Introduction to the Navigator's Way
Navigators and military people use a coordinate plane similar to the one preferred by mathematicians. The radius is called the range, and realworld units are commonly specified, such as meters (m) or kilometers (km). The angle, or direction, is called the azimuth, heading, or bearing, and is measured in degrees clockwise from geographic north. The basic scheme is shown in Fig. 514. The azimuth is symbolized α (the lowercase Greek alpha), and the range is symbolized r. The position of a point is definable by an ordered pair ( α , r ).
What Is North?
There are two ways of defining “north,” or 0°. The more accurate, and thus the preferred and generally accepted, standard uses geographic north. This is the direction you should travel if you want to take the shortest possible route over the earth’s surface to the north geographic pole. The less accurate standard uses magnetic north. This is the direction indicated by the needle in a magnetic compass.
For most locations on the earth’s surface, there is a difference between geographic north and magnetic north. This difference, measured in degrees, is called declination. (This, by the way, is not the same thing as the declination used in celestial coordinates!) Navigators in olden times had to know the declination for their location when they couldn’t use the stars to determine geographic north. Nowadays, there are electronic navigation systems such as the Global Positioning System (GPS) that are far more accurate than any magnetic compass, provided the equipment is in working order.
Strict Restrictions
In navigator’s polar coordinates, the range can never be negative. No navigator ever talks about traveling –20 km on a heading of 270°, for example, when they really mean to say they are traveling 20 km on a heading of 90°. When working out certain problems, it’s possible that the result might contain a negative range. If this happens, the value of r should be multiplied by –1 and the value of α should be increased or decreased by 180° so the result is at least 0° but less than 360°.
The azimuth, bearing, or heading must also conform to certain values. The smallest possible value of α is 0° (representing geographic north). As you turn clockwise as seen from above, the values of α increase through 90° (east), 180° (south), 270° (west), and ultimately approach, but never reach, 360° (north again). We therefore have these restrictions on the ordered pair ( α , r ):
0° ≤ α < 360°
r ≥ 0
Mathematician’s Polar Vs Navigator’s Polar
Sometimes it is necessary to convert from mathematician’s polar coordinates (let’s call them MPC for short) to navigator’s polar coordinates (NPC), or vice versa. When making the conversion, the radius of a particular point, r _{0} , is the same in both systems, so no change is necessary. But the angles differ.
If you know the direction angle θ _{0} of a point in MPC and you want to find the equivalent azimuth α _{0} in NPC, first be sure θ _{0} is expressed in degrees, not radians. Then you can use either of the following conversion formulas, depending on the value of θ _{0} :
α _{0} = 90° – θ _{0} if 0° ≤ θ _{0} ≤ 90°
α _{0} = 450° – θ _{0} if 90° < θ _{0} < 90°
If you know the azimuth α _{0} of a distant point in NPC and you want to find the equivalent direction angle θ _{0} in MPC, then you can use either of the following conversion formulas, depending on the value of α _{0} :
θ _{0} = 90° – α _{0} if 0° ≤ α _{0} ≤ 90°
θ _{0} = 450° – α _{0} if 90° < α _{0} < 90°

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