Introduction to projective geometry

The earliest origin of projective geometry is painting. The vigorous development of perspective in European Renaissance provided good conditions for the growth of projective geometry. Comrades who know a little about art know what perspective is.

In fig. 26, the two power rails are initially parallel to each other. Through perspective, the farther they go, the closer they get. Finally, it intersects at a point at infinity.

In projective geometry, two parallel straight lines intersect at infinity, which is called infinity. The trajectory of the point at infinity is a straight line at infinity, which is different from Euclidean geometry. After all the figures in Euclidean geometry are transformed by rigid bodies (such as translation and rotation), the length of line segments, the size of angles, figures, shapes and areas will not change. The geometry that studies the invariance of plane or space geometry under rigid body transformation is Euclidean geometry. If a projection cone is emitted from the center o, the cross section of the rectangular ABCD on the plane p is A'B'C'D'. Section A'B'C'D' is not necessarily rectangular. Intuitively, it is easy to see that its size and shape have changed with ABCD. So after this projective transformation, what are the geometric properties of graphs A'b'c'd' and ABCD? The geometry that studies the invariant properties of graphs under projective transformation is called projective geometry.

One of the most basic concepts in projective geometry is cross ratio. In a graph, S is the center point, and four rays are drawn from S to form a fixed wire harness. The other line intersects the harness at a, b, c and d respectively. AB/CD: AD/BC or AB CD/BC AD is called cross ratio on this wire harness. No matter how the straight line L is taken (such as L'), as long as the wire harness is fixed, the value of the cross ratio is always the same. Cross ratio invariance is one of the most basic invariants under projective transformation. Many important properties in projective geometry are derived from cross ratio properties.

If the content of geometry is concerned, projective geometry