Английская Википедия:Euler's rotation theorem
Шаблон:Short description Шаблон:Refimprove
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 Шаблон:Math. Its product by the rotation angle is known as an axis-angle 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 one eigenvalue is 1 and the other two are both complex, or both equal to −1. 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:[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.
Proof
Euler's original proof was made using spherical geometry and therefore whenever he speaks about triangles they must be understood as spherical triangles.
Previous analysis
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 Шаблон:Math is one of the two intersection points of that axis with the sphere. Then he considers an arbitrary great circle that does not contain Шаблон:Math (the blue circle), and its image after rotation (the red circle), which is another great circle not containing Шаблон:Math. He labels a point on their intersection as point Шаблон:Math. (If the circles coincide, then Шаблон:Math can be taken as any point on either; otherwise Шаблон:Math is one of the two points of intersection.)
Now Шаблон:Math is on the initial circle (the blue circle), so its image will be on the transported circle (red). He labels that image as point Шаблон:Math. Since Шаблон:Math 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 Шаблон:Math (see Figure 2). Then he considers the two arcs joining Шаблон:Math and Шаблон:Math to Шаблон:Math. These arcs have the same length because arc Шаблон:Math is mapped onto arc Шаблон:Math. Also, since Шаблон:Math is a fixed point, triangle Шаблон:Math is mapped onto triangle Шаблон:Math, so these triangles are isosceles, and arc Шаблон:Math bisects angle Шаблон:Math.
Construction of the best candidate point
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 the Figure 1. Let point Шаблон:Math be a point of intersection of those circles. If Шаблон:Math’s image under the transformation is the same point then Шаблон:Math is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing Шаблон:Math is the axis of rotation and the theorem is proved.
Otherwise we label Шаблон:Math’s image as Шаблон:Math and its preimage as Шаблон:Math, and connect these two points to Шаблон:Math with arcs Шаблон:Math and Шаблон:Math. These arcs have the same length. Construct the great circle that bisects Шаблон:Math and locate point Шаблон:Math on that great circle so that arcs Шаблон:Math and Шаблон:Math have the same length, and call the region of the sphere containing Шаблон:Math and bounded by the blue and red great circles the interior of Шаблон:Math. (That is, the yellow region in Figure 3.) Then since Шаблон:Math and Шаблон:Math is on the bisector of Шаблон:Math, we also have Шаблон:Math.
Proof of its invariance under the transformation
Now let us suppose that Шаблон:Math is the image of Шаблон:Math. Then we know Шаблон:Math and orientation is preserved,Шаблон:Efn so Шаблон:Math must be interior to Шаблон:Math. Now Шаблон:Math is transformed to Шаблон:Math, so Шаблон:Math. Since Шаблон:Math is also the same length as Шаблон:Math, then Шаблон:Math and Шаблон:Math. But Шаблон:Math, so Шаблон:Math and Шаблон:Math. Therefore Шаблон:Math is the same point as Шаблон:Math. In other words, Шаблон:Math is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing Шаблон:Math is the axis of rotation.
Final notes about the construction
Euler also points out that Шаблон:Math can be found by intersecting the perpendicular bisector of Шаблон:Math with the angle bisector of Шаблон:Math, a construction that might be easier in practice. He also proposed the intersection of two planes:
- the symmetry plane of the angle Шаблон:Math (which passes through the center Шаблон:Math of the sphere), and
- the symmetry plane of the arc Шаблон:Math (which also passes through Шаблон:Math).
- Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for.
- Proof. Let us call Шаблон:Math either of the endpoints (there are two) of this diameter over the sphere surface. Since Шаблон:Math is mapped on Шаблон:Math and the triangles have the same angles, it follows that the triangle Шаблон:Math is transported onto the triangle Шаблон:Math. Therefore the point Шаблон:Math 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 points Шаблон:Math, Шаблон:Math, Шаблон:Math lie. The rotation axis is obviously orthogonal to this plane, and passes through the center Шаблон:Math 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.
Matrix proof
A spatial rotation is a linear map in one-to-one correspondence with a Шаблон:Nowrap rotation matrix Шаблон:Math that transforms a coordinate vector Шаблон:Math into Шаблон:Math, that is Шаблон:Math. Therefore, another version of Euler's theorem is that for every rotation Шаблон:Math, there is a nonzero vector Шаблон:Math for which Шаблон:Math; this is exactly the claim that Шаблон:Math is an eigenvector of Шаблон:Math associated with the eigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue of Шаблон:Math; the rotation axis of Шаблон:Math will be the line Шаблон:Math, where Шаблон:Math is the eigenvector with eigenvalue 1.
