Product of Rotations

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The product, or the sum as its often called, of two rotations is either a rotation or a translation, both of which can be trivial. Depending on the context, one may treat the product of two rotations which is trivial as either a rotation or a translation.

The product of two rotations through angles α and β is a rotation through angle (α + β). Its center, which exists if the product is not trivial, can be found constructively.

Let there be given two rotations g_P and g_Q with centers P and Q. Assume g_P is executed first, g_Q second. g_P leaves P in place. Denote g_Q(P) = P'. g_P carries some point Q' onto Q which is then left in place by g_Q. The point O that lies at the intersection of perpendicular bisectors of QQ' and PP' is equidistant from P and P' and also from Q and Q'.

Now prove that ∠POP' = ∠Q'OQ = ∠PQP' + ∠Q'PQ. Finally, consider what happens with an arbitrary point (or two) under two successive rotations.

(The fact just established provides a powerful tool for solving geometric problems, see, for example, proofs of Bottema's theorem and an analogue.)

Geometric Transformations

Plane Isometries
- Reflection in Point
- Reflection in Line
  - Geometric Problems by Reflection
- Translation Transform
- Product of Rotations
  - Rotation Transform
- Glide Reflection
Affine Transformations (definition)
Inversion
- Stereographic Projection and Inversion
- Möbius Transformations
Projective transformation

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