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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