right of the operator, the crosshairs of the threads and the selected reference point are combined.

The readings of the horizontal circle (KP) are recorded. Then the theodolite pipe is unrolled, the alidad is detached, and readings are taken at the left position of the measuring wheel (C). The magnitude of the measurement error is determined by the formula:

C = ((Kl-Kp) ± 180 °) / 2

The sign in front of 180 ° is determined depending on the sign of the difference between K and KP. If the difference in readings is positive, then the sign “-“ is taken, otherwise - “+”.

The results of the verification are evaluated by the obtained value of C. If it does not exceed the value of the double precision of the scale, then the theodolite is serviceable. Excess signals the need for correction. To do this, loosen one of the vertical screws in the wire mesh. Adjusting the side screw, combine the intersection of the threads of the measuring grid with the intended reference;

- Check the relative position of the axis of rotation of the optical tube and the vertical axis of the theodolite. The fulfillment of this condition ensures the vertical position of the measuring plane during operation.

On the surface of the wall, located approximately 10 m, a point is selected that is located at an angle of 40-50 ° relative to the horizon. It is guided by a pipe and fixed alidade. After that, the pipe is transferred to a horizontal position and the projection of the point is marked on the wall by rotating the alidade 180 ° and by moving the pipe through the zenith position on the wall. At the horizon level, the projection of the initially selected point is remarked. Both derived projection points must match. In case of discrepancy, this error in field conditions cannot be compensated for and the theodolite is subject to repair.

Back to table of contents## Preparation for measurement / testing

Centering theodolite: 1 - theodolite; 2.3 - tripod legs, 4 - plumb.

If all the checks provided for in the regulations have been successfully passed, we can assume that the tool is in good condition and they can work. Before starting the measurement of vertical angles theodolite must first be prepared for work.

Preparation consists in performing the following operations:

- Checking the position of the measuring device - setting the center position of the dial above the set point using a standard plumb line. The center point of a horizontal or vertical angle is usually used as the center point.
- Setting the surface of the measuring dial in the horizontal plane. The procedure is carried out by adjusting the lifting screws until the position of the bubble in the level ampoule coincides with the zero point.
- Adjusting visibility optics — rotate the diopter adjustment ring until you get the best visibility of the grid.
- Adjustment of optics in the measured point; Cremaliera rotation adjusts the image clarity of the object.

## Measurement and processing of results

The scheme of measurement of vertical angles by theodolite.

In practice, measurements of vertical angles are performed using a vertical limb. A characteristic feature is the rotation of the limb in the process of working together with the optical tube at the fixed position of the alidade.

The zero diameter of the alidade is given in a horizontal position, controlled by the position of the bubble in a cylindrical level. As soon as zero is set, the vertical angles are determined by the position reading on the vertical measuring circle.

In order to measure the vertical angles, the theodolite must be ready for operation in accordance with the methods described above. At any position of the vertical circle (left or right), the horizontal thread of the measuring grid is aimed at the measuring point. If necessary, a small passing adjustment of the position of the bubble in the level relative to the zero point is made. The obtained values on the limb are recorded in the measurement log.

Next, a similar measurement procedure is performed when the vertical circle is shifted in the opposite direction and the results are recorded.

The calculation of the measured vertical angles.

To calculate the magnitude of the measured vertical angles, it is necessary to calculate the intermediate value - the place of zero. This value characterizes the counting on the limb of a vertical circle with the horizontal position of the sighting axis of the telescope and the presence of an air bubble in the middle of the scale of a cylindrical level.

Place zero is determined by the formula:

Mo = (CL + KP) / 2

where, Mo is the place of zero;

Kl, Kp - readings at the location of the measuring wheel to the left and right, respectively.

Vertical angles are determined by the formulas:

V = Kl-Mo or V = Mo-Kp

where, V - the value of the measured vertical angle.

It is worth noting that in order to increase accuracy, it is recommended to carry out measurements several times with the subsequent processing of the results obtained using methods of mathematical statistics.