The Snellius–Pothenot problem is a problem in planar surveying. Given three known points A, B and C an observer at an unknown point P observes that the segment AC subtends an angle α and the segment CB subtends an angle β; the problem is to determine the position of the point P. (See figure; the point denoted C is between A and B as seen from P).
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 Snellius, he found a solution around 1615.
Contents 
First equation
Denoting the (unknown) angles CAP as x and CBP as y we get:
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 we can express PC in two different ways:
A useful trick at this point is to define an auxiliary angle φ such that
(A minor note: we should be concerned about division by zero, but consider that the problem is symmetric, so if one of the two given angles is zero we can, if needed, rename that angle alpha and call the other (nonzero) angle beta, reversing the roles of A and B as well. This will suffice to guarantee that the ratio above is well defined. An alternative approach to the zero angle problem is given in the algorithm below.)
With this substitution the equation becomes
We can use two known trigonometric identities, namely
to put this in the form of the second equation we need:
We now need to solve these two equations in two unknowns. Once x and y are known the various triangles can be solved straightforwardly to determine the position of P.^{[1]} The detailed procedure is shown below.
We are given two lengths AC and BC, and three angles α, β and C. The solution proceeds as follows:
If we know the coordinates of A: x_{A},y_{A} and C: x_{C},y_{C} in some appropriate coordinate system then we can find the coordinates of P as well.
By the inscribed angle theorem the locus of points from which AC subtends an angle α is a circle having its center on the midline of AC; from the center O of this circle AC subtends an angle 2α. Similarly the locus of points from which CB subtends an angle β is another circle. The desired point P is at the intersection of these two loci.
Therefore on a map or nautical chart showing the points A, B, C, the following graphical construction can be used:
This method of solution is sometimes called Cassini's method.
When the point P happens to be located on the same circle as A, B and C, the problem has an infinite number of solutions; the reason is that from any other point P' located on the arc APB of this circle the observer sees the same angles alpha and beta as from P (inscribed angle theorem). Thus the solution in this case is not uniquely determined.
The circle through ABC is known as the "danger circle", and observations made on (or very close to) 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 APBC is cyclic iff a pair of opposite angles (such as the angle at P and the angle at C) are supplementary i.e. iff . If this condition is observed the calculations should be stopped and an error message ("indeterminate case") returned.
(Adapted form Bowser^{[2]}, exercise 140, page 203). A, B and C are three objects such that AC = 435 (yards), CB = 320, and C = 255.8 degrees. From a station P it is observed that APC = 30 degrees and CPB = 15 degrees. Find the distances of P from A, B and C. (Note that in this case the points C and P are on the same side of the line AB, a different configuration from the one shown in the figure).
Answer: PA = 790, PB = 777, PC = 502.
A slightly more challenging test case for a computer program uses the same data but this time with CPB = 0. The program should, without blowing up, return the answers 843, 1157 and 837.
The British authority on geodesy, George Tyrrell McCaw (18701942) wrote^{[3]} that the proper term in English was Snellius problem, while SnelliusPothenot was the continental European usage.
McCaw thought the name of Laurent Pothenot (16501732) did not deserve to be included as he had made no original contribution, but merely restated Snellius 75 years later.
