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Ingeometry,Euler's rotation theorem states that, inthree-dimensional space, any displacement of arigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through thefixed 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 arotation group.

The theorem is named afterLeonhard Euler, who proved it in 1775 by means ofspherical geometry. The axis of rotation is known as anEuler axis, typically represented by aunit vectorê. Its product by the rotation angle is known as anaxis-angle vector. The extension of the theorem tokinematics yields the concept ofinstant axis of rotation, a line of fixed points.

In linear algebra terms, the theorem states that, in 3D space, any twoCartesian 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-identityrotation matrix oneeigenvalue is 1 and the other two are both complex, or both equal to −1. Theeigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

Euler states the theorem as follows:^[1]

Theorema.Quomodocunque sphaera circa centrum suum conuertatur, semper assignari potest diameter,cuius directio in situ translato conueniat cum situ initiali.

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.

Euler's original proof was made usingspherical geometry and therefore whenever he speaks about triangles they must be understood asspherical triangles.

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 inFigure 1 goes through the center of the sphere and is the axis of rotation we are looking for, and pointO is one of the two intersection points of that axis with the sphere. Then he considers an arbitrary great circle that does not containO (the blue circle), and its image after rotation (the red circle), which is another great circle not containingO. He labels a point on their intersection as pointA. (If the circles coincide, thenA can be taken as any point on either; otherwiseA is one of the two points of intersection.)

NowA is on the initial circle (the blue circle), so its image will be on the transported circle (red). He labels that image as pointa. SinceA 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 asα (seeFigure 2). Then he considers the two arcs joiningα anda toA. These arcs have the same length because arcαA is mapped onto arcAa. Also, sinceO is a fixed point, triangleαOA is mapped onto triangleAOa, so these triangles are isosceles, and arcAO bisects angle∠αAa.

Let us construct a point that could be invariant using the previous considerations. We start with the blue great circle and its image under the transformation, which is the red great circle as in theFigure 1. Let pointA be a point of intersection of those circles. IfA’s image under the transformation is the same point thenA is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containingA is the axis of rotation and the theorem is proved.

Otherwise we labelA’s image asa and its preimage asα, and connect these two points toA with arcsαA andAa. These arcs have the same length. Construct the great circle that bisects∠αAa and locate pointO on that great circle so that arcsAO andaO have the same length, and call the region of the sphere containingO and bounded by the blue and red great circles the interior of∠αAa. (That is, the yellow region inFigure 3.) Then sinceαA =Aa andO is on the bisector of∠αAa, we also haveαO =aO.

Now let us suppose thatO′ is the image ofO. Then we know∠αAO = ∠AaO′ and orientation is preserved,^[a] soO′ must be interior to∠αAa. NowAO is transformed toaO′, soAO =aO′. SinceAO is also the same length asaO, thenaO =aO′ and∠AaO = ∠aAO. But∠αAO = ∠aAO, so∠αAO = ∠AaO and∠AaO = ∠AaO′. ThereforeO′ is the same point asO. 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 containingO is the axis of rotation.

Euler also points out thatO can be found by intersecting the perpendicular bisector ofAa with the angle bisector of∠αAa, a construction that might be easier in practice. He also proposed the intersection of two planes:

• the symmetry plane of the angle∠αAa (which passes through the centerC of the sphere), and
• the symmetry plane of the arcAa (which also passes throughC).

Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for.

Proof. Let us callO either of the endpoints (there are two) of this diameter over the sphere surface. SinceαA is mapped onAa and the triangles have the same angles, it follows that the triangleOαA is transported onto the triangleOAa. Therefore the pointO has to remain fixed under the movement.

Corollaries. This also shows that the rotation of the sphere can be seen as two consecutive reflections about the two planes described above. Points in a mirror plane are invariant under reflection, and hence the points on their intersection (a line: the axis of rotation) are invariant under both the reflections, and hence under the rotation.

Another simple way to find the rotation axis is by considering the plane on which the pointsα,A,a lie. The rotation axis is obviously orthogonal to this plane, and passes through the centerC of the sphere.

Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis.

A spatial rotation is a linear map in one-to-one correspondence with a3 × 3rotation matrixR that transforms a coordinatevectorx intoX, that isRx =X. Therefore, another version of Euler's theorem is that for every rotationR, there is a nonzero vectorn for whichRn =n; this is exactly the claim thatn is aneigenvector ofR associated with theeigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue ofR; the rotation axis ofR will be the lineμn, wheren is the eigenvector with eigenvalue 1.

A rotation matrix has the fundamental property that its inverse is its transpose, that is

${\displaystyle {\mathbf{R}}^{\mathsf{T}}\mathbf{R}=\mathbf{R}{\mathbf{R}}^{\mathsf{T}}=\mathbf{I},}$

whereI is the3 × 3 identity matrix and superscript T indicates the transposed matrix.

Compute the determinant of this relation to find that a rotation matrix hasdeterminant ±1. In particular,

A rotation matrix with determinant +1 is a proper rotation, and one with a negative determinant −1 is animproper rotation, that is a reflection combined with a proper rotation.

