二次曲面和平面位置关系的判式

在解析几何中有二次曲线与直线位置关系的讨论、二次曲面与直线位置关系的讨论,而二次曲面与平面相关位置关系的探讨较少.本文给出二次曲面a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)和平面Ax+By+Cz+D=0(2)的相对位置的判别式Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DA B C D0(aij=aji).(3)并证明了:若Δ>0,则二次曲面(1)与平面(2)相交;若Δ=0,则(1)和(2)相切;若Δ<0,则(1)和(2)相离.

A Discriminant of Relative Position between a Quadric and a Plane

In analytic geometry,there was the discussion with respect to the relative position between a conic and a line,and there was the discussion with respect to the relative position between a quadrics and a line,but there is no discussion with respect to the relative position between a quadric and a plane.In this paper,we give a discriminant Δ with respect to the relative position between a quadrica11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)and a planeAx+By+Cz+D=0,(2)Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DABCD0(aij=aji).(3)And we proved that if Δ>0,then the quadric surface intersects the plane;if Δ=0,then the plane is tangential to the quadric surface,if Δ<0,then the plane and the quadric surface are disjoint.