ge3 glides
DESCRIPTION
1. q[g, c, θ] = t[g]q[c, θ] 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] q[c, θ] is called the reflection component of g t[g] is called the translation component of g g can be expressed in two ways: This is the Standard Form from Block 1. Watchpoint: Everyone forgets this! q[c, θ] q[θ]TRANSCRIPT
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. q[c, θ] is called the reflection component of g t[g] is called the translation component of g 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.
Glide Reflections in Lattices Shirleen Stibbe [email protected]
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]
This is the Standard Form from Block 1. 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.
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.
2. q[g, c, θ] = t[d]q[θ]
q[c, θ]
g t[g] c
q[θ]
t[d] d
Watchpoint: Everyone
forgets this!