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