Projective correspondence and projective transformation of projective geometry

One-dimensional basic shapes can be transformed into each other through projection and silhouette.

Use {p} to represent the point column on the straight line L, where p represents any point in the point column. Let s be a point not on L and make a straight line p=SP, then when P changes on L, we will get a light beam {p} centered on S, which is called the projection of point sequence {p}, {p} is called the silhouette of light beam {p}, and P and P are called corresponding elements (Figure 2).

Let S 1 be a point whose space is not on the {p} plane, and make a plane π pass through S 1 and p, then we can get the polygon cluster {π} with SS 1 as the axis, which is the projection of {p}, and {p} is the silhouette of {π}, p and p. After a series of projections and silhouettes, from one one-dimensional basic shape to another, these two basic shapes are called projection correlation, and the corresponding relationship between their elements is called projection correspondence. The two transformations contained in projective correspondence are called projective transformations, and they are reciprocal transformations.

In space, through projection and silhouette, point field and line handle, line field and surface handle can be transformed into each other, so point field, line handle, line field and surface handle can also be transformed into each other. As for other transformations between two-dimensional primitives, such as the transformation between point field and line field, it can be determined by the algebraic method described below. Similarly, the transformation between three-dimensional basic shapes should also be carried out by algebraic method. In short, there can be projective correspondence and projective transformation between two two-dimensional basic forms or two three-dimensional basic forms.

It is pointed out how to establish homogeneous coordinate system in point sequence, point field, point space, line field and surface space. In fact, homogeneous (or projective) coordinates can be based on any basic form of one-dimensional, two-dimensional or three-dimensional (see projective coordinates). In this way, projective correspondence or projective transformation can be expressed by full-rank homogeneous linear transformation between homogeneous coordinates. For example, (x) and () are the homogeneous coordinates of two point fields, and the projective transformation (x)→ () can use the homogeneous linear transformation of three variables.

(2)

Where det stands for determinant; ρ is a non-zero proportional constant. By solving this set of equations, the inverse transformation equation (x')→(x) can be obtained.

One of the basic properties of projective transformation is to maintain correlation, which means that it changes linearly related elements into linearly related elements. For example, the transformation between point fields (2) turns a point sequence into a point sequence, that is, a straight line into a straight line, so it also turns a wire harness into a wire harness. It can be seen that every theorem (such as Dezag theorem) that only involves correlation must represent a projective property, that is, a property that is invariant after projective transformation. In other words, this theorem is a projective theorem.

There is a basic theorem about projective correspondence. If one-dimensional, two-dimensional and three-dimensional are summarized together, that is, if a group of n+2 elements is specified by two basic forms of n dimensions (n= 1, 2,3), and each n+ 1 element in each group is linearly independent, there is a unique projective correspondence between the two basic forms, thus two groups of elements are given. In fact, this theorem is applicable to projective correspondence of any dimension. The so-called "linear independence" can be illustrated by examples: two linearly independent points are not coincident, three linearly independent points are not collinear, and four linearly independent points are not coplanar.

Projective transformation can also act on the enlarged space, but after projective transformation, infinite elements can become non-infinite elements, and non-infinite elements can become infinite elements (for example, parallel planes can become non-parallel and non-parallel planes can become parallel). Therefore, in the unexpanded Euclidean or affine space, the projective transformation is not completely one-to-one.