1、Flying over the North Pole AreaAbstractWe 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 wit
2、h the radius of 6371 .kmAs 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
3、 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
4、 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 kwe multiplied it by the route that we got in the first case, which is the route that we consider th
5、e 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 l
6、atitude,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 tw
7、o 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 of on the spherical (or on the ),(iizyx iAellipsoid):the longitude and latitude of on the spherical (or
9、 on the ellipsoid)(iiA: the flight timet: the saved flight time:sum of sub-flying aircraft, the total journey distance总l:Radius of the earthR:aircraft altitudeh: the minor axis and the long axle of the prolate spheroid),(ba3、The earth as a ballAssuming the earth is a sphere with the radius of 6371 .
10、At this point A, B is km10 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:111sin,sin
11、,coshRhhRA 222coRBThe flight distance from A to B: OBAhlABarsThe flight time from A to B: 980BltSolution 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,vecto
12、r of each point on the sphere can be expressed as follow: iiiiii kjhROAsncosncos 1111 iiiiiiThe distance between the two points can be expressed as follow:11arcosiii OAhRAThe total time of flight routes: ;Direct from Beijing, the sailing time for 1098iitDetroit: 01AtSo the time saved by direct fligh
13、t: According to the results calculated by MATLAB program:(unit:km)The distance between Beijing and A1:1113.2The distance between A1 and 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 dista
14、nce between A6 and A7:651.5The distance between A7 and A8:497.6The distance between A8 and A9:227.8The distance between A9 and A10:2810.9The 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 fl
15、ight from Beijing to Detroit total time flight:10.9hTime saved: 3.9htBy comparing the results of the above methods and the actual situation,they are consistent with each other.That means the report of “the time saved from Beijing to Detroit is 4 hours“has scientific basis.4、The earth as an ellipsoid
16、When 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 axis Rkma6371,which is differ from . We can assume ellipsoid along the minor axis b5by the compression of the sphere obtained, if we ignore the small gap between t
17、he Rand and the sphere of the original points such as ,which are in the a 1321A、ellipsoid surface after compression. Whats more, the greater the latitude difference between two points on the arc length of the compressed the greater the points on the same latitude, little change in the compressed arc
18、 length.bThe conditions given by the subject shows: the latitude of the difference between adjacent points of are , the 1321A、 .40534975.8、difference of latitude between Beijing and Detroit is ,and its a small difference.So 3here the compression method we use is effective and the compression ratio i
19、s constant ;Here comes another situation:when assuming the 97806.16375hRbkEarth is spherical,we have calculated the length of the route,and put this result multiplied by ,so we can get the routes for the long ellipsoid. Here is the results of kthis approach:Segment Flight :the total length of the ro
20、ute , the total flight kml9614.53time , the route lengthfrom Beijing to Detroit is , its flight ht79.14 l.07time is ,so we can see ,the shorten of the distance is t825.0,and the shorten of the time is .kml36.8 ht9265.35、The earth as a rotating ellipsoidAs we know, the earth is a rotating ellipsoid,
21、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 normal PN and equator plane.3、Latitude of naturalization :Y-poi
22、nt p on the line upward, and to a large radius arc intersect at ,the angle of line and X-axis.NAP PO4、Geodetic line:the shortest curve between two points on the ellipsoid plane.Generally,using the latitude of naturalization to calculate the length of Geodetic line between the two known points.Figure
23、(1)Figure(2)Figure 2,the equation of meridian oval at P: (1)2byaxDerivation from the above equation:yaxbd2Curve at point p at the first derivative:(2)Bdxycot90tanObtained by the formula 1 and 2: Babxytn2Drawing from the principles of geometrical oval: ,aPObThus: ubpoyaxsinicoSo the relationship of L
24、atitude of naturalization and Geodetic line is Babutnt(3)The differential relationship of geodetic line between two points 21,ULPand great circle of Auxiliary spherical:duads2cos1Because the flat rate of the earth is ,hence,we can omit the 03.ab2item ,thus:(4)duduads 22 sin1cos1The Spherical trigono
25、metry formula of Spherical polar angle triangle issubstituting the formula 1 into the formula 4:csinsu duaduad n 2cos1sio21 22Integral equation both ends: 1222 siinsi41sinuSWhile are unknown,so we use and to 21,nu 21,uexpress, ,obtained by the Spherical trigonometry formula 2211cosinsiuOrder , so:1i
26、U21siniuVcos12cos12sinVUu cos1iniii 12 VUnis the great circle of Auxiliary spherical,Approximate formula is as follows:212121coscssicos LuuAccording to the approximate treatment to : 1222 siniin4i naaSSo: VUcos1icos1i4If order , ,baHinMcos1inN则: NVSiiiiii Luuscoscosncos 1112iiU2nii4baHcos1Mcos1NAlon
27、g the flight path of the original total time of flight 36.1405STime saved: 04.98036.5.6VStThis 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
28、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
29、yclopedia,reduced latitude, http:/ %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 (210/180)*pi (220/1
30、80)*pi (225/180)*pi (230/180)*pi (235/180)*pi (238/180)*pi (273/180)*pi (276/180)*pi;%anglefor i=1:1:12x(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=1:1:11A(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