Specifically, if the origin of the coordinate system is taken as the center of symmetry, a straight line is drawn through point P, and point P is divided into two equal parts, then point P' is the symmetrical point of point P about the origin of the coordinate system.
By the following steps, the symmetrical point of the point about the origin can be determined:
1. express the coordinates of point p as (x, y);
2. Inverse each coordinate of p to obtain the coordinate of point p' as (-x,-y);
Symmetry of origin is often used to solve problems and prove theorems in geometry and mathematics. It has some important properties, such as the properties of symmetric graphs and the symmetry of equations. Origin symmetry is also widely used in computer graphics and physics.
The origin of symmetry
The concept of origin symmetry comes from the study of coordinate system and geometry. It is gradually formed in the process of mathematical development.
The concept of coordinate system was first put forward by Descartes, a French mathematician. He put forward the concept of Cartesian coordinate system in his work Geometry in the17th century. Cartesian coordinate system is a system that uses numerical values to describe the position of the midpoint in space. It consists of X axis and Y axis, where X axis and Y axis are perpendicular to each other and intersect at the origin.
Symmetry is an important concept in coordinate system. The earliest concept of symmetry can be traced back to Euclid's Elements of Geometry, in which he discussed the symmetry properties of points, lines and surfaces in detail.
As a special way of symmetry, origin symmetry refers to the symmetry about the origin of coordinate system. Its concept and properties have been studied and popularized by many mathematicians, including the later development of algebraic geometry and linear algebra, and gradually improved, further deepening the understanding of symmetry.
Today, the concept of origin symmetry has become a basic concept in many mathematical fields, such as geometry, algebra and physics, and has been widely used in various fields. It is not only of great significance to the study of geometry and motion, but also provides a powerful tool for solving problems and developing new mathematical theories.
Origin symmetry is widely used in mathematics and other fields.
1. Geometry: origin symmetry is often used to study the properties and relationships of geometry. By finding the symmetry point of the graph origin, the symmetry axis of the graph can be determined, the symmetry property can be judged and the calculation can be simplified. For example, the symmetry of the origin can prove the nature of the square and judge whether the ellipse is symmetrical about the origin.
2. Symmetry of the equation: the symmetry of the origin is also applicable to solving the equation. For quadratic functions, polynomial functions, etc. The symmetry of origin can be used to judge the symmetry of equation, simplify calculation and draw images. For example, if a function is symmetrical about the origin, the range of the solution of the equation can be reduced by half by using symmetry.
3. Physics: Symmetry of origin is also applied in physics. For example, in the study of electromagnetic field, considering the symmetry of the origin of charge distribution can simplify the analysis and calculation process of the problem. In mechanics, the symmetry of origin can be used to study the equilibrium conditions and kinetic energy of rigid bodies.
4. Computer graphics: In computer graphics, origin symmetry is often used for graphic transformation and rendering. By making the original graph symmetrical about the origin, the mirror image effect of the image can be realized. This is widely used in image processing, computer games and virtual reality.
The following is an example of symmetry about the origin.
Question: What is the symmetry point of point A (2, -3) about the origin of the coordinate system?
Solution: Find the symmetrical point of point A about the origin, which can be calculated according to the definition of origin symmetry. The coordinates of point A are (2, -3).
According to the nature of origin symmetry, the coordinate of point A' about the origin is (-2,3), that is, the point A is symmetrical about the origin is (-2,3).
So the symmetry point of point A (2, -3) with respect to the origin of the coordinate system is (-2, 3).