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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 structure known as a ''rotation group''. The theorem is named after Leonhard Euler, who proved it in 1775 by an elementary geometric argument. The axis of rotation is known as an Euler axis, typically represented by a unit vector . 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 it must happen that: one of its eigenvalues is 1 and the other two are -1, or it has only one real eigenvalue which is equal to unity. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems. ==Euler's theorem (1776)== Euler states the theorem as follows:〔Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478)〕
or (in English): :''When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position.'' 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.) 200pxO (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 ɑ (see Figure 2). Then he considers the two arcs joining ɑ and a to A. These arcs have the same length because arc ɑA is mapped onto arc Aa. Also, since O is a fixed point, triangle ɑOA is mapped onto triangle AOa, so these triangles are isosceles, and arc AO bisects angle ɑAa. So here is the actual proof: We start with the blue great circle and its image under the transformation, which is the red great circle as in the Figure 1. Let point A be a point of intersection of those circles. If A’s image under the transformation is the same point then A is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing A is the axis of rotation and the theorem is proved. Otherwise we label A’s image as a and its preimage as ɑ, and connect these two points to A with arcs ɑA and Aa. These arcs have the same length. Construct the great circle that bisects angle ɑAa and locate point O on that great circle so that arcs AO and aO have the same length, and call the region of the sphere containing O and bounded by the blue and red great circles the "interior" of angle ɑAa. (That is the yellow region in Figure 3.) Then since ɑA = Aa and O is on the bisector of angle ɑAa, we also have ɑO = aO. Now suppose O' is the image of O. Then we know angle ɑAO = angle AaO' and orientation is preserved *, so O' must be interior to angle ɑAa. Now AO is transformed to aO', so AO = aO'. Since AO is also the same length as aO, angle AaO = angle aAO. But angle aAO = angle AaO', so angle AaO = angle AaO' and therefore O' is the same point as O. In other words, O is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing O is the axis of rotation. Euler also points out that O can be found by intersecting the perpendicular bisector of Aa with the angle bisector of angle ɑAO. *Note: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euler's rotation theorem」の詳細全文を読む スポンサード リンク
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