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Types of triangles; first-order specification

Short answer

Triangulation uses chains of single triangles (weak, for narrow strips), braced quadrilaterals (strongest, with diagonals observed, for first-order long strips), or central-point polygons (wide-area coverage). First-order specification: baseline 5–15 km, triangle sides 30–150 km, average triangular closure <1″, maximum 3″, probable baseline error 1 in 1,000,000, baseline discrepancy 5 mm√K, computed distance probable error 1 in 60,000–250,000.

Explain briefly the different types of triangles used in triangulation system with sketches. Write down the specification of 1st order triangulation.

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). It is the least costly and needs the fewest stations, but it gives the fewest checks and is the weakest figure.

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

2. Chain of braced quadrilaterals Each quadrilateral has both diagonals observed, so there are many checks (side and angle conditions). It is the strongest and most accurate arrangement and is used for first order triangulation over long strips.

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

3. Chain of polygons (central-point figures) A polygon (pentagon or hexagon) with a station at the centre O. Used to cover wide areas extending in all directions; it gives good checks, but progress is slow because many stations and observations are needed.

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

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 an isosceles triangle the best shape has base angles of about 56 degrees 14 minutes, which keeps the error in the computed side to a minimum.

Specification of first order triangulation

ItemFirst order
Length of base line5 to 15 km
Length of sides of triangles30 to 150 km
Average triangular error (closure)less than 1"
Maximum triangular error3"
Probable error of base line1 in 1,000,000
Discrepancy between two measures of the base5 mm K\sqrt{K} (KK in km)
Probable error of computed distance1 in 60,000 to 1 in 250,000
InstrumentPrecise (1") direction theodolite; observations in several sets

✓ Answer checked by Claude Opus 5.5 (high effort)

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