飞越北极数学建模方案Ⅰ

对“飞机从北京出发，飞越北极直达底特律所需时间可比原航线节省多少”的问题进行讨论，并将航线的选择归结为寻求曲面上的最短弧，应用“曲面上最短弧为测地线”的事实，分两种情况展开讨论：模型一：假设地球是球体，则测地线恰好是大圆，而球面上两点间的最短弧，就是其所在大圆对应的劣弧，我们可通过单位向量的点乘与夹角的关系加以解决；模型二：假设地球是旋转椭球体．我们利用微分几何学中测地线方程及曲线的弧长公式，并且把球面的纬度转化为旋转椭球面纬度，对于4组较特殊的点，纬度几乎相等或相近，或者两者之间的经度差过大，用测地线计算比较困难，我们用椭圆弧长代替测地线，结合数学软件Mathematica可求得测地线长．

Mathematical Model of Flight Couse I

The article discusses how much time would be saved for a flight from Beijing to Detroit directly over arctic pole instead of the original route. The best choice is defined as searching for the shortest arc on the curved surface. The discussion is based on the fact that the shortest arc on surface is geodesic.Model l is on the assumption that the Earth is a sphere. It can be solved by the relation between point product and angle of two unit vectors.Model 2 is on the assumption that the Earth is a revolving elliptical sphere It can be solved by geodesic equation in differential geometry,which turns latitude of the Earth into that of elliptical sphere. For the 4 pairs of special points, their latitudes or longitudes is too close to calculate geodesic,so we replace geodesic with ellipse arc,and use software Mathematica to obtain the length.