Find one of the two glide reflections of the plane lattice L.

Hi,

This is probably more of an algebra/geometry question so I hope I've posted it in the right place. I hope I have the answer to this right but as I'm not 100% sure I thought I'd ask for some feedback. The question is: I have the set of vectors {a=(1,-1), b=(1,2)} as a reduced basis for the plane lattice L. I need to find one of the two glide reflections of L that map the point (1,2) to (-1,1), in the standard form t[d]q[θ] and then in the form q[g,c,θ]. I have selected in standard form t[(-3,0)]q[∏/4] which is reflection in the line ∏/4 followed by translation of (-3,0), and in the other form this is t[(-3/2, -3/2)]q[(3/4,9/4)] ∏/4, which is reflection in ∏/4 through the point (3/4, 9/4) followed by a translation of (-3/2, -3/2), which is in the direction of the reflection line.

Could anyone offer any advice with this question, whether my thinking is correct or am I way off (which wouldn't surprise me!).

Thanks.

Pat

Re: Find one of the two glide reflections of the plane lattice L.

ok, this is what my calculations show:

your first mapping does this:

(1,2) → (2,1) → (2,1) + (-3,0) = (-1,1)

your second mapping does this:

(1,2) = (3/4,9/4) + (1/4,-1/4) → (3/4,9/4) + (-1/4,1/4) = (1/2,5/2) → (1/2,5/2) + (-3/2,-3/2) = (-1,1)

so that all looks fine. of course, i would be tempted to verify they are actually the same map, like so:

$\displaystyle \begin{bmatrix}x\\y \end{bmatrix} \to \begin{bmatrix}y\\x \end{bmatrix}\to \begin{bmatrix}y-3\\x \end{bmatrix}$

versus:

$\displaystyle \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}\frac{3}{4}\\ \frac{9}{4} \end{bmatrix} + \begin{bmatrix}x-\frac{3}{4}\\y - \frac{9}{4}\end{bmatrix} \to\begin{bmatrix}\frac{3}{4}\\ \frac{9}{4} \end{bmatrix} + \begin{bmatrix}y-\frac{9}{4}\\x - \frac{3}{4}\end{bmatrix}$

$\displaystyle =\begin{bmatrix}y-\frac{3}{2}\\x +\frac{3}{2}\end{bmatrix} \to \begin{bmatrix}y-3\\x \end{bmatrix}$

i do not know how your "standard forms" are defined (in particular which θ and c you are allowed to use), but as near as i can tell, your second form is "kosher" in that the final translation is indeed parallel to the axis of reflection.

Re: Find one of the two glide reflections of the plane lattice L.

Hey Deveno, thanks for the reply. I think I get what you are saying. Just to clarify when I say standard form, that to me is reflection in the line at the specified angle through the origin followed then by the translation. The other form is reflection in the line at the specified angle through a point c and then translation parallel to the line of reflection. There are only two reflection symmetries for this lattice, in ∏/4 and 3∏/4, so only two possible glide reflections. I think the one above is fine then but any advice on getting the other one, just for completion?

Re: Find one of the two glide reflections of the plane lattice L.

ok, well reflection about the line at angle 3π/4 is this mapping:

$\displaystyle \begin{bmatrix}x\\y \end{bmatrix} \to \begin{bmatrix}-y\\-x \end{bmatrix}$

(this is actually a linear map).

this takes (1,2) → (-2,-1), so we need to follow with a translation by (1,2) (note that (1,2) is indeed in the lattice L).

if you'll tell me how you determine "c", i'll tell you the second form.