Reflection

Reflection is a symmetry operation.

If we draw a square in 2D space (x,y) centered about the origin with its sides parallel to the x and y axis, then there are two possible reflections:

  • reflection across the x axis
    • (x2,y2) = (x1,-y1)
  • reflection across the y axis
    • (x2,y2) = (-x1,y1)

Matrices are shown below for reflections in 3D space:

\sigma_{xy} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}

\sigma_{xz} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

\sigma_{yz} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

For each of the three reflections mentioned above, it is a reflection across a plane containing two axes. Notice that for the first example, the plane contains the x and y axes and z coordinate becomes the negative of what it was originally.