Math : Geometric Transformations

When you discuss symmetry and groups, it is important to remember that the elements of those groups are often geometric transformations. A geometric transformation is an act - it is something you do to a system: a rotation, a reflection, etc. Here are a few of the more important types of geometric transformations:

Translation
Translation, also known as spatial displacement, takes an image of the initial object and moves it in a certain direction for a certain distance. This type of transformation appears in things with repeating patterns like wallpaper.
Rotation
Rotation takes place about a fixed common axis by a designated angle. The axis is sometimes called a symmetry axis and is often named after the symmetry group (e.g. a C4 axis).
Plane Reflection
This is the transformation of reflection through a double-sided plane mirror, and for this reason the plane of reflection is often called the mirror plane. If a point is located in the plane, it will remain in the same place after the transformation.
Line Reflection
Similar to the plane reflection, the line reflection is the inversion through a straight line. It is actually the same transformation as a rotation of 180 degrees.
Point Inversion
This is also a type of reflection and can be realized by drawing a straight line from the initial point through the inversion point to the other side for the same amount of distance. For example, in a Cartesian coordinate system with the inversion point at the origin, the point (x, y, z) will become (-x, -y, - z).
Glide Transformation
This is a combination of translation and reflection; it consists of the consecutive operation of a displacement parallel to a plane and then reflection through the plane (called a glide plane). These can also be performed in reverse order.
Screw Transformation
This transformation combines rotation and translation and can also be performed in either order. It consists of a rotation about an axis and a displacement parallel to that axis (called a screw axis).

Page author: Sasie Sealy