Hansen's problem explained

In trigonometry, Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795 - 1874), who worked on the geodetic survey of Denmark. There are two known points, and two unknown points . From and an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of and . See figure; the angles measured are .

Since it involves observations of angles made at unknown points, the problem is an example of resection (as opposed to intersection).

Solution method overview

Define the following angles:As a first step we will solve for and .The sum of these two unknown angles is equal to the sum of and, yielding the equation

$$\backslash phi\; +\; \backslash psi\; =\; \backslash beta\_1\; +\; \backslash beta\_2.$$

A second equation can be found more laboriously, as follows. The law of sines yields

$$\backslash frac\; =\; \backslash frac,\; \backslash qquad\; \backslash frac\; =\; \backslash frac.$$

Combining these, we get

$$\backslash frac\; =\; \backslash frac.$$

Entirely analogous reasoning on the other side yields

$$\backslash frac\; =\; \backslash frac.$$

Setting these two equal gives

$$\backslash frac\; =\; \backslash frac\; =\; k.$$

Using a known trigonometric identity this ratio of sines can be expressed as the tangent of an angle difference:

\tan \tfrac 12 (\phi - \psi) = (k -1)/(k +1) \tan\tfrac 12 (\phi + \psi).

Where

k = (\sin \phi)/(\sin \psi) .

This is the second equation we need. Once we solve the two equations for the two unknowns, we can use either of the two expressions above for

\tfrac{\overline{AB}}{\overline{P _{1} P _{2}}} to find since is known. We can then find all the other segments using the law of sines.^[1]

Solution algorithm

We are given four angles and the distance . The calculation proceeds as follows:

• Calculate $$\backslash begin$$

\gamma &= \pi-\alpha_1-\beta_1-\beta_2, \\ \delta &= \pi-\alpha_2-\beta_1-\beta_2.\end

• Calculate $$$$

k = \frac.

• Let $$$$

s = \beta_1+\beta_2, \quad d = 2 \arctan \left(\frac \tan\tfrac12 s \right) and then $$\backslash phi\; =\; \backslash frac,\; \backslash quad\; \backslash psi\; =\; \backslash frac.$$

Calculate $$\backslash overline\; =\; \backslash overline\; \backslash \; \backslash frac$$ or equivalently $$\backslash overline\; =\; \backslash overline\; \backslash \; \backslash frac.$$ If one of these fractions has a denominator close to zero, use the other one.

Solutions via Geometric Algebra

In addition to presenting algorithms for solving the problem via Vector Geometric Algebra and Conformal Geometric Algebra, Ventura et al.^[2] review previous methods, and compare the various methods' computational speeds and sensitivity to measurement error.

See also

• Solving triangles
• Snell's problem

Notes and References

1. Udo Hebisch: Ebene und Sphaerische Trigonometrie, Kapitel 1, Beispiel 4 (2005, 2006)http://www.mathe.tu-freiberg.de/~hebisch/skripte/sphaerik/strigo.pdf
2. Ventura . Jorge . Martinez . Fernando . Zaplana . Isiah . Eid . Ahmad Hosny . Montoya . Francisco G. . Smith . James . January 2024 . Revisiting the Hansen Problem: A Geometric Algebra Approach . Mathematics . en . 12 . 13 . 1999 . 10.3390/math12131999 . free . 2227-7390. 2117/411951 . free .