A rotation matrix has the fundamental property that its inverse is its transpose, that is
- <math>
\mathbf{R}^\mathsf{T}\mathbf{R} = \mathbf{R}\mathbf{R}^\mathsf{T} = \mathbf{I}, </math> where Шаблон:Math is the Шаблон:Nowrap identity matrix and superscript T indicates the transposed matrix.
Compute the determinant of this relation to find that a rotation matrix has determinant ±1. In particular,
- <math>\begin{align}
1 = \det(\mathbf{I}) &= \det\left(\mathbf{R}^\mathsf{T}\mathbf{R}\right) = \det\left(\mathbf{R}^\mathsf{T}\right)\det(\mathbf{R}) = \det(\mathbf{R})^2 \\
\Longrightarrow\qquad \det(\mathbf{R}) &= \pm 1.
\end{align}</math>
A rotation matrix with determinant +1 is a proper rotation, and one with a negative determinant −1 is an improper rotation, that is a reflection combined with a proper rotation.
It will now be shown that a proper rotation matrix Шаблон:Math has at least one invariant vector Шаблон:Math, i.e., Шаблон:Math. Because this requires that Шаблон:Math, we see that the vector Шаблон:Math must be an eigenvector of the matrix Шаблон:Math with eigenvalue Шаблон:Math. Thus, this is equivalent to showing that Шаблон:Math.
Use the two relations
- <math> \det(-\mathbf{A}) = (-1)^{3} \det(\mathbf{A}) = - \det(\mathbf{A}) \quad</math>
for any Шаблон:Nowrap matrix A and
- <math> \det\left(\mathbf{R}^{-1} \right) = 1 \quad</math>
(since Шаблон:Math) to compute
- <math>\begin{align}
&\det(\mathbf{R} - \mathbf{I}) = \det\left((\mathbf{R} - \mathbf{I})^\mathsf{T}\right) \\
{}={} &\det\left(\mathbf{R}^\mathsf{T} - \mathbf{I}\right) = \det\left(\mathbf{R}^{-1} - \mathbf{R}^{-1}\mathbf{R}\right) \\
{}={} &\det\left(\mathbf{R}^{-1}(\mathbf{I} - \mathbf{R})\right) = \det\left(\mathbf{R}^{-1}\right) \, \det(-(\mathbf{R} - \mathbf{I})) \\
{}={} &-\det(\mathbf{R} - \mathbf{I}) \\[3pt]
\Longrightarrow\ 0 ={} &\det(\mathbf{R} - \mathbf{I}).
\end{align}</math>
This shows that Шаблон:Math is a root (solution) of the characteristic equation, that is,
- <math>
\det(\mathbf{R} - \lambda \mathbf{I}) = 0\quad \hbox{for}\quad \lambda=1. </math>
In other words, the matrix Шаблон:Math is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say Шаблон:Math, for which
- <math>
(\mathbf{R} - \mathbf{I}) \mathbf{n} = \mathbf{0} \quad \Longleftrightarrow \quad \mathbf{R}\mathbf{n} = \mathbf{n}. </math>
The line Шаблон:Math for real Шаблон:Mvar is invariant under Шаблон:Math, i.e., Шаблон:Math is a rotation axis. This proves Euler's theorem.
Equivalence of an orthogonal matrix to a rotation matrix
Two matrices (representing linear maps) are said to be equivalent if there is a change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:
- <math>
\mathbf{R} \sim \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\\ \end{pmatrix}, \qquad 0\le \phi \le 2\pi. </math>
Then, any orthogonal matrix is either a rotation or an improper 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.
If Шаблон:Math has more than one invariant vector then Шаблон:Math and Шаблон:Math. Any vector is an invariant vector of Шаблон:Math.
Excursion into matrix theory
In order to prove the previous equation some facts from matrix theory must be recalled.