It will now be shown that a proper rotation matrixR has at least one invariant vectorn, i.e.,Rn =n. Because this requires that(R −I)n = 0, we see that the vectorn must be aneigenvector of the matrixR with eigenvalueλ = 1. Thus, this is equivalent to showing thatdet(R −I) = 0.

Use the two relations

${\displaystyle det(-\mathbf{A})=(-1{)}^{3}det(\mathbf{A})=-det(\mathbf{A})}$

for any3 × 3 matrixA and

${\displaystyle det\left({\mathbf{R}}^{-1}\right)=1}$

(sincedet(R) = 1) to compute

This shows thatλ = 1 is a root (solution) of thecharacteristic equation, that is,

${\displaystyle det(\mathbf{R}-\lambda \mathbf{I})=0\text{for}\lambda =1.}$

In other words, the matrixR −I is singular and has a non-zerokernel, that is, there is at least one non-zero vector, sayn, for which

${\displaystyle (\mathbf{R}-\mathbf{I})\mathbf{n}=\mathbf{0}⟺\mathbf{R}\mathbf{n}=\mathbf{n}.}$

The lineμn for realμ is invariant underR, i.e.,μn is a rotation axis. This proves Euler's theorem.

Two matrices (representing linear maps) are said to be equivalent if there is achange of basis that makes one equal to the other. A properorthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:

Then, any orthogonal matrix is either a rotation or animproper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

IfR has more than one invariant vector thenφ = 0 andR =I.Any vector is an invariant vector ofI.

In order to prove the previous equation some facts from matrix theory must be recalled.

Anm ×m matrixA hasm orthogonal eigenvectors if and only ifA isnormal, that is, ifA^†A =AA^†.^[b] This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

${\displaystyle \mathbf{A}\mathbf{U}=\mathbf{U}\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_m)⟺{\mathbf{U}}^{†}\mathbf{A}\mathbf{U}=\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_m),}$

andU is unitary, that is,

${\displaystyle {\mathbf{U}}^{†}={\mathbf{U}}^{-1}.}$

The eigenvaluesα₁, ...,α_m are roots of the characteristic equation. If the matrixA happens to be unitary (and note that unitary matrices are normal), then

${\displaystyle {\left({\mathbf{U}}^{†}\mathbf{A}\mathbf{U}\right)}^{†}=\mathrm{diag}\left({\alpha }_1^{\ast },\dots ,{\alpha }_m^{\ast }\right)={\mathbf{U}}^{†}{\mathbf{A}}^{-1}\mathbf{U}=\mathrm{diag}\left(\frac{1}{{\alpha }_1},\dots ,\frac{1}{{\alpha }_m}\right)}$

and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:

${\displaystyle {\alpha }_k^{\ast }=\frac{1}{{\alpha }_k}⟺{\alpha }_k^{\ast }{\alpha }_k={\left|{\alpha }_k\right|}^2=1,k=1,\dots ,m.}$

Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (anmth order polynomial inλ) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, ifα is a root then so isα^∗. There are 3 roots, thus at least one of them must be purely real (+1 or −1).

After recollection of these general facts from matrix theory, we return to the rotation matrixR. It follows from its realness and orthogonality that we can find aU such that:

If a matrixU can be found that gives the above form, and there is only one purely real component and it is −1, then we define${\displaystyle \mathbf{R}}$ to be an improper rotation. Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1). The third column of the3 × 3 matrixU will then be equal to the invariant vectorn. Writingu₁ andu₂ for the first two columns ofU, this equation gives

${\displaystyle \mathbf{R}{\mathbf{u}}_1=e^{i\varphi }{\mathbf{u}}_1\text{and}\mathbf{R}{\mathbf{u}}_2=e^{-i\varphi }{\mathbf{u}}_2.}$

Ifu₁ has eigenvalue 1, thenφ = 0 andu₂ has also eigenvalue 1, which implies that in that caseR =I. In general, however, as${\displaystyle (\mathbf{R}-e^{i\varphi }\mathbf{I}){\mathbf{u}}_1=0}$ implies that also${\displaystyle (\mathbf{R}-e^{-i\varphi }\mathbf{I}){\mathbf{u}}_1^{\ast }=0}$ holds, so${\displaystyle {\mathbf{u}}_2={\mathbf{u}}_1^{\ast }}$ can be chosen for${\displaystyle {\mathbf{u}}_2}$. Similarly,${\displaystyle (\mathbf{R}-\mathbf{I}){\mathbf{u}}_3=0}$ can result in a${\displaystyle {\mathbf{u}}_3}$ with real entries only, for a proper rotation matrix${\displaystyle \mathbf{R}}$. Finally, the matrix equation is transformed by means of a unitary matrix,

which gives

The columns ofU′ are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that${\displaystyle {\mathbf{u}}_1}$ is the complex conjugate of${\displaystyle {\mathbf{u}}_2}$ and that${\displaystyle {\mathbf{u}}_3}$ is a vector with real-valued components. The third column is still${\displaystyle {\mathbf{u}}_3=}$n, the other two columns ofU′ are perpendicular ton. We can now see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrixR corresponding to a proper rotation is equivalent to a rotation over an angleφ around an axisn.

