Direct transformation and affine transformation of projective geometry, projective group

Consider the two-dimensional projective transformation on the plane. A plane is the bottom of both a point field and a line field, so the projective transformation on it can turn a point into a point (or a line into a line) or a point into a line (or a line into a point). The former is called direct transformation and the latter is called affine transformation.

The inverse transformation of direct transformation and their product (that is, the transformation formed by the continuous action of two direct transformations) are both direct transformations. Therefore, all direct transformations on a plane form a group, which is called a plane direct group. The characteristic of direct transformation is to change the point of * * * line into the point of * * * line, so it can be said that the straight line is also changed into a straight line. The direct transformation can be represented by the linear transformation (2) about the point coordinates. If it changes the straight line (□) into (□), it can be obtained through relevant conditions.

□ (4) where□□ is the cofactor (□□) of□□ in a square matrix, and□ is the proportionality constant. It can be considered that (2) and (4) represent the same direct transformation, and the difference between them is only that one uses point coordinates and the other uses line coordinates.

Similarly, injective transformation changes a straight line of * * * points into points of * * * lines, and changes points of * * * lines into straight lines of * * * points, that is, lines become points and points become lines. The product of two injective transformations is a direct transformation. The injective transformation does not form a group, but all the direct transformations and injective transformations on the plane form a group, which is called a projective group. A direct group is a subgroup of a projective group. But sometimes the term projective group is also used to refer to direct groups.

Because plane affine transformation turns points into lines and lines into points, it embodies the duality principle on the plane while maintaining correlation.

Similarly, there are direct transformations and affine transformations in space. The former turns points into points and faces into faces, while the latter turns points into faces and faces into points. Is to turn straight lines into straight lines. All direct transformations in space form a direct group, and all direct transformations and affine transformations form a projective group. Spatial injective transformation embodies the principle of spatial duality.

The projective transformation of all points on a straight line constitutes a projective group on a straight line.

Other basic forms have their own projective groups.