Snellius–Pothenot problem

The Snellius–Pothenot problem is a trigonometry problem first described in the context of planar surveying where known points are used to solve an unknown one. Given three known points A, B, C, can the location of an observer at an unknown point P be found?

Given these points, and that C is between A and B as seen from P, an observer at P can resolve that the line segment AC subtends an angle α and the segment CB subtends an angle β; the solution to establishing the position of the point P can be variously found through graphical geometry, rational trigonometry, and geometric algebra.

An indeterminate case exists when all four points fall on the same circle, giving an infinite number of solutions. Thus the circle through ABC is known as the "danger circle", and observations made on (or very close to) this circle should be avoided.

Since it involves the observation of known points from an unknown point, the problem is an example of resection. Historically it was first studied by Willebrord Snellius, who found a solution around 1615.

Formulating the equations

First equation

Denoting the (unknown) angles ∠CAP as x and ∠CBP as y gives:

x

+

y

=

2

π

−

α

−

β

−

C

{\displaystyle x+y=2\pi -\alpha -\beta -C}

by using the sum of the angles formula for the quadrilateral PACB. The variable C represents the (known) internal angle in this quadrilateral at point C. (Note that in the case where the points C and P are on the same side of the line AB, the angle ∠C will be greater than π).

Second equation

Applying the law of sines in triangles △PAC and △PBC, PC can be expressed in two different ways:

P

C

¯

=

A

C

¯

sin

⁡

x

sin

⁡

α

=

B

C

¯

sin

⁡

y

sin

⁡

β

.

{\displaystyle {\overline {PC}}={\frac {{\overline {AC}}\sin x}{\sin \alpha }}={\frac {{\overline {BC}}\sin y}{\sin \beta }}.}

A useful trick at this point is to define an auxiliary angle φ such that

tan

⁡

ϕ

=

B

C

sin

⁡

α

A

C

sin

⁡

β

.