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.

\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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