The Global Grid Help (page 2)
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°.
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