The Global Grid Help (page 3)
All the trigonometry we’ve dealt with until now has been on flat surfaces, or in space where all the lines are straight. But in the real world—in particular, on the surface of the earth—lines are not always straight. The route an airline pilot takes to get from New York to Rome is not a straight line; if it were, the aircraft would have to cut through the interior of the planet. The paths of over-the-horizon radio signals are not straight lines. In this chapter, we’ll see how trigonometry works on the surface of the earth.
When the geometric universe is confined to the surface of a sphere, there is no such thing as a straight line or line segment. The closest thing to a straight line in this environment is known as a great circle or geodesic. The closest thing to a straight line segment is an arc of a great circle or geodesic arc.
The surface of a sphere is the set of all points in space that are equidistant from some center point P. All paths on the surface of a sphere are curved. If Q and R are any two points on the surface of a sphere, then the straight line segment QR cuts through the interior of the sphere. Navigators and aviators cannot burrow through the earth, and radio waves can’t do it either. (Electrical currents at extremely low frequencies can. This is how land-based stations communicate with submarines.)
The shortest distance between any two points Q and R on the surface of the sphere is an arc that lies in a plane passing through P, the center point (Fig. 11-1). The arc QR, representing the shortest on-the-surface distance between the two points, is always part of a great circle, which is a circle on the sphere that has P as its center point. It never fails, as long as the sphere is perfectly round. The surface of the earth, averaged to sea level, is close enough to a perfect sphere that this principle holds quite well. Henceforth in this chapter, when we say “the surface of the earth,” it should be understood that we mean “the sphere corresponding to the surface of the earth at sea level.” We won’t be concerned with local irregularities such as hills, mountains, or buildings.
Latitude And Longitude Revisited
Latitude and longitude refer to angles that can be used to uniquely define the position of a point on a sphere, given certain references.
Latitude is defined as an angle, either north (positive) or south (negative), with respect to a great circle representing the equator. The equator is the set of points on the surface of the sphere equidistant from the north geographic pole and the south geographic pole. The geographic poles are the points at which the earth’s rotational axis intersects the surface. The latitude, commonly denoted θ, can be as large as +90° or as small as –90°, inclusive. That is:
–90° ≤ θ ≤ +90°
90°S ≤ θ ≤ 90°N
where “S” stands for “south” and “N” stands for “north.”
Longitude is defined as an angle, either east (positive) or west (negative), with respect to a great circle called the prime meridian. Longitude is always measured around the equator, or around any circle on the surface of the earth parallel to the equator. The prime meridian has its end points at the north pole and the south pole, and it intersects the equator at a right angle. Several generations ago, it was decided by convention that the town of Greenwich, England, would receive the distinction of having the prime meridian pass through it. For that reason, the prime meridian is also called the Greenwich meridian. (When the decision was made, as the story goes, people in France were disappointed, because they wanted the officials to choose the prime meridian so it would pass through Paris. If that had happened, we would be discussing the Paris Meridian right now.) Angles of longitude, denoted ø , can range between –180° and +180°, not including the negative value:
–180° < ø ≤ +180°
180°W < ø ≤ 180°E
where “W” stands for “west” and “E” stands for “east.”
For any given angle θ between and including -90° and +90°, there is a set of points on the earth’s surface such that all the points have latitude equal to θ . This set of points is a circle parallel to the equator; for this reason, all such circles are called parallels (Fig. 11-2A). The exceptions are at the extremes θ = -90° and θ = +90°; these correspond to points at the south geographic pole and the north geographic pole, respectively.
The radius of a given parallel depends on the latitude. When θ = 0°, the parallel is the equator, and its radius is equal to the earth’s radius. The earth is not quite a perfect sphere—it is slightly oblate—but it is almost perfect. If we imagine the earth as a perfect sphere with the oblateness averaged out, then we can regard the radius of the earth as equal to 6371 kilometers. That is the radius of the parallel corresponding to θ = 0°. For other values of θ, the radius r (in kilometers) of the parallel can be found according to this formula:
The earth’s circumference is approximately 6371 × 2 π, or 4.003 × 10 4 kilometers. Therefore, the circumference k (in kilometers) of the parallel whose latitude is θ can be found using this formula:
k = (4.003 × 10 4 ) cos θ
For any given angle ø such that –180° < ø ≤ + 180°, there is a set of points on the earth’s surface such that all the points have longitude equal to ø . This set of points is a half-circle (not including either of the end points) whose center coincides with the center of the earth, and that intersects the equator at a right angle, as shown in Fig. 11-2B.