An Шаблон:Math matrix Шаблон:Math has Шаблон:Math orthogonal eigenvectors if and only if Шаблон:Math is normal, that is, if Шаблон:Math.Шаблон:Efn This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:
- <math>
\mathbf{A}\mathbf{U} = \mathbf{U}\; \operatorname{diag}(\alpha_1,\ldots,\alpha_m)\quad \Longleftrightarrow\quad \mathbf{U}^\dagger \mathbf{A}\mathbf{U} = \operatorname{diag}(\alpha_1,\ldots,\alpha_m), </math> and Шаблон:Math is unitary, that is,
- <math>
\mathbf{U}^\dagger = \mathbf{U}^{-1}. </math> The eigenvalues Шаблон:Math are roots of the characteristic equation. If the matrix Шаблон:Math happens to be unitary (and note that unitary matrices are normal), then
- <math>
\left(\mathbf{U}^\dagger\mathbf{A} \mathbf{U}\right)^\dagger = \operatorname{diag}\left(\alpha^*_1,\ldots,\alpha^*_m\right) = \mathbf{U}^\dagger\mathbf{A}^{-1} \mathbf{U} = \operatorname{diag}\left(\frac{1}{\alpha_1},\ldots,\frac{1}{\alpha_m}\right) </math> and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:
- <math>
\alpha^*_k = \frac{1}{\alpha_k} \quad\Longleftrightarrow\quad \alpha^*_k\alpha_k = \left|\alpha_k\right|^2 = 1,\qquad k=1,\ldots,m. </math> Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an Шаблон:Mvarth order polynomial in Шаблон:Mvar) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if Шаблон:Mvar is a root then so is Шаблон:Math. 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 matrix Шаблон:Math. It follows from its realness and orthogonality that we can find a Шаблон:Math such that:
- <math>
\mathbf{R} \mathbf{U} = \mathbf{U}
\begin{pmatrix} e^{i\phi} & 0 & 0 \\ 0 & e^{-i\phi} & 0 \\ 0 & 0 & \pm 1 \\ \end{pmatrix} </math> If a matrix Шаблон:Math can be found that gives the above form, and there is only one purely real component and it is −1, then we define <math>\mathbf{R}</math> 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 the Шаблон:Nowrap matrix Шаблон:Math will then be equal to the invariant vector Шаблон:Math. Writing Шаблон:Math and Шаблон:Math for the first two columns of Шаблон:Math, this equation gives
- <math>
\mathbf{R}\mathbf{u}_1 = e^{i\phi}\, \mathbf{u}_1 \quad\hbox{and}\quad \mathbf{R}\mathbf{u}_2 = e^{-i\phi}\, \mathbf{u}_2.
</math> If Шаблон:Math has eigenvalue 1, then Шаблон:Math and Шаблон:Math has also eigenvalue 1, which implies that in that case Шаблон:Math. In general, however, as <math> (\mathbf{R}-e^{i\phi}\mathbf{I})\mathbf{u}_1 = 0 </math> implies that also <math> (\mathbf{R}-e^{-i\phi}\mathbf{I})\mathbf{u}^*_1 = 0 </math> holds, so <math> \mathbf{u}_2 = \mathbf{u}^*_1 </math> can be chosen for <math> \mathbf{u}_2 </math>. Similarly, <math> (\mathbf{R}-\mathbf{I})\mathbf{u}_3 = 0 </math> can result in a <math> \mathbf{u}_3 </math> with real entries only, for a proper rotation matrix <math>\mathbf{R}</math>. Finally, the matrix equation is transformed by means of a unitary matrix,
- <math>
\mathbf{R} \mathbf{U}
\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} = \mathbf{U} \underbrace{ \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{-i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} }_{=\;\mathbf{I}} \begin{pmatrix} e^{i\phi} & 0 & 0 \\ 0 & e^{-i\phi} & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} </math> which gives
- <math>
\mathbf{U'}^\dagger \mathbf{R} \mathbf{U'} = \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\\ \end{pmatrix} \quad\text{ with }\quad \mathbf{U'} = \mathbf{U} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} . </math> The columns of Шаблон:Math are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that <math> \mathbf{u}_1 </math> is the complex conjugate of <math> \mathbf{u}_2 </math> and that <math> \mathbf{u}_3 </math> is a vector with real-valued components. The third column is still <math> \mathbf{u}_3 =</math> Шаблон:Math, the other two columns of Шаблон:Math are perpendicular to Шаблон:Math. 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 matrix Шаблон:Math corresponding to a proper rotation is equivalent to a rotation over an angle Шаблон:Mvar around an axis Шаблон:Math.
Equivalence classes
The trace (sum of diagonal elements) of the real rotation matrix given above is Шаблон:Math. Since a trace is invariant under an orthogonal matrix similarity transformation,
- <math>
\mathrm{Tr}\left[\mathbf{A} \mathbf{R} \mathbf{A}^\mathsf{T}\right] = \mathrm{Tr}\left[ \mathbf{R} \mathbf{A}^\mathsf{T}\mathbf{A}\right] = \mathrm{Tr}[\mathbf{R}]\quad\text{ with }\quad \mathbf{A}^\mathsf{T} = \mathbf{A}^{-1}, </math> it follows that all matrices that are equivalent to Шаблон