Thetrace (sum of diagonal elements) of the real rotation matrix given above is1 + 2 cosφ. Since a trace is invariant under an orthogonal matrix similarity transformation,

${\displaystyle \mathrm{T}\mathrm{r}\left[\mathbf{A}\mathbf{R}{\mathbf{A}}^{\mathsf{T}}\right]=\mathrm{T}\mathrm{r}\left[\mathbf{R}{\mathbf{A}}^{\mathsf{T}}\mathbf{A}\right]=\mathrm{T}\mathrm{r}[\mathbf{R}]\text{ with }{\mathbf{A}}^{\mathsf{T}}={\mathbf{A}}^{-1},}$

it follows that all matrices that are equivalent toR by such orthogonal matrix transformations have the same trace: the trace is aclass function. This matrix transformation is clearly anequivalence relation, that is, all such equivalent matrices form an equivalence class.

In fact, all proper rotation3 × 3 rotation matrices form agroup, usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. All elements of such an equivalence classshare their rotation angle, but all rotations are around different axes. Ifn is an eigenvector ofR with eigenvalue 1, thenAn is also an eigenvector ofARA^T, also with eigenvalue 1. UnlessA =I,n andAn are different.

Suppose we specify an axis of rotation by a unit vector[x,y,z], and suppose we have aninfinitely small rotation of angleΔθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrixΔR is represented as:

A finite rotation through angleθ about this axis may be seen as a succession of small rotations about the same axis. ApproximatingΔθ as⁠θ/N⁠ whereN is a large number, a rotation ofθ about the axis may be represented as:

${\displaystyle R={\left(\mathbf{1}+\frac{\mathbf{A}\theta }{N}\right)}^N\approx e^{\mathbf{A}\theta }.}$

It can be seen that Euler's theorem essentially states thatall rotations may be represented in this form. The productAθ is the "generator" of the particular rotation, being the vector(x,y,z) associated with the matrixA. This shows that the rotation matrix and theaxis–angle format are related by the exponential function.

One can derive a simple expression for the generatorG. One starts with an arbitrary plane (in Euclidean space) defined by a pair of perpendicular unit vectorsa andb. In this plane one can choose an arbitrary vectorx with perpendiculary. One then solves fory in terms ofx and substituting into an expression for a rotation in a plane yields the rotation matrixR which includes the generatorG =ba^T −ab^T.

To include vectors outside the plane in the rotation one needs to modify the above expression forR by including twoprojection operators that partition the space. This modified rotation matrix can be rewritten as anexponential function.

Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as theLie algebra of the rotation group.

It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called aquaternion.

While the quaternion described above does not involvecomplex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutativequaternion algebra derived byWilliam Rowan Hamilton through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use ofdirection cosines in aerospace applications through their reduction of the required calculations, and their ability to minimizeround-off errors. Also, incomputer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

In higher dimensions, any rigid motion that preserves a point in dimension2n or2n + 1 is a composition of at mostn rotations in orthogonalplanes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any rigid motion that preservesn linearly independent points, which span ann-dimensional body in dimension2n or2n + 1, is a singleplane of rotation. To put it another way, if two rigid bodies, with identical geometry, share at leastn points of 'identical' locations within themselves, the convex hull of which isn-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension2n or2n + 1.

A rigid motion in three dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; seescrew axis. This gives rise toscrew theory.

• Euler angles
• Euler–Rodrigues formula
• Rotation formalisms in three dimensions
• Angular velocity
• Rotation around a fixed axis
• Matrix exponential
• Axis–angle representation
• Chasles' theorem (kinematics), for an extension concerning general rigid body displacements.

This article incorporates material from theCitizendium article "Euler's theorem (rotation)", which is licensed under theCreative Commons Attribution-ShareAlike 3.0 Unported License but not under theGFDL.

• Euler's theorem and its proof are contained in paragraphs 24–26 of the appendix (Additamentum. pp. 201–203) of L. Eulero(Leonhard Euler),Formulae generales pro translatione quacunque corporum rigidorum (General formulas for the translation of arbitrary rigid bodies), presented to the St. Petersburg Academy on October 9, 1775, and first published inNovi Commentarii academiae scientiarum Petropolitanae20, 1776, pp. 189–207 (E478) and was reprinted inTheoria motus corporum rigidorum, ed. nova, 1790, pp. 449–460 (E478a) and later in his collected worksOpera Omnia, Series 2, Volume9, pp. 84–98.
• Palais, Bob; Palais, Richard; Rodi, Stephen (2009). "A Disorienting Look at Euler's Theorem on the Axis of a Rotation".American Mathematical Monthly.116 (10):892–909.doi:10.4169/000298909x477014.

• Euler's original treatise inThe Euler Archive: entry onE478, first publication 1776 (pdf)
• Euler's original text (in Latin) and English translation (by Johan Sten)
• Wolfram Demonstrations Project for Euler's Rotation Theorem (by Tom Verhoeff)

• Euclidean symmetries
• Theorems in geometry
• Eponymous theorems of geometry
• Rotation in three dimensions
• Leonhard Euler

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