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, can the location of an observer at an unknown point be found?
Given these points, and that is between and as seen from, an observer at P can resolve that the line segment subtends an angle and the segment subtends an angle ; the solution to establishing the position of the point 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 is known as the "danger circle", and observations made on 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 angles as and as gives:
by using the sum of the angles formula for the quadrilateral. The variable represents the internal angle in this quadrilateral at point..

Second equation

Applying the law of sines in triangles and, can be expressed in two different ways:
A useful trick at this point is to define an auxiliary angle such that
With this substitution the equation becomes
Now two known trigonometric identities can be used, namely
to put this in the form of the second equation;
Now these two equations in two unknowns must be solved. Once and are known the various triangles can be solved straightforwardly to determine the position of. The detailed procedure is shown below.

Solution algorithm

Given are two lengths, and three angles, the solution proceeds as follows.
  • calculate where atan2 is a computer function, also called the arctangent of two arguments, that returns the arctangent of the ratio of the two values given. Note that in Microsoft Excel the two arguments are reversed, so the proper syntax would be = atan2, BC*\sin). The atan2 function correctly handles the case where one of the two arguments is zero.
  • calculate
  • calculate
  • find
  • find
  • find
  • find
If the coordinates of and are known in some appropriate Cartesian coordinate system then the coordinates of can be found as well.

Geometric (graphical) solution

By the inscribed angle theorem the locus of points from which subtends an angle is a circle having its center on the midline of ; from the center of this circle, subtends an angle. Similarly the locus of points from which subtends an angle is another circle. The desired point is at the intersection of these two loci.
Therefore, on a map or nautical chart showing the points, the following graphical construction can be used:
  • Draw the segment, the midpoint and the midline, which crosses perpendicularly at. On this line find the point such that Draw the circle with center at passing through and.
  • Repeat the same construction with points and the angle.
  • Mark at the intersection of the two circles
This method of solution is sometimes called Cassini's method.

Rational trigonometry approach

The following solution is based upon a paper by N. J. Wildberger. It has the advantage that it is almost purely algebraic. The only place trigonometry is used is in converting the angles to spreads. There is only one square root required.
  • define the following:
  • now let:
  • the following equation gives two possible values for :
  • choosing the larger of these values, let:
finally:

Solution via geometric algebra

Ventura et al. solve the planar and three-dimensional Snellius-Pothenot problem via vector geometric algebra and conformal geometric algebra. The authors also characterize the solutions' sensitivity to measurement errors.

The indeterminate case

When the point happens to be located on the same circle as, the problem has an infinite number of solutions; the reason is that from any other point located on the arc of this circle the observer sees the same angles and as from . Thus the solution in this case is not uniquely determined.
The circle through is known as the "danger circle", and observations made on this circle should be avoided. It is helpful to plot this circle on a map before making the observations.
A theorem on cyclic quadrilaterals is helpful in detecting the indeterminate situation. The quadrilateral is cyclic iff a pair of opposite angles are supplementary i.e. iff. If this condition is observed the computer/spreadsheet calculations should be stopped and an error message returned.

Solved examples

. are three objects such that = 435, = 320, and = 255.8 degrees. From a station it is observed that = 30 degrees and = 15 degrees. Find the distances of from..
Answer: = 790, = 777, = 502.
A slightly more challenging test case for a computer program uses the same data but this time with = 0. The program should return the answers 843, 1157 and 837.

Naming controversy

The British authority on geodesy, George Tyrrell McCaw wrote that the proper term in English was Snellius problem, while Snellius-Pothenot was the continental European usage.
McCaw thought the name of Laurent Pothenot did not deserve to be included as he had made no original contribution, but merely restated Snellius 75 years later.