Hansen's problem is a problem in planar surveying

Surveying

See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

, named after the astronomer Peter Andreas Hansen

Peter Andreas Hansen

Peter Andreas Hansen was a Danish astronomer, was born at Tønder, Schleswig.-Biography:The son of a goldsmith, Hansen learned the trade of a watchmaker at Flensburg, and exercised it at Berlin and Tønder, 1818–1820...

 (1795–1874), who worked on the geodetic survey of Denmark. There are two known points A and B, and two unknown points P₁ and P₂. From P₁ and P₂ 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 P₁ and P₂. See figure; the angles measured are (α₁, β₁, α₂, β₂).

Since it involves observations of angles made at unknown points, the problem is an example of resection

Resection (orientation)

Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...

 (as opposed to intersection).

Solution method overview

Define the following angles:
γ = P₁AP₂, δ = P₁BP₂, φ = P₂AB, ψ = P₁BA.
As a first step we will solve for φ and ψ.
The sum of these two unknown angles is equal to the sum of β₁ and β₂, yielding the following equation:


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

Law of sines

In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

 yields
and

combining these together we get


An entirely analogous reasoning on the other side yields


Setting these two equal gives


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


This is the second equation we need. Once we solve the two equations for the two unknowns and , we can use either of the two expressions above for to find P₁P₂ since AB is known. We can then find all the other segments using the law of sines.

Solution algorithm

We are given four angles (α₁, β₁, α₂, β₂) and the distance AB. The calculation proceeds as follows:

• Calculate

• Calculate

• Let , and then ,

• Calculate

or equivalently,

If one of these fractions has a denominator close to zero, use the other one.