Mirror Constructions and Isometry Type Identifications
Translations
Given only a figure
F
and its image figure
F
'
under the translation
T
PQ
,
appropriate mirror lines L
1
and L
2
can be constructed as follows:
1)
Locate a point
A in the figure
F
and its image
A' in the image figure
F
' .
2)
Construct vector
AA'
.
3)
Perform the construction of
L
1
and L
2
using
AA'
rather than PQ
.
This suffices because
PQ
and
AA'
both serve as defining vectors for the
same translation:
T
PQ
=
T
AA'
.
In fact, any two lines L
1
and L
2
will suffice as long as:
1) they are parallel lines which are perpendicular to PQ
,
and
2) the distance between them is d = (1/2) (PQ), and
3) the direction from
L
1
to L
2
is the same as the direction
from
P to Q .
L
2
L
1
d
Construct:
1) The midpoint
M
of
PQ
.
2) Line L
1
through P and
⊥
PQ
3) Line L
2
through M and
⊥
PQ
Then,
T
PQ
=
R
L
2
°
R
L
1
.
Given:
Construction:
Given the defining vector PQ
, construct two parallel mirror lines
L
1
and L
2
such that
T
PQ
=
R
L
2
°
R
L
1
.
Note: The distance
d
between the two lines is d
=
(1/2) (PQ) .
Every translation
T
PQ
is a direct isometry which is the composition
of two reflections with parallel mirror lines.
M
Q
P
Q
P
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View Full DocumentMirror Constructions and Isometry Type Identifications
Rotations
α
If
θ
is undetermined, set
θ
=
∠
APA' , viewed as a directed angle.
L
2
L
1
n
m
Given only a figure
F
and its image figure
F
'
under the rotation
R
P ,
θ
,
the points
A
and
A'
can easily be located.
However, we must first construct
the center of rotation P .
Construction: Given a figure
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 Spring '10
 Shirley
 Trigraph, Line segment, Elementary geometry, glide reflection

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