A. Geometry
1) Introduction to Vectors to Represent a Journey.
If a displacement has taken place, this movement can be written as a displacement vector.
The position of the plane in the picture relative to the airport can be described as 5 miles, bearing 053°.
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If the same picture is drawn on graph paper:
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The position of the plane could be described as 4 miles East and 3 miles North. Instead of saying 4 miles east and 3 miles north we could write (4/3). This is a vector representation of the position of the plane. The distance to the east is written first followed by the distance to the north.
If the plane moves from position S to position Q and the movement is 2 miles east and 6 miles north the total journey can be represented as a sum (4/3) + (2/6) = (6/9). Vectors can be added by adding the top components 4 + 2 and the bottom components 3 + 6.
2) Angles and Bearings
Bearings are measured from the north line, the line or axis about which the earth rotates. It cuts the surface of the earth at what are called the north and south poles. The pole star is almost on this line and so appears to be fixed in the heavens, while other stars seems to rotate about the axis. The pole star gives a fixed direction from which navigators used to set their course. Today the magnetic compass which points in approximately the same direction is used to set a course.
To set the bearing i) always start from the north line and ii) always measure the angle clockwise.
The following example will be used to demonstrate how to draw a model of a journey.
A pilot on an aircraft made a two stage journey.
Stage A: 500 miles, bearing 060°
Stage B: 300 miles, bearing 150°
1st step: Draw the north line (NA), and measure the 060° angle clockwise from the north line. Draw a line to represent 500 miles from the point A to B. This represents the first leg of the journey.
2nd step: At point B draw another north, line measure the angle of 150°, and draw the length 300 miles.
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If the pilot had flown from A to C, we could measure the bearing of C from A and measure the length of the line segment AC. The stages of this journey can also be written as a displacement vector.
(500,060°) followed by (300,150°).
3) Navigation and Spherical Geometry
The following topic is being introduced with the intent of providing an enrichment topic for students in the higher mathematics courses, since some experience with trigonometry, logarithms and rotational symmetry would be required.
There are several methods of navigation: Pilotage is the method of flying an aircraft from one point to another by the observation of landmarks either already known or recognized from a map. This method has limitations, because if the flight is over poorly mapped country, over large bodies of water, or at night when visibility is poor, it is difficult to use the landmarks. This method is most efficient when used with other forms of navigation.
The method Dead Reckoning is the basic method of navigation. It uses known or established factors such as wind direction, wind velocity, and air speed to compute a position from a known position. Lindbergh used Dead Reckoning on his flight from New York to Paris.
Radio Navigation is method of directing an aircraft from one point to another by radio waves. Its major feature is that one does not wait to see the ground to make approaches and landings.
Celestial Navigation is the oldest method of navigation. This is the determination of an aircraft by the observation of the celestial bodies to determine position.
Using a globe is the only accurate means of representing the spherical surface of the earth9. To make a mathematical model of the earth choose the diameter NS passing through the north and south poles as axis of rotation. If O is the midpoint of NS, and OQ is perpendicular to NS then the locus of Q is the equator.
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To make a two dimensional diagram of a sphere, the equator and the parallels of latitude and the meridians are drawn.
Latitude and Longitude. The coordinates used to describe points on the Earth’s surface.
The position of a point A on the surface can be determined by stating:
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a) Which circular section perpendicular to NS contains A.
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b) Which position of the semi-circle rotating about NS contains A.
These pieces of information are given by specifying
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i) the latitude and
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ii) the longitude of A.
Figure III shows the equator and the circles, or parallels of latitude 60°N and 50°S. The range of latitude is from 0° to 90° north or south of the equator.
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Figure IV shows the equator, the Greenwich meridian NGS and the semicircles, or meridian of longitude 40°W and 20°E. The range of longitude is from 0° to 180° East and West of Greenwich.
Figure V shows point A with latitude to N, and longitude go W.
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Figure VI shows the point P with latitude 60°N and longitude 50°W.
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Figure VII shows the point Q with latitude 30° S and longitude 50°E.
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The Solution of Right Spherical Triangles
A spherical angle is formed by intersecting arcs of two great circles. The three important properties of spherical triangles are:
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a) the sum of the lengths of any two sides exceeds the length of the third side;
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b) the sum of the lengths of the three sides is less than 360°;
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c) the sum of the angles is greater than 180° but less than 540°.
To find the great circle distance between two points A and B, the triangle of reference is constructed as follows:
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i) The great circle joining A and B form one side of the triangle;
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ii) The meridians through both A and B form the other two sides.
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Two such triangles are formed and either can be used to obtain the great circle distance from A to B.
Example: Suppose A has latitude 60°N, longitude 55°E, and B has latitude 60°N, longitude 13°W. The length of arc AB can be calculated as follows.
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* The radius of a circle of latitude t° = R cos t° where R the radius of the earth is 6400 km.
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= 68 x2 x 3.14 x 6400 x cos 60°
= 3800 km. 360°
If a plane or ship follows a great circle path, its course is the angle the path makes with the meridian of the ship and is measured from north through east to the path of the ship.
