Geometric Transformations: Properties, Types, and Applications

Geometric transformations are an important topic in geometry. In plane and space geometry, if there exists a congruent transformation between two figures, then these two figures are congruent; if there exists a similar transformation between two figures, then these two figures are similar. Both of these transformations are special affine transformations in Euclidean space. Affine transformations have two main characteristics: they preserve lines and are uniform, meaning that the transformation coefficients are independent of spatial position. In the complex plane, there is a special type of transformation called a conformal transformation, which preserves angles. For curved surfaces, there are two special types of transformations, both of which are non-uniform. Two special types of transformations for curved surfaces are isometric transformations and conformal transformations. This article aims to discuss and summarize the properties and applications of these geometric transformations.