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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
15、n,they are consistent with each other.That means the report of the time saved from Beijing to Detroit is 4 hourshas scientific basis.4、 The earth as an ellipsoidWhen we assume that when the earth is rotating ellipsoid，its told that the major semi axis (is the radius of the earth), and semi-minor axi
16、s ,which is differ from . We can assume ellipsoid along the minor axis by the compression of the sphere obtained, if we ignore the small gap between the and and the sphere of the original points such as ,which are in the ellipsoid surface after compression. Whats more, the greater the latitude diffe
17、rence between two points on the arc length of the compressed the greater the points on the same latitude, little change in the compressed arc length.The conditions given by the subject shows: the latitude of the difference between adjacent points of are , the difference of latitude between Beijing a
18、nd Detroit is ,and its a small difference.So here the compression method we use is effective and the compression ratio is constant ;Here comes another situation:when assuming the Earth is spherical,we have calculated the length of the route,and put this result multiplied by ,so we can get the routes
19、 for the long ellipsoid. Here is the results of this approach:Segment Flight :the total length of the route, the total flight time , the route lengthfrom Beijing to Detroit is, its flight time is ,so we can see ,the shorten of the distance is ,and the shorten of the time is.5、 The earth as a rotatin
20、g ellipsoidAs we know, the earth is a rotating ellipsoid, of which the equator radius is 6378 km and the meridian radius is 6357 km.Some basic concept：1、 Longitude of geodetic:figure1, Dihedral angle L constitute by meridian plane NPS and NGS.2、 Latitude of geodetic:figure1,angle B constitute by nor
21、mal PN and equator plane.3、 Latitude of naturalization ：Y-point p on the line upward, and to a large radius arc intersect at ,the angle of line and X-axis.4、 Geodetic line:the shortest curve between two points on the ellipsoid plane.Generally,using the latitude of naturalization to calculate the len
22、gth of Geodetic line between the two known points.Figure(1)Figure(2)Figure 2，the equation of meridian oval at P： （1）Derivation from the above equation：Curve at point p at the first derivative: （2）Obtained by the formula 1 and 2： Drawing from the principles of geometrical oval:,Thus：So the relationsh
23、ip of Latitude of naturalization and Geodetic line is （3）The differential relationship of geodetic line between two points and great circle of Auxiliary spherical： Because the flat rate of the earth is,hence，we can omit the item ,thus： （4）The Spherical trigonometry formula of Spherical polar angle t
24、riangle is substituting the formula 1 into the formula 4:Integral equation both ends：While are unknown，so we use and to express，obtained by the Spherical trigonometry formula Order ， so： is the great circle of Auxiliary spherical，Approximate formula is as follows：According to the approximate treatme
25、nt to :So：If order，则： Along the flight path of the original total time of flightTime saved：This result explains the original title of saving 4 hours, .6、 References1 JiangQiyuan, mathematic model, Beijing:Higher Education Press, 1987,12 2MeiXiangming HuangJinzhi, differential geometry, Beijing:Highe
26、r Education Press 1988.63Department of Mathematics, Tongji University,higher mathematics, Beijing:Higher Education Press,19954Department of Surveying and Mapping, Wuhan Institute of Control Survey, Control Surveying(the next volume ), Beijing:Mapping Press,19965Baidu Encyclopedia, Bessel, http:/ Enc
27、yclopedia,reduced latitude, http:/ appendixR=6371; %earth radiush=10; %terrain clearancefai=(40/180)*pi (31/180)*pi (36/180)*pi (53/180)*pi (62/180)*pi (59/180)*pi (55/180)*pi (50/180)*pi (47/180)*pi (47/180)*pi (42/180)*pi (43/180)*pi; %角度sita=(116/180)*pi (122/180)*pi (140/180)*pi (195/180)*pi (21
28、0/180)*pi (220/180)*pi (225/180)*pi (230/180)*pi (235/180)*pi (238/180)*pi (273/180)*pi (276/180)*pi;%anglefor i=1:1:12 x(i)=(R+h)*cos(fai(i)*cos(sita(i); y(i)=(R+h)*cos(fai(i)*sin(sita(i); z(i)=(R+h)*sin(fai(i);endfor j=1:1:11 A(j)=x(j)*x(j+1)+y(j)*y(j+1)+z(j)*z(j+1);%cross productB(j)=(sqrt(x(j)2+y(j)2+z(j)2)*(sqrt(x(j+1)2+y(j+1)2+z(j+1)2); %product of the distance of OA and OBL(j)=(R+h)*acos(A(j)/B(j); %the distance between two pointsendAA=x(1)*x(12)+y(1)*y(12)+z(1)*z(12);BB=(sqrt(x(1)2+y(1)2+z(1)2)*(sqrt(x(12)2+y(12)2+z(12)2);LL=(R+h)*acos(AA/BB); TT=LL/980;S=sum(L)T=S/980dT=T-TT