Complex Numbers and Transformations of the Plane

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Complex Numbers and Transformations of the Plane

• Lesson
• 9.4

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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)

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Definitions

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

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Isometry Theorem

• 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)

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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

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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

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Example 2

• Find a formula that maps PQR (with P 1, Q
  2 i, R 4 i) onto PQR (with P -1, Q
  -2 i , R -4 i).
• rotation 180
• z(cos 180 i sin 180)
• - z

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Example 3

• Describe the transformation with the given rule
  as a composite of rotations, reflections, or
  translations.
• iz 3 5i
• T3-5i ? Rp/2 ? rR

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Homework

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