Transformationsgeometrie

Transformationsgeometrie is a branch of geometry that studies geometric figures and their properties under various transformations. Transformations are mappings that move points in a plane or space according to specific rules. These rules can involve translation, rotation, reflection, and scaling. The core idea is to analyze how geometric shapes change, or remain invariant, when subjected to these operations.

Key concepts in transformationsgeometrie include the study of congruence and similarity. Congruent figures are identical in shape and size, meaning one can be transformed into the other through a sequence of rigid motions (translations, rotations, reflections). Similar figures have the same shape but not necessarily the same size; they can be transformed into one another through a similarity transformation, which involves scaling in addition to rigid motions.

The field has applications in various areas, including computer graphics, where transformations are fundamental for manipulating objects on screen. It also plays a role in understanding symmetry, tessellations, and other geometric patterns. The axiomatic approach to geometry often begins with the concept of transformations as a way to define geometric equivalence and properties. Studying transformations helps in understanding the underlying structure and relationships within geometric spaces.