Circle inversion is a geometric transformation that takes points in a plane and maps them to new locations based on their distance from a specific circle. This transformation is defined with respect to a circle, called the inversion circle, which has a center and a radius. When a point lies inside the circle, it is mapped to a point outside the circle, and vice versa. The distance from the center of the inversion circle to the new point is inversely proportional to the distance from the center to the original point. In circle inversion, points on the inversion circle itself remain unchanged. This transformation preserves angles, making it useful in various fields of mathematics, including complex analysis and geometry. Circle inversion can simplify problems by transforming complex shapes into more manageable forms, allowing mathematicians to analyze properties and relationships more easily.