# Tienstra formula

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The Tienstra formula is used to solve the resection problem in surveying, by which the location of a given point is determined by observations of angles to known landmarks from the unknown point.

Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land surveyor. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership, locations, such as building corners or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales.

## Contents

J.M.Tienstra (1895-1951) was a professor of the Delft university of Technology where he taught the use of barycentric coordinates in solving the resection problem. It seems most probable that his name became attached to the procedure for this reason, though when, and by whom, the formula was first proposed is unknown. [1]

In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices. Coordinates also extend outside the simplex, where one or more coordinates become negative. The system was introduced in 1827 by August Ferdinand Möbius.

## Tienstra formula

The resection problem consists in finding the location of an observer by measuring the angles subtended by lines of sight from the observer to three known points. Tienstra’s formula provides the most compact and elegant solution to this problem. [2]

${\displaystyle E_{p}={\frac {K_{1}E_{a}+K_{2}E_{b}+K_{3}E_{c}}{K_{1}+K_{2}+K_{3}}}}$

${\displaystyle N_{p}={\frac {K_{1}N_{a}+K_{2}N_{b}+K_{3}N_{c}}{K_{1}+K_{2}+K_{3}}}}$

Where:
${\displaystyle K_{1}={\frac {1}{cot(A)-cot(\alpha )}}}$
${\displaystyle K_{2}={\frac {1}{cot(B)-cot(\beta )}}}$
${\displaystyle K_{3}={\frac {1}{cot(C)-cot(\gamma )}}}$

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## References

1. Philip Howard (2006) Archaeological Surveying and Mapping: Recording and Depicting the Landscape page 51 Routledge ISBN   1134400861 Retrieved February 2015
2. Porta, J. and Thomas, F. (2009). Concise Proof of Tienstra’s Formula. J. Surv. Eng., 135(4), 170–172. Retrieved February 2015