Hansen's problem

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 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 (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:

\phi+\psi=\beta_1+\beta_2

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

\frac{AB}{P_2 B}=\frac{\sin \alpha_2}{\sin \phi}

and

\frac{P_2 B}{P_1 P_2}=\frac{\sin \beta_1}{\sin \delta}

combining these together we get

\frac{AB}{P_1 P_2}=\frac{\sin \alpha_2 \sin \beta_1}{\sin \phi \sin \delta}

An entirely analogous reasoning on the other side yields

\frac{AB}{P_1 P_2}=\frac{\sin \alpha_1 \sin \beta_2}{\sin \psi \sin \gamma}

Setting these two equal gives

\frac{\sin \phi}{\sin \psi}=\frac{\sin \gamma \sin \alpha_2 \sin \beta_1}{\sin \delta \sin \alpha_1 \sin \beta_2} = k

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

\tan \frac{\phi - \psi}{2}=\frac{k-1}{k+1}\tan\frac{\phi+\psi}{2}

This is the second equation we need. Once we solve the two equations for the two unknowns $\phi$ and $\psi$ , we can use either of the two expressions above for $\frac{AB}{P_1 P_2}$ to find P₁P₂ since AB is known. We can then find all the other segments using the law of sines.^[1]

Solution algorithm

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

Calculate $\gamma=\pi-\alpha_1-\beta_1-\beta_2,\quad \delta=\pi-\alpha_2-\beta_1-\beta_2$

Calculate $k=\frac{\sin \gamma \sin \alpha_2 \sin \beta_1}{\sin \delta \sin \alpha_1 \sin \beta_2}$

Let $s=\beta_1+\beta_2$ , $d=2 \arctan \left[ \frac{k-1}{k+1}\tan(s/2)\right]$ and then $\phi=(s+d)/2$ , $\psi=(s-d)/2.$

Calculate

P_1 P_2 = AB \frac{\sin \phi \sin \delta}{\sin \alpha_2 \sin \beta_1},

or equivalently,

P_1 P_2 = AB \frac{\sin \psi \sin \gamma}{\sin \alpha_1 \sin \beta_2}.

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

References

↑ Udo Hebisch: Ebene und Sphaerische Trigonometrie, Kapitel 1, Beispiel 4 (2005, 2006)

This article is issued from Wikipedia - version of the Monday, May 05, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.