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Abstract geometry. Reflections and products of reflection.

### Transcript of Geometry Transformation

• 1. Geometry transformations Reflectionsandproducts of reflection By: SR. TA

2. Geometry transformations ? Geometry ? A one-to-one mapping transformations ? l l OPQ OPQ Collineation 3. More example : Product TS is not equal to the product ST T: rotation 120* Clockwise about O T S S: reflection across line AO 4. ST S T 5. Commute :if 2 transformations S,T happen to have the property ST=TS Commutative : a collection of transformations in whichevery pair commute 6. Reflections: Rm The fundamental type of motion. m Q A A^m AQ = QA^m m :line of reflectionfor a A^m :it own image. A : is the reflection of A^m with respect to Q Q:point of reflection 7. Reflections: Rm R R R m m Rm = (Rm) -1 RmRm =I A reflection: ( or flip) is anisometryin 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. 8. Reflections preserve :collinearity, betweeness of points m S X T Y U Z 9. Reflections preserve :Angle measureand distance measure y x A B B C C A ABC ABC Proposition 9.5 10. Isometries As Products of Reflections

• The four Euclidean isometries:
• Reflection
• Translation
• Rotation
• Glide reflection

11. 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 12. Proposition 9.7 A motion T =Iis a rotation if and only if T has exactly one fixed point 13. 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 14. Proposition 9.9: Given a point A, the set of rotations about A is a commutative group. 15. 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