Inversion plays an important role in mathematics (some geometric proofs). If you put [1, +∞) "into" (0, 1), you can take the reciprocal method. This is one-dimensional inversion. Two-dimensional inversion is based on a specific inversion circle: the center of the circle is O, the radius of the circle is constant K, and the point P is inverted.

Given a circle o with a radius of r on a plane, if A' is a point on a straight line OA passing through the point o, and the directed line segments OA and OA' satisfy OA' = k 2 (k is a non-zero constant), then this transformation is called the inverse transformation about ⊙O(r), which is called inversion for short. Let A' be the inversion point of A about ⊙O(r), and similarly, A is the inversion point of A' about ⊙O(r); The center o is called the inversion center or inversion pole; The radius r of a circle is called the inverse radius; ⊙O(r) is called the inverse (base) circle. K is called the inverse power, 1) When k = r 2 (the square of r) > 0, the directed line segments OA and OA' are in the same direction, and A and A' are on the same side of the inverse pole. This inverse evolution is called positive power inverse, also known as hyperbolic inverse transformation. 2) When k =-r 2 < 0, the directed line segments OA and OA' are opposite, and A and A' are on opposite sides of the inversion. This kind of inverse evolution is called negative power inverse, also called elliptic inverse evolutionary transformation. In inverse transformation, two corresponding figures are inverse figures or inverse images.