Thursday, April 17, 2014
A Couple of Practical Positioning Examples
The problem presented in the last blog was somewhat contrived but it illustrates two methods that can be used to find a line of position in a plane. The straight line assumed that we can determine the direction of north at the unknown position in order to measure the angle. If we can do this for one point we can also do it for another and we can find the position by determining the point of intersection for two lines.
If the direction of north is not readily available one can use three known positions and two angular separations to determine the unknown position. These are shown in the following figure.
For each angle we can find the center of a circle of position using the procedure described in the last blog then we can draw arcs from the common point to determine the location of X.
The line P2X turns out to be perpendicular to the line between the centers Z12 and Z23 and the point X is the same distance from this line as P2.
One uses two circles to determine a position in Celestial Navigation. The latitude is a circle for which the distance from the North Pole is a constant. Latitude can be found by measuring the altitude of the Pole. Measuring the altitude of a star gives a second circle of position. The point on the Earth's surface directly beneath the star at the time of the altitude's measurement is needed and can be found using the star's right ascension and declination and sidereal time. The altitude of the star determines the distance of the second circle from the point beneath the star. And the intersection of these two circles gives an estimate of position. One needs the altitude of at least two points of the celestial sphere and the time of their measurement to determine a position on the surface of the Earth. A third star measurement or a computed position based on the heading, speed and time from the last known position is needed to determine which of the two possible points of intersection to use for the estimated position.