飞跃北极数学模型英文板.doc
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1、Flying over the North Pole AreaAbstract We use three models to explain mathematically the meaning of “The flying time from Beijing to Detroit will be four hours shorter”,considering two conditions:the earth is a ball and it is a rotating ellipsoid.In the first case ,we assum the earth is a sphere wi
2、th the radius of 6371.As we know,e the shortest distance between two points on a sphere is the length of the inferior arc. So we put the center of the earth as the origin of coordinates to set a three-dimensional Cartesian coordinate system. By calculating the relevant vector,we got the expression o
3、f the flight time from Beijing to Detroit and the save time ,too. Finally got the saved time of 3.92 hours through running the C programming language,and compare it with the time that given by a geographic website ,US. It turned out that our result is exact.The second case,we consider the earth as a
4、n ellipsoid. As it is given, the difference of latitude between two adjacent points is very small, so we can use the compression ratio method and the compression ratio=0.998,is a fixed value. Then we multiplied it by the route that we got in the first case, which is the route that we consider the ea
5、rth as an ellipsoid, and then we got the saved time is 3.9265hours.The third case, we regard the earth as a rolling ellipsoid. The main idea is using the reduced latitude to find out the geodesic distance between two points on an ellipsoid. Firstly, we turn the geodetic latitude to the reduced latit
6、ude,then we use the Bessel to deduce the approximate length between two points expressed by reduced latitude and get the final answer that the saved fight time is 4.041 hours.1、 Assumptions and Hypotheses (1) 、After each of the twos adjacent to the voyage,calculate the shortest distance between two
7、points when plane flies through two places that is adjacent;(2) 、The plane fly without refueling,ignoring the lifting and lowering time,either; (3) 、The total aircraft flying in the Earths gravitational field, and the Earths rotation and revolution of the aircrafts absolute speed and size of flight
8、is negligible;(4) 、The flight speed of only consider the speed of the aircraft relative to the earth;(5) 、The speed remains unchanged.2、 Symbol Description:the three-dimensional cartesian coordinates ofon the spherical (or on the ellipsoid):the longitude and latitude ofon the spherical (or on the el
9、lipsoid): the flight time: the saved flight time:sum of sub-flying aircraft, the total journey distance:Radius of the earth:aircraft altitude: the minor axis and the long axle of the prolate spheroid3、 The earth as a ballAssuming the earth is a sphere with the radius of 6371.At this point A, B is 10
10、 kilometers above the Earths two points away from the ground,by using the knowledge of differential geometry we know:over A, B two points worse than the great circle arc length is the shortest distance between the two points.A, B are the coordinates of two points:The flight distance from A to B: The
11、 flight time from A to B:Solution of the model:The center of the earth as the origin of coordinates,the equatorial plane as plane XOY,the meridian plane as plane XOZ,establish a three dimensional Cartesian coordinate system,vector of each point on the sphere can be expressed as follow:The distance b
12、etween the two points can be expressed as follow:The total time of flight routes:;Direct from Beijing, the sailing time for Detroit:So the time saved by direct flight:According to the results calculated by MATLAB program:(unit:km)The distance between Beijing and A1:1113.2The distance between A1 and
13、A2:1758.8The distance between A2 and A3:4624.4The distance between A3 and A4:1339.1The distance between A4 and A5:641.2The distance between A5 and A6:538.6The distance between A6 and A7:651.5The distance between A7 and A8:497.6The distance between A8 and A9:227.8The distance between A9 and A10:2810.
14、9The distance between A10 and Detroit:331.9Direct from Beijing to Detroit:10684.9Along the flight path of the original total time of flight:14.8hAlong the direct flight from Beijing to Detroit total time flight:10.9hTime saved:3.9hBy comparing the results of the above methods and the actual situatio
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