## 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:


```
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:


```
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:

$Op_m = p_t$

$OMp = Tp$

$OM = T$

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


```
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:

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


```
 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**.


```

```

4c5344d241f5952368d0579399a23574d1e5b4e6
ipynb
Jupyter Notebook
problem_sets/PS1 - Qn 9.ipynb
ksyang2013/nano106
222f2c6142f43fc4395d9f9ee833ec88c5106e8d
BSD-3-Clause
2015-04-19T23:24:25.000Z
2022-02-10T15:21:12.000Z
problem_sets/PS1 - Qn 9.ipynb
materialsvirtuallab/nano106
222f2c6142f43fc4395d9f9ee833ec88c5106e8d
BSD-3-Clause
problem_sets/PS1 - Qn 9.ipynb
materialsvirtuallab/nano106
222f2c6142f43fc4395d9f9ee833ec88c5106e8d
BSD-3-Clause
2016-06-30T21:37:14.000Z
2021-02-15T16:36:41.000Z
Qwen/Qwen-72B
1. YES
2. YES