Triangulation and Trilateration

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Differences

Advantages of triangulation: needs only one base line; theodolite angle measurement is reliable; suitable for large areas; established practice. Disadvantages of triangulation: base line measurement is laborious; angular observation is slow and needs signals/towers; errors accumulate. Advantages of trilateration: quick, no angle measurement or base-line extension; EDM gives high accuracy; works with low visibility (microwave); fewer observers. Disadvantages of trilateration: needs costly EDM; atmospheric corrections necessary; fewer checks on gross errors unless the figure is redundant; no azimuth.

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Principle

The base line AB is measured accurately. At A and B the angles to a third station C are measured. In triangle ABC, the sides AC and BC are computed by the sine rule:

The computed side BC (or AC) is then the base for the next triangle BCD, and so on. The whole area is covered by a chain or net of triangles. At the end, a second base line (check base) is measured and compared with its computed length; the difference shows the accuracy of the work. Azimuth is observed at the ends to control the direction.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F
   base                   check base

General specifications of the different orders of triangulation

($K$ is the distance in km. The figures are the common textbook values; Survey Department practice may differ slightly.)

Primary (first order) triangulation gives the main control of a country, secondary triangulation fills in the primary net, and tertiary triangulation provides closely spaced stations for engineering and topographic surveys.

The triangulation system is classified into first order (primary), second order (secondary) and third order (tertiary) triangulation, according to the accuracy of the work. The classification depends on the length of the base line and sides, the closure of triangles and the probable error of the computed lengths.

1. First order (primary) triangulation

• It forms the main framework of a country and is used for geodetic work, determining the shape and size of the earth, and for control of the whole area.
• Highest accuracy: long sides (30 to 150 km), base line 5 to 15 km, average triangular error less than 1".
• Stations are at the top of hills/towers; signals and precise theodolites are used.

2. Second order (secondary) triangulation

• Based on the primary stations as control; the network is divided into smaller parts.
• Sides 8 to 30 km, base line 1.5 to 5 km, average triangular error 3". Used for regional surveys and for fixing points for topographic mapping.

3. Third order (tertiary) triangulation

• Based on the secondary net; gives stations close together, for engineering surveys (towns, roads, dams, bridges, tunnels).
• Sides 1.5 to 10 km, base line 0.5 to 3 km, average triangular error 6". Lower-cost instruments are acceptable.

($K$ is the distance in km. The figures are the common textbook values; Survey Department practice may differ slightly.)

Classification by figure

The triangulation system can also be seen by the type of figure used: (i) a chain of single triangles, (ii) a chain of braced quadrilaterals, and (iii) a chain of polygons with a central station. The best-conditioned figures have angles between 30 and 120 degrees.

Principle of triangulation

The base line AB is measured accurately. At A and B the angles to a third station C are measured. In triangle ABC, the sides AC and BC are computed by the sine rule:

The computed side BC (or AC) is then the base for the next triangle BCD, and so on. The whole area is covered by a chain or net of triangles. At the end, a second base line (check base) is measured and compared with its computed length; the difference shows the accuracy of the work. Azimuth is observed at the ends to control the direction.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F
   base                   check base

General specifications of first order trilateration

(The first order values are as per the common geodetic standards; they may be revised by the Survey Department.)

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule. Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Purposes of triangulation and trilateration

• To establish accurate horizontal control (position of stations) for topographic and engineering surveys of large areas, with plane tabling, tacheometry or traverses based on the control stations.
• To determine the size and shape of the earth (geodetic surveys) and to fix positions on the earth by latitude and longitude.
• To provide control for photogrammetric maps and aerial surveys.
• To set out the exact alignments and centre lines of large engineering works, such as long bridges, tunnels, dams and canals.
• To monitor movement or deformation (of dams, landslides, the crust) by repeating the survey.
• To fix the boundaries of states or countries and provide a reference framework for cadastral surveys.

General specification of tertiary (third order) triangulation

Tertiary triangulation is based on the secondary stations and is used for closely spaced engineering control.

Principle of triangulation

The base line AB is measured accurately. At A and B the angles to a third station C are measured. In triangle ABC, the sides AC and BC are computed by the sine rule:

The computed side BC (or AC) is then the base for the next triangle BCD, and so on. The whole area is covered by a chain or net of triangles. At the end, a second base line (check base) is measured and compared with its computed length; the difference shows the accuracy of the work. Azimuth is observed at the ends to control the direction.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F
   base                   check base

General specification of third order (tertiary) triangulation

Advantages of trilateration over triangulation

1. Speed: only distances are measured with EDM, which takes minutes per side, while the angle observation in several sets is slow.
2. No base line: no separate base line, extension of base, or check base is needed; every side is measured with high accuracy.
3. Accuracy: modern EDM gives 1 part in 100,000 to 1,000,000, and there is no accumulation of angular error.
4. Fewer observers and less cost: the work can be done by a small party, with no tall towers for signals in many cases.
5. Weather: it can work in poor visibility or at night, particularly with microwave instruments.
6. Simple computation: angles are computed from the sides by the cosine rule, and the figure is adjusted by the least squares method.
7. Very long sides and a strong network are possible with the same field effort.

General specifications of primary (first order) triangulation

($K$ is the distance in km. The figures are the common textbook values; Survey Department practice may differ slightly.)

