What does the one-dimensional projective transformation diagram show?

The cross ratio is invariant to the projective coordinate rotation of a straight line.

One-dimensional projective transformation refers to the projective correspondence between overlapping one-dimensional primitives. If two one-dimensional basic shapes (both points or beams) of the same kind on the plane are the same base or concentric, they are called overlapping one-dimensional basic shapes, and the projective correspondence between the two overlapping one-dimensional basic shapes, that is, the projective correspondence between the one-dimensional basic shapes and themselves, is called one-dimensional projective transformation.

Projective transformation is a special case of projective correspondence, so its algebraic expression is also nonsingular linear transformation (see "projective transformation"), and projective correspondence can also be expressed by parameter method, that is, the necessary and sufficient conditions for projective correspondence of two one-dimensional basic forms (point and beam) A+λB and A+λ′ b are that the bilinear equation A λ′+B λ+C λ′+is satisfied between the parameter λ′ of any element and the parameter λ′ of the corresponding element. On the other hand, if λ and λ′ satisfy the bilinear equation, the correspondence between A+λB and a+λ′ b is projective.

We call it the parametric representation of one-dimensional projective transformation. For one-dimensional projective correspondence, parameter representation is often more convenient than algebraic expression. Bilinear equation can be used as the definition of one-dimensional projective transformation, and from the algebraic point of view, this definition can not only be limited to point series, but also hold true for wire harness. In the process of proving theorem 1, if the overlapping condition is abandoned, the conclusion still holds.