Contents of projective geometry

In projective geometry, the point at infinity is regarded as the "ideal point". A Euclidean straight line plus an infinite point is a straight line in projective geometry. If two straight lines are parallel in a plane, then the two straight lines will intersect at the infinite point of the two straight lines. All straight lines through the same point at infinity are parallel.

After the introduction of infinite points and infinite straight lines, the original combination relationship between ordinary points and ordinary straight lines still holds. In the past, the restriction that the intersection point could only be found when the two straight lines were not parallel disappeared.

Since straight lines passing through the same infinite point are parallel, central projection and parallel projection can be unified. A parallel projection can be regarded as a central projection passing through a point at infinity. In this way, any mapping that uses central projection or parallel projection to map one graphic into another graphic can be called a projective transformation.

Projective transformation has two important properties: first, projective transformation changes a point sequence into a point sequence, a straight line into a straight line, and a wire harness into a wire harness. The combination of points and straight lines is the invariance of projective transformation; secondly, Under projective transformation, the cross ratio remains unchanged. Cross ratio is an important concept in projective geometry, which can be used to illustrate the projective correspondence between two plane points.

In projective geometry, points and straight lines are called dual elements, and "drawing a straight line through a point" and "taking a point on a straight line" are called dual operations. In two figures, if they are both composed of points and straight lines, if each element in one figure is changed to its dual element, and each operation is changed to its dual operation, the other figure will be obtained. These two figures are called dual figures. The content described in a proposition is only about the positions of points, straight lines and planes. When each element is changed to its dual element and each operation is changed to its dual operation, the result is another proposition. These two propositions are called dual propositions. Point A straight line passes through... A straight line connecting two points The intersection of two straight lines *** point *** line tangent tangent point trajectory envelope... This is the duality principle unique to projective geometry. On the projective plane, if a proposition is true, then its dual proposition is also true. This is called the principle of plane duality. Similarly, in projective space, if a proposition is true, then its dual proposition is also true, which is called the spatial duality principle.

Studying the invariant properties of conic curves under projective transformation is also an important part of projective geometry.

In 1872, the German mathematician F. Klein (Felix Klein) proposed the famous "Erlangen Plan" at the University of Erlangen, which proposed using transformation groups to classify geometry. That is, for any kind of transformation, all its components can form a "group", and there will be corresponding geometry. In each kind of geometry, the main study is the invariants and invariance under the corresponding transformation.