关于椭圆抛物面圆形截线的研究

The Study on Circular Transversal of Elliptic Paraboloid

用形数结合的方法得出了椭圆抛物面圆形截线的截平面的位置、圆心和半径等参数 ,并得出了一些重要结论 :(a) 椭圆抛物面的圆形截线的截平面都平行于该曲面的横断面椭圆的长轴方向 ,且垂直于长轴方向的对称面 .(b) 截平面的斜率由式K =tanθ=± a2 -b2b 所决定 ,其截平面方程为z =± a2 -b2b y +h .若b >a ,则方程为z =±b2 -a2a x+h.(c) 椭圆抛物面有且仅有两族平行的圆形截线 ,它们对称于oz轴 .(d) 截交线圆的中心坐标为x=0 , y =±b a2 -b2 , z =a2 -b2 +h , R=a a2 -b2 +2h .(e) 两族圆形截线的圆心轨迹为两条平行于oz轴的直线 ,其方程为x =0 , y=±b a2 -b2 .该方法具有普遍意义 .

The author has worked out some parameters,such as the section′s position,center and radias of circular transversal of elliptic paraboloid by the way of combination of graph and digit,and concluded some important conclusion:(a) the sections of circular transversal of elliptic paraboloid x 2a 2+y 2b 2=2z (a>b>0) ,parallel the direction of the major axis of the sectiou′s ellipse,and perpendicular to the symmetroid of direction of the major axis.(b) the seetion′s slop depends on the formula K= tan θ=±a 2-b 2b ,and the section′s equation is: z=±a 2-b 2by+h ,if b>a,then the equation is z=±b 2-a 2ax+h .(c) the elliptic paraboloid has and only has two families of circular transversal which parallel with eachother and are symmetry on oz axis.(d) the center coordinate of circular transversal is: x=0, y=±ba 2-b 2, z=a 2-b 2+h, R=aa 2-b 2+2h .(e) the center track of two families of circular transversal are two lines which are both parallel to oz axis,and it′s equation is x=0, y=±ba 2-b 2 . The way hsa universal meaning.