Selection of triangulation / trilateration stations

1. Intervisibility: every station should be visible from the adjacent stations; hills, high buildings or towers can be used, and the line of sight should clear the ground, trees and buildings (at least a few metres, to avoid refraction near the ground).
2. Well-conditioned triangles: the angles should lie between 30 and 120 degrees (preferably 45 to 90); avoid long thin triangles.
3. Stability and accessibility: stations should be on firm ground that will not settle, and easily reached with the instrument and signals, with enough space for signal erection.
4. Altitude: stations should be high enough to see over obstructions, but not so high that the cost of towers or accessibility suffer; keep the number of stations to a minimum.
5. Base line: a site that is nearly level and free from obstacles for the base line, near the centre of the area, so the base may be extended into the net.
6. For trilateration: line of sight clear for the EDM beam, free from strong reflecting surfaces, power lines or heat sources close to the beam, and stations where the temperature and pressure can be measured at both ends.
7. Permanence: the stations should be placed so that they may be used for future survey work and be safe from disturbance and construction.

Field applications of triangulation

• Providing control points for topographic mapping and large-scale surveys.
• Fixing accurate positions of towns, boundaries, bridges and dams.
• Setting out long bridges, tunnels, and the centre line of dams or canals.
• Geodetic work: size and shape of the earth, latitude, longitude and azimuth.
• Hydrographic surveys: fixing soundings and shore points from shore stations.
• Photogrammetric control for aerial mapping.

Types of figures (triangles) in a triangulation system

The triangles are arranged in one of the following figures:

1. Chain of single triangles Used for a narrow strip of country (roads, rivers, railways). The least costly, with the least number of stations, but the weakest in checking.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F

2. Chain of braced quadrilaterals Each quadrilateral has both diagonals observed, so that there are many checks (side and angle equations). Used for first order triangulation over long strips; stronger than a chain of single triangles.

   C--------D
   | \    / |
   |   \/   |
   |   /\   |
   | /    \ |
   A--------B

3. Chain of polygons (central-point figures) A polygon (hexagon or pentagon) with a station at the centre. Used for large, open areas; this gives the most checks and is the strongest figure, but needs many observations.

        B
      /   \
     A  o   C
      \ | /
        F--D

Well-conditioned triangles

For a good result, a triangle should have no angle less than 30 degrees or more than 120 degrees. The ideal triangle is equilateral; for a chain of triangles the best shape has base angles of about 56 degrees 14 minutes, which gives the weakest side the least error.

Specification of first order triangulation

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Principle of EDM for measuring a side

An EDM instrument sends out a modulated electromagnetic wave (infrared, light or microwave) from one end of the line to a reflector (prism) at the other end. The wave returns to the instrument, and the distance is found from the phase difference between the transmitted and the received signals.

If the wavelength of the modulated wave is $\lambda$, the double path (to the reflector and back) contains $n$ whole waves and a part wave $\Delta\lambda$:

where $\phi$ is the measured phase difference (angle). The instrument measures $\phi$ accurately, so $\Delta\lambda$ is known, while the integer $n$ is resolved by using several modulation frequencies (a coarse frequency to give approximate distance and finer ones for precise value). The velocity of the wave depends on temperature, pressure and humidity, so meteorological corrections are applied. Alternatively, in a pulse instrument, the time $t$ taken by a pulse for the double path gives $D = \frac{c\,t}{2}$.

   EDM  ====== modulated wave =====>  reflector (prism)
   unit <=====   reflected wave ====
   phase difference -> distance D

Typical accuracy is $\pm (3\text{ to }5\ \text{mm} + 1\text{ to }2\ \text{ppm})$. The sides of the trilateration figure are measured this way, and the angles are computed with the cosine rule.

Principle of triangulation

The base line AB is measured accurately. At A and B the angles to a third station C are measured. In triangle ABC, the sides AC and BC are computed by the sine rule:

The computed side BC (or AC) is then the base for the next triangle BCD, and so on. The whole area is covered by a chain or net of triangles. At the end, a second base line (check base) is measured and compared with its computed length; the difference shows the accuracy of the work. Azimuth is observed at the ends to control the direction.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F
   base                   check base

Principle of trilateration

If the three sides a, b, c of a triangle ABC are measured, any angle is found with the cosine rule:

The sides are measured by EDM, so no angles are observed and no base line extension is needed. The figure is solved from the lengths alone.

Trilateration is a method of horizontal control survey in which the lengths of all the sides of a chain or net of triangles are measured directly with an EDM instrument, and the angles of the triangles are then computed from the measured sides using the cosine rule. No angles are observed in the field.

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Graphical intersection in plane tabling

Graphical intersection is the plane table method in which a point is located by drawing rays to it from two plotted stations at the ends of a base line, and marking the point at the intersection of the rays. No distance to the point needs to be measured.

1. Set the table at A, level it, centre and orient it; plot a and the base line ab to scale.
2. Sight the point P from a and draw the ray.
3. Shift to B, orient the table by back-sighting along ba, sight P and draw the ray from b.
4. The two rays meet at p, the plotted position of P. A third ray from another station checks it.

            p
          / |  \
        /   |    \
      a ---------- b

Graphical intersection is the same idea as triangulation (the point is fixed by two angles from a known base), done graphically instead of by computation.