Geometry and Navigation Help (page 2)
Introduction to Geometry and Navigation
Navigators and military people use a form of coordinate plane similar to that used by mathematicians. The radius is more often called the range , and real-world units are commonly specified, such as meters (m) or kilometers (km). The angle, or direction, is more often called the azimuth, heading , or bearing , and is measured in degrees clockwise from north. The basic scheme is shown in Fig. 10-14. The azimuth is symbolized α (the lowercase Greek alpha), and the range is symbolized 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 would travel if you wanted 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 some difference between geographic north and magnetic north. This difference, measured in degrees, is called the declination . 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 make the magnetic compass irrelevant, provided the equipment is in working order. (Most oceangoing vessels still have magnetic compasses on board, just in case of a failure of the more sophisticated equipment.)
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 likewise 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 < 360°
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 < 360°
Navigator’s Polar Vs Cartesian
Now suppose that you want to convert from NPC to Cartesian coordinates.
Here are the conversion formulas for translating the coordinates for a point (a 0 , r 0) in NPC to a point (x 0 , y 0) in the Cartesian plane:
x 0 = r 0 sin α 0
y 0 = r 0 cos α 0
These are similar to the formulas you use to convert MPC to Cartesian coordinates, except that the roles of the sine and cosine function are reversed.
In order to convert the coordinates of a point (x 0 , y 0) in Cartesian coordinates to a point (α 0 , r 0) in NPC, use these formulas:
α 0 = arctan ( x 0 / y 0 ) if y 0 > 0
α 0 = 180° + arctan ( x 0 / y 0 ) if y 0 < 0
r 0 = ( x 2 0 + y 2 0 ) 1/2
We can’t have y 0 = 0, because that produces an undefined quotient. If a value of α 0 thus determined happens to be negative, add 360° to get the “legitimate” value. These are similar to the formulas for converting Cartesian coordinates to MPC.
Geometry and Navigation Practice Problems
Suppose a radar set, that uses NPC, indicates the presence of a hovering object at a bearing of 300° and a range of 40 km. If we say that a kilometer is the same as a “unit,” what are the coordinates ( θ 0 , r 0 ) of this object in mathematician’s polar coordinates? Express θ 0 in both degrees and radians.
We are given coordinates (α 0 , r 0 ) = (300°,40). The value of r 0 , the radius, is the same as the range, in this case 40 units. As for the angle θ 0 , remember the conversion formulas given above. In this case, because α 0 is greater than 90° and less than 360°:
θ 0 = 450° − α 0
= 450° − 300° = 150°
Therefore, (θ 0 , r 0 ) = (150°,40). To express θ 0 in radians, recall that there are 2π radians in a full 360° circle, or π radians in a 180° angle. Note that 150° is exactly 5/6 of 180°. Therefore, θ 0 = 5π/6 rad, and we can say that (θ 0 , r 0 ) = (150°,40) = (5π/6,40). We can leave the “rad” off the angle designator here. When units are not specified for an angle, radians are assumed.
Suppose you are on an archeological expedition, and you unearth a stone on which appears a treasure map. The map says “You are here” next to an X, and then says, “Go north 40 paces and then west 30 paces.” Suppose that you let west represent the negative x axis of a Cartesian coordinate system, east represent the positive x axis, south represent the negative y axis, and north represent the positive y axis. Also suppose that you let one “pace” represent one “unit” of radius, and also one “unit” in the Cartesian system. If you are naïve enough to look for the treasure and lazy enough so you insist on walking in a straight line to reach it, how many paces should you travel, and in what direction, in navigator’s polar coordinates? Determine your answer to the nearest degree, and to the nearest pace.
Determine the ordered pair in Cartesian coordinates that corresponds to the imagined treasure site. Consider the origin to be the spot where the map was unearthed. If we let ( x 0 , y 0 ) be the point where the treasure should be, then 40 paces north means y 0 = 40, and 30 paces west means x 0 = −30:
( x 0 , y 0 ) = (−30,40)
Because y 0 is positive, we use this formula to determine the bearing or heading α 0 :
α 0 = arctan ( x 0 / y 0 )
= arctan (−30/40)
= arctan –0.75
This is a negative angle, so to get it into the standard form, we must add 360°:
α 0 = −37° + 360° = 360° − 37°
To find the value of the range, r 0 , use this formula:
r 0 = ( x 2 0 + y 2 0 ) 1/2
= (30 2 + 40 2 ) 1/2
= (900 + 1600) 1/2
= 2500 1/2
This means (α 0 , r 0 ) = (323°,50). Proceed 50 paces, approximately north by northwest. Then, if you have a shovel, go ahead and dig!
Practice problems for these concepts can be found at: Polar Coordinates Practice Test.
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