What is the projective theorem?

Projective geometry theorem is a basic theorem describing the concept of two-dimensional projection geometry, also known as projection theorem.

It is an important theorem in geometric foundation, which explains the preservation of the relationship between lines under perspective projection transformation. Simply put, the projective theorem means that if there is a straight line L and a point P on a plane that is not on L, P can be mapped to another straight line L' through perspective projection, so that any point on the original straight line L can get the corresponding point in the mapping of L', and the intersection and parallelism between the straight lines remain unchanged. This mapping is projective transformation.

Projective theorem has a wide range of applications, such as in computer graphics, it is often necessary to project objects in three-dimensional space onto a two-dimensional screen, and this operation can be completed by projective transformation. In addition, the related theories of projective geometry are helpful to popularize Euclidean geometry and provide an important theoretical basis for the study of geometry.

Although the projective theorem is the basic theorem in two-dimensional geometry, it can also be extended to high-dimensional geometry. Specifically, if we consider the projective space P n in n+ 1 dimensional space, there is also a projective theorem in this space, which also describes the projective mapping relationship between lines, but the lines and points at this time have higher abstract properties.

The use of projective theorem

Projective theorem is often used in computer vision, robotics, physics or other fields to realize the processing, analysis and observation of three-dimensional models or objects. The following are some common usage scenarios of the projective theorem:

1, 3D reconstruction: Projective theorem can help realize 3D reconstruction of image data obtained by multiple cameras from different perspectives. Based on the perspective projection transformation of images, we can match the feature points in multiple images, determine their three-dimensional coordinates in the real world, and then build a three-dimensional model.

2. Attitude estimation: In robotics, projection theorem can be used to calculate the attitude estimation of objects in three-dimensional space. By collecting images of a three-dimensional object from multiple perspectives, the corresponding two-dimensional points and camera parameters can be calculated, and the three-dimensional posture can be calculated by using projective theorem.

3. Geometric transformation: In computer graphics, projective transformation is often used for geometric transformation, such as rotation, scaling and translation of objects in three-dimensional space. At the same time, in image processing, projective transformation can also be used to adjust the shape of the image. For example, any shape in the target image can be transformed into any other shape by projection transformation.