geometry transformation
DESCRIPTION
Abstract geometry. Reflections and products of reflection.TRANSCRIPT
Geometry transformations
Reflections and
products of reflection
By: SR. TA
Geometry transformations ?
Geometry ?
A one-to-one mapping
transformations ?
l
l’
O P Q
O’ P’ Q’
Collineation
More example: Product TS is not equal to the product ST
T: rotation 120* Clockwise about O
T
S
S: reflection across line AO
ST
S
T
Commute: if 2 transformations S,T happen to have the property ST=TS
Commutative: a collection of transformations in which every pair commute
Reflections: Rm
The fundamental type of motion.
mQ
A
A^m
AQ = QA^m
m : line of reflection for a
A^m : it own image.
A : is the reflection of A^m with respect to Q
Q: point of reflection
Reflections: Rm …
m
m
Rm = (Rm)-1
RmRm = I
A reflection: ( or flip) is an isometry in which a figure and its images have opposite orientations.
Isometry: ( or motion) A transformation T of the entire plane onto itself, it length is invariant under T.
Reflections preserve : collinearity, betweeness of points
m
S X
T Y
U Z
Reflections preserve : Angle measure
and distance measure
y
x
A
B’B
C C’
A’
ABC ≡
A’B’C’
≡
Proposition 9.5
Isometries As Products of Reflections
The four Euclidean isometries:
1. Reflection
2. Translation
3. Rotation
4. Glide reflection
Translation and Reflection
X units
Translate 2X units to the right
m n
-Translation is equivalent to the composition of 2 reflections, one across m and the other across n
-- A composition of reflection in 2 parallel lines is a translation
Proposition 9.12. Given a line t, the set of translation along t is a commutative group
Proposition 9.7
A motion T = I is a rotation if and only if T has exactly one fixed point
Rotation and Reflection
m
n
C
A
B
< ACB = 2 < mCn
- Rotation is then a composition of the 2 reflections over m and n
- A composition of reflections in 2 intersecting lines is a rotation
Proposition 9.8
Proposition 9.9: Given a point A, the set of rotations about A is a commutative group.
Glide and Reflection
A glide reflection: is the composition of translation and a reflection in a line parallel to glide vector
X units
Translate 2X units to the right
m n