Reflection (mathematics)

In mathematics, a reflection is a transformation that "flips" a figure or object across a line (in 2D) or a plane (in 3D). This line or plane is called the axis of reflection or mirror line/plane. The reflected image is a mirror image of the original.

Key properties of reflections include:

• Distance Preservation: The distance between any point on the original figure and the axis of reflection is the same as the distance between the corresponding point on the reflected image and the axis of reflection.
• Size and Shape Preservation: Reflections are isometric transformations, meaning they preserve the size and shape of the object. The reflected image is congruent to the original figure.
• Orientation Reversal: Reflections change the orientation or "handedness" of a figure. For example, if you reflect your right hand, the result looks like a left hand. This is a crucial distinction from rotations and translations, which preserve orientation.
• Involution: Performing a reflection twice results in the original figure. In other words, a reflection is its own inverse.
• Fixed Points: Points that lie on the axis of reflection are unchanged by the transformation; they are called fixed points.

Reflections are fundamental transformations in geometry and are used in various areas of mathematics, including Euclidean geometry, linear algebra, and group theory. They are often studied in the context of symmetry, where a figure has reflection symmetry if it is invariant under a reflection about a line or plane.