Positional Info from Bearings

From: Lewis Jardine <jardine_at_rmcs.cranfield.ac.uk>
Date: Wed, 07 May 1997 16:13:39 +0100


Hi All

I have noticed that there is an on-going rumble over whether you can derive your position from the bearings of two fixed points.

I am pleased to see that most people have come to the correct conclusion. However, for those who would like it I have found a simple mathmatical method for showing this.

\begin{boringoldmaths}

There is a fairly well known trigonometric rule for triangles known as the Sine Rule. It states:

In any triangle: a/(sin A) = b/(sin B) = c/(sin C) = 2R

Where a, b & c are the lengths of the sides and A, B & C are the angles of the opposite corners.
(R is the length of the circumcirle for those who are interested.)

                  (1)          a          (2)
                    +---------------------+
                     \ B               C /
                      \                 / 
                       \               /
                        \             / 
                      c  \           /  b
                          \         / 
                           \       / 
                            \     /
                             \ A /
                              \ /
                               +
                            Observer


What this means for our example is that for any two fixed points (1 & 2) which are a known distance (a) apart, observed from a third point. There will be a specific angle (A) between them. However, as can be seen from the Sine Rule this only constrains the formula to:

a/(sin A) = fixed value = b/(sin B) = c/(sin C) = 2R
            ^^^^^^^^^^^

Thus, there are an infinite number of possible values for b and sin B, provided that b/(sin B) = the constant. These values have coresponding values of c & sin C (the pairs are limited by the geometry of triangles) which describe an arc as Loren stated.

Basically, the equation has 4 unknowns a, A and b, B or c, C. By knowning the distance a and the angle A we only known 2 of these thus it is impossible to solve without fixing one of the unknowns...

\end{boringoldmaths}

Navigators normally get round this problem by cheating. They use their compass to determine where north is which gives them a direction (of a third point) too.

Cheers

        Lewis


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