In mathematics, a reflection is a type of geometrical transformation where an object is flipped to create a mirror or congruent image. This transformation is a mirror image of the original shape over a line, known as the line of reflection, or over a point, known as the reflection point. A reflection maintains the size of the shape and is a type of rigid transformation. For example, in a two-dimensional plane, when the line of reflection is the x-axis, the point (x, y) transforms into (x, -y), and when the line of reflection is the y-axis, the point (x, y) transforms into (-x, y)
. A reflection is an involution, meaning that when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term "reflection" is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points that is an affine subspace, but is possibly smaller than a hyperplane
. In practical terms, a reflection can be visualized as a "flip" over a line, and the central line over which the reflection occurs is called the mirror line. For example, when reflecting over the x-axis, each point (x, y) is transformed into (x, -y), and when reflecting over the y-axis, each point (x, y) is transformed into (-x, y)
. This concept is often taught in geometry and is an important topic in the study of transformations in mathematics.
