Final notes about the construction Euler's rotation theorem

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euler s original drawing


euler points out o can found intersecting perpendicular bisector of aa angle bisector of ∠αao, construction might easier in practice. proposed intersection of 2 planes:



the symmetry plane of angle ∠αaa (which passes through center c of sphere), and
the symmetry plane of arc aa (which passes through c).


proposition. these 2 planes intersect in diameter. diameter 1 looking for.


proof. let call o either of endpoints (there two) of diameter on sphere surface. since αa mapped on aa , triangles have same angles, follows triangle oαa transported onto triangle oaa. therefore point o has remain fixed under movement.


corollaries. shows rotation of sphere can seen 2 consecutive reflections 2 planes described above. points in mirror plane invariant under reflection, , hence points on intersection (a line: axis of rotation) invariant under both reflections, , hence under rotation.

another simple way find rotation axis considering plane on points α, a, lie. rotation axis orthogonal plane, , passes through center c of sphere.


given rigid body movement leaves axis invariant rotation, proves arbitrary composition of rotations equivalent single rotation around new axis.







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