All such open half-circles are called meridians. The end points of any meridian, which technically are not part of the meridian, are the south geographic pole and the north geographic pole. (The poles themselves have undefined longitude.)
All meridians have the same radius, which is equal to the radius of the earth, approximately 6371 kilometers. All the meridians converge at the poles. The distance between any particular two meridians, as measured along a particular parallel, depends on the latitude of that parallel. The distance between equal-latitude points on any two meridians ø 1 and ø 2 is greatest at the equator, decreases as the latitude increases negatively or positively, and approaches zero as the latitude approaches -90° or +90°.
Distance Per Unit Latitude
As measured along any meridian (that is, in a north-south direction), the distance d lat-deg per degree of latitude on the earth’s surface is always the same. It can be calculated by dividing the circumference of the earth by 360. If d lat-deg is expressed in kilometers, then:
d lat-deg = (4.003 × 10 4 )/360 = 111.2
The distance d lat-min per arc minute of latitude (in kilometers) can be obtained by dividing this figure by exactly 60:
d lat-min = 111.2/60.00 = 1.853
The distance d lat-sec per arc second of latitude (in kilometers) is obtained by dividing by exactly 60 once again:
d lat-sec = 1.853/60.00 = 0.03088
This might be better stated as d lat-sec = 30.88 meters. That’s a little more than the distance between home plate and first base on a major league baseball field.
Distance Per Unit Longitude
As measured along the equator, the distances d lon-deg (distance per degree of longitude), d lon-min (distance per arc minute of longitude), and d lon-sec (distance per arc second of longitude), in kilometers, can be found according to the same formulas as those for the distance per unit latitude. That is:
d lon-deg = (4.003 × 10 4 )/360 = 111.2
d lon-min = 111.2/60.00 = 1.853
d lon-sec = 1.853/60.00 = 0.03088
These formulas do not work when the east-west distance between any two particular meridians is determined along a parallel other than the equator. In order to determine those distances, the above values must be multiplied by the cosine of the latitude θ at which the measurement is made. Thus, the formulas are modified into the following:
d lon-deg = 111.2 cos θ
d lon-min = 1.853 cos θ
d lon-sec = 0.03088 cos θ
The last formula can be modified for d lon-sec in meters, as follows:
d lon-sec = 30.88 cos θ
The Global Grid Practice Problems
Imagine that a certain large warehouse, with a square floor measuring 100 meters on a side, is built in a community at 60° 0′ 0″; north latitude. Suppose that the warehouse is oriented “kitty-corner” to the points of the compass, so its sides run northeast-by-southwest and northwest-by-southeast. What is the difference in longitude, expressed in seconds of arc, between the west corner and the east corner of the warehouse?
The situation is diagrammed in Fig. 11-3. Let be the difference in longitude between the west and east corners of the warehouse. (The Δ symbol in this context is an uppercase Greek letter delta, which means “the difference in”; it’s not the symbol for a geometric triangle.)
First, we must find the distance in meters between corners of the warehouse. This is equal to 100 × 2 1/2 , or approximately 141.4, meters. Now let’s find out how many meters there are per arc second at 60° 0′ 0″ north latitude:
In order to obtain , the number of arc seconds of longitude between the east and west corners of the warehouse, we divide 141.4 meters by 15.44 meters per arc second, obtaining:
We round off to three significant figures because that is the extent of the accuracy of our input data (100 meters along each edge of the warehouse). If we want to express this longitude difference in degrees, minutes, and seconds, we write:
What is the difference in latitude, expressed in seconds of arc, between the north and the south corners of the warehouse described above?
We already know that the distance between corners of the warehouse is 141.4 meters. We also know that there are 30.88 meters of distance per arc second, as measured in a north-south direction, at any latitude. Let Δ θ wh be the difference in latitude between the north and the south corners. We divide 141.4 meters by 30.88 meters per arc second, obtaining:
Again, we round off to three significant figures, because that is the extent of the accuracy of our input data (100 meters along each edge of the warehouse). If we want to express this longitude difference in degrees, minutes, and seconds, we write:
Practice problems for these concepts can be found at: Global Trigonometry Practice Test
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