Glide Reflections in Lattices
Shirleen Stibbe s.stibbe@open.ac.uk
Watchpoint: Everyone forgets this!
Let g = q[g, c, θ] be a glide reflection in the lattice L. Then g is a reflection in the line lying at angle θ to the x-axis which passes through c, followed by a translation through the vector g, where g is parallel to the reflection axis. g can be expressed in two ways:
1.
q[g, c, θ] = t[g]q[c, θ]
q[c, θ] is reflection in the line at angle θ to the x-axis passing through c, and t[g] is translation through g.
g
t[g] c
q[c, θ] is called the reflection component of g t[g] is called the translation component of g
q[c, θ]
This is the form you look at to decide whether the glide is essential (i.e. neither component is a symmetry of L) or inessential (i.e. both components are symmetries of L). [The glide in the diagram above is essential.] Note the all-or-nothing property of the components; either both are symmetries of L, or neither is a symmetry of L. You can't have one component a symmetry and the other not a symmetry of L.
2.
q[g, c, θ] = t[d]q[θ]
This is the Standard Form from Block 1.
t[d]
q[θ] is a linear reflection in the line at angle θ to the x-axis (Note: linear means the axis passes through the origin) and t[d] is translation through d, where d is the image of the origin under g. Note 1: d is not necessarily parallel to the reflection axis.
q[θ]
Note 2: If c is perpendicular to the reflection axis, then d = g + 2c by Equation (15) of the Isometry Toolkit, Note 3: The quick way to find d is to track what happens to the origin under the glide reflection – look at the Proofs & Strategies Booklet, 3.2.1, p14 to see why. q[θ] is called the linear part of g - recall that a linear transformation fixes the origin t[d] is called the translation part of g - d is not generally parallel to the reflection axis Note that both parts will always be symmetries of L.
Final note: If c = ( 0, 0) in q[g, c, θ] - i.e. the reflection axis passes through the origin - then there's no difference between components and parts.
d