Positional Info from Bearings

From: Lewis Jardine (jardine@rmcs.cranfield.ac.uk)
Date: Wed 07 May 1997 - 18:13:39 EEST

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.


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 /
                              \ /

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


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.



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