9. Show mathematically that the combination of a mirror plane and a translation vector $\mathbf{t}$ perpendicular to the mirror plane implies the existence of a mirror plane at $\mathbf{t}$/2.

Solution:

Without loss of generality, let's define the translation vector be to the c-basis vector (note that we can always redefine basis vectors to make this the case).

The mirror operation is therefore given by:



In [1]:

    
import numpy as np
M = np.eye(4).astype(np.int)
M[2, 2] = -1
print M









    



[[ 1  0  0  0]
 [ 0  1  0  0]
 [ 0  0 -1  0]
 [ 0  0  0  1]]

The translation operation is given by:



In [2]:

    
from sympy import symbols
t = symbols("t")
T = [[1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 1, t],
     [0, 0, 0, 1]]

Consider the relationship between a reflected point and a translated point. The operation that maps the reflected point to the translated point is given by:

$Op_m = p_t$

$OMp = Tp$

$OM = T$

=> $O = TM^{-1}$



In [3]:

    
O = np.dot(T, np.linalg.inv(M))
print "O = %s" % str(O)









    



O = [[1.0 0.0 0.0 0.0]
 [0.0 1.0 0.0 0.0]
 [0 0 -1.00000000000000 1.0*t]
 [0.0 0.0 0.0 1.0]]

Now consider a mirror plane at t/2. The symmetry operation is given by:

$O' = T_{t/2}MT_{-t/2}$



In [4]:

    
T__t2 = [[1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 1, -t/2],
     [0, 0, 0, 1]]
T_t2 = [[1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 1, t/2],
     [0, 0, 0, 1]]
O = np.dot(T_t2, np.dot(M, T__t2))
print "O' = %s" % str(O)









    



O' = [[1 0 0 0]
 [0 1 0 0]
 [0 0 -1 t]
 [0 0 0 1]]

We may observe that $O' = O$, i.e., a reflection with a perpendicular translation t implies the existence of a mirror plane at t/2.



In [4]: