Find the content of projective geometry

Find the content of projective geometry

In projective geometry, points and straight lines are called dual elements, and "draw a straight line through a point" and "take a point on a straight line" It's called the dual operation. 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.

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 terms of the content of geometry, projective geometry<

affine geometry<

Euclidean geometry, that is to say, Euclidean geometry Geometry is the richest in content, while projective geometry is the poorest. For example, in Euclidean geometry, we can discuss the objects of affine geometry (such as simple ratios, parallelism, etc.) and the objects of projective geometry (such as the intersection ratio of four points, etc.), but conversely, we cannot discuss them in projective geometry. The affine properties of graphics, and the metric properties of graphics cannot be discussed in affine geometry.