page_title: Euler's rotation theorem
section_title: Previous analysis
hierarchical_section_title: Euler's rotation theorem / Euler's theorem (1776) / Proof / Previous analysis
caption_attribution_description: English: Arcs connecting preimage alpha and image a of A with bisector AO of the angle at A
context_page_description: In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
  The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê. Its product by the rotation angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points.
  In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1.
context_section_description: To arrive at a proof, Euler analyses what the situation would look like if the theorem were true. To that end, suppose the yellow line in Figure 1 goes through the center of the sphere and is the axis of rotation we are looking for, and point O is one of the two intersection points of that axis with the sphere. Then he considers an arbitrary great circle that does not contain O (the blue circle), and its image after rotation (the red circle), which is another great circle not containing O. He labels a point on their intersection as point A. (If the circles coincide, then A can be taken as any point on either; otherwise A is one of the two points of intersection.) 
  Now A is on the initial circle (the blue circle), so its image will be on the transported circle (red). He labels that image as point a. Since A is also on the transported circle (red), it is the image of another point that was on the initial circle (blue) and he labels that preimage as α (see Figure 2). Then he considers the two arcs joining α and a to A. These arcs have the same length because arc αA is mapped onto arc Aa. Also, since O is a fixed point, triangle αOA is mapped onto triangle AOa, so these triangles are isosceles, and arc AO bisects angle ∠αAa.
caption: Figure 2: Arcs connecting preimage α and image a of A with bisector AO of the angle at A.
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