4.3: Algebra of Transformations
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4.3.1 Explore
An transformation T is a translation if and only if there exists a non-zero constant vector v such that T(P) - P = v for all points P.
An transformation T is a rotation if and only if there exists a fixed point C and constant angle α such that m∠PCT(P) = α and llCPll = llCT(P)ll for all points P.
An transformation T is a reflection if and only if there exists a fixed line ℓ such that the line perpendicular to ℓ through P contains T(P) and the distances from P and T(P) to the ℓ are equal.
Translations, rotations, and reflections are isometries.
Draw an arbitrary triangle △ABC. Draw the result △A'B'C' of some translation. Draw the result △A''B''C'' of some translation applied to △A'B'C' Determine which type of isometry would transform △ABC to △A''B''C''.
Complete the following table of composition of isometries.
Translate | Reflect | Rotate | |
Translate | |||
Reflect | |||
Rotate |
4.3.2 Prove
How many isometry types are there?
How many isometry types are needed to generate all isometry types?
How many isometries are needed to generate all isometries?