Thursday, April 17, 2014

Determining One's Position on a Map

  With the recent difficulties encountered over the borders of Ukraine and the search for the black boxes of Flight 370 it might be useful to review some surveying problems. How can one determine one's position on a map? First of all one needs to determine the positions of locations relative known positions in a coordinate system to draw the map. This can be done by using triangulation in which one measures the angles of the unknown locations relative to pairs of known positions then computes the unknown positions using geometry or trigonometry. One can do something similar to determine one's position, P, on the map. Only two angles are needed. One can use the angle of one, A, relative to north and the angle between it and a second point, B.

The angle, φ, of the line PA relative to north is the same for all points on it including the point A so we have one line of position for P. A second line of position uses the fact that all points for which the angle between points A and B is the same angle θ lie on a circle. This is the inscribed angle theorem which is Prop. XXI in Bk I of Euclid's Elements.The trick is to find the center and radius of this circle. The center is equidistant from points A and B so it lies on a line bisecting the one between them.

The position for P on the line bisecting the line AB forms an isosceles triangle and we can assume that the center of the circle, Z, is at a distance x from the line AB. The exterior angle theorem (Euclid, Bk III, Prop. XXXII) tells us that the angle of A relative to the perpendicular bisector of line AB is equal to the sum of the two opposite angle of triangle APZ which is θ. From plane trigonometry we know that b/x = 2 tan(θ).

 We can also determine P geometrically using the following construction.

The point D is the intersection of two lines, one perpendicular to the line AB and a second drawn through the midpoint, C, of line AB at an angle θ relative to the perpendicular bisector, CZ. It gives us the distance x from the line AB. The intersection of the circle about Z through A and B with the line at angle φ from north through A is the required position.

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