二次曲面和平面位置关系的判式

A Discriminant of Relative Position between a Quadric and a Plane

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.

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 quadries and a line, but there is no discussion with respect to the relative position between a quadrie and a plane. In this paper, we give a discriminant A with respect to the relative position between a quadrie a11x2＋a22y2＋a33z2＋2a12xy＋2a13zz＋2a23yz＋2a14x＋2a24y＋2a34z＋a44=0（1） and a plane And we proved that if △〉0, then the quadric surface intersects the plane;if △=0, then the plane is tangential to the quadric surface, if △〈0, then the plane and the quadric surface are disjoint.