1  TRANSLATIONS  REFLECTIONS  ROTATIONS

2  In lay terms, a transformation is a change.  Instinctively, in geometry a transformation means movement.  In the formal sense, a transformation of a plane is a one-to one mapping (function) of a plane into itself. Sun F(Sun) = Right_Sun Sun RightMove Right_Sun

3 A translation “slides” the original figure (or preimage) to a new position creating a new one called image. A B A’ B’ D CD’ C’ A” B” D” C”

4 In other words, a translation of a plane shifts all points on the plane in the same direction and in the same distance. A B A’ B’ D CD’ C’ A” B” D”’ C”’ A’” B’” D” C”

5 An arrow (or vector) also shows a translation. The vector AA’ (1,4) translates point A to point A’ as for every point of the figure, in particular the vertices. C A’ BA D B’ C’ D’ What does this mean? Given a vector, v, the image P’ of a point P is the point for what PP’ is parallel to v and PP’= v

6 Let’s practice Translations!

7 Do you see yourself in the mirror every morning? That image is a reflection of your face. When a shape is reflected, it does so over a reflecting line, “flipping” over it.

8 Then, a reflection is a transformation where each point in a shape appears at an equal distance on the opposite side of a given line - the line of reflection.

9 In Geometric terms, a reflection is described like this: Given a point P, let m’ be the straight line through P that is perpendicular to m. Then P’ is the point on m’ on the opposite side of m to P that is equidistant from m. m’

10 Now, an experiment: Reflecting your hand! And some practice too!

11 A rotation can be seen as a ‘spin’ around a center.

12 So, a rotation is a transformation that turns a figure about a center of rotation. Said ‘turn’ is made in a specific angle, called angle of rotation.

13 A formal definition defines rotation about a point P through angle α is a transformation such that: (1) If a point A is different from P, then PA=PA’ and the measure of <APA’= α and (2) If point A is the same as point P, then A’=A (2) Point rotating over itself is transformed in the same place. α α

14 Now take turns practicing rotations!

15 Native American Art and Video Games