Geometry Transformation

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