Complex Numbers and Transformations of the Plane

1. Complex Numbers and Transformations of the Plane Lesson 9.4

2. Definitions • Distance between complex numbers – The distance between two complex numbers z and w is |z – w| • A transormation T of the plane is an isometry iff |z – w| = |T(z) – T(w)|

3. Definitions • Complex Number Transformation Theorem 1. The translation Ta,b a units horizontally and b units vertically can be expressed as Ta+bi: z  z + a + bi 2. The reflection over the real axis is rR = z  z 3. The rotation Rθ of mag. θ about the origin can be expressed as Rθ: z  (cos θ + i sin θ) * z 4. The size change of Sk of mag k and centered at the origin can be expressed as Sk: z  k ∙ z

4. IsometryTheorem The transformation T is an isometry iff there exist complex numbers c and d with |c| = 1 st T(z) = cz + d (direct isometry) T(z) = cz + d (opposite isometry)

5. Plain and Simple… • It is an isometry if it is only a translation, rotation, and/or reflection. • It cannot be an isometry if there is a scale change. Isometry Direct Indirect Translation rotation reflection glide reflection

6. Example 1 Find a formula for the congruence transformation T that maps black onto blue: *rotation across imaginary axis * real : 0, im: down 5 - z - 5i

7. Example 2 Find a formula that maps PQR (with P = 1, Q = 2 + i, R = 4 + i) onto P’Q’R’ (with P = -1, Q = -2 – i , R = -4 – i). rotation 180 z(cos 180 + i sin 180) - z

8. Example 3 Describe the transformation with the given rule as a composite of rotations, reflections, or translations. iz + 3 – 5i T3-5i ◦ Rπ/2 ◦ rR

9. Homework Page 545 1 - 9