飞跃北极数学模型(有垄文板).doc

Flying over the North Pole Area Abstract 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 with 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 of 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 an 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 earth 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 latitude,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. Assumptions and Hypotheses 、After each of the twos adjacent to the voyage，calculate the shortest distance between two points when plane flies through two places that is adjacent; 、The plane fly without refueling,ignoring the lifting and lowering time,either; 、The total aircraft flying in the Earths gravitational fi