平面双线性映射的逆的几何意义

The Geometric Meaning of the Inverse of the Planar Bilinear Mapping

揭示了平面双线性映射的逆的几何意义.对于梯形的4个顶点所定义的平面双线性映射,梯形两腰的交点的逆有无穷多个,对于过梯形两腰延长线的交点且与底边平行的直线上除交点外的点,其逆不存在,对于平面上其它的点,其逆是唯一的;对于其它平面四边形的4个顶点所定义的双线性映射,存在1条与4条边所在直线都相切的抛物线,对于抛物线上的点,其逆唯一,对于抛物线外侧的点,其逆有2个,对于抛物线内侧的点,其逆不存在.

The paper reveals the geometric meaning of the inverse of planar bilinear mapping. For the planar bilinear mapping defined by the four vertices of the trapezoid, there are infinite numbers inverses on the intersection point between two trapezoid waists. For the line which is parallel to bottom edges of the trapezoid and passes the intersection point between two trapezoid waists except the intersection, the inverse does not exist. And there is the unique inverse of the bilinear mapping for other points. For the planar bilinear mapping defined by the four vertices of the planar quadrilateral, there is a parabola which is tangent to the lines on the four edges. For the points on the parabola, there is the unique inverse. For the points out of the parabola, there are two inverses. And for the points inside of the parabola, the inverse doesn＇t exist.