透视和透视投影变换 --论图形变换和投影的若干问题之三

讨论了透视变换的基本原理．由于与画面成一角度的平行线簇经透视变换后交于灭点，因此可采用两种不同的方法来获得透视图：一是保持画面铅垂，通过旋转物体使之与画面构成角度达到透视变换效果，得到了三种最佳透视变换矩阵；二是通过倾斜投影画面达到透视变换效果，给出了通过倾斜画面得到三灭点透视图的齐次透视变换矩阵．两种方法的灭点都是可预先控制(即可先决定灭点再决定变换矩阵)的，比较彻底地解决了透视变换矩阵元素的产生方法．给出了“对一个空间物体，一定存在另一个空间物体，使前者在画面上的透视投影与后者的平行投影是一样的，且保留了深度方向的对应关系”的证明．这个性质可使复杂的透视投影转化成简单的平行投影，使得立体图形的处理大为简化．

Perspective and Its Projection Transformation

Fundamental principles of perspective transformation are discussed in the paper. Based on the fact that parallel-lines in some angle with view plane intersect at a vanishing-point, two methods are presented to produce a perspective view: one is to keep the view plane vertical while rotating objects to some angle, to achieve perspective transformation effect, and the other is by means of inclining the projective view to achieve the effect. To the first method, three best perspective transformation matrices are presented, and with regard to the second method, homogeneous perspective transformation matrix is presented, for generating 3-vanishing-point drawing through the inclining view. Both methods are vanishing-point pre-controllable. It means that the vanishing-point could be pre-determined, then the transformation matrix is generated afterwards, As a result, the generation theory of perspective transformation is relatively well solved. We prove that for each 3D object there must be a corresponding 3D object, with its parallel projection the same as the perspective projection of the former counter object, and at the same time, the corresponding depth relationship is well preserved. With this useful property, a complicated perspective projection can be converted to a simple parallel projection, so that the complexity of 3D graphics processing could be greatly reduced.