**From:** Lewis Jardine (*jardine@rmcs.cranfield.ac.uk*)

**Date:** Wed 07 May 1997 - 18:13:39 EEST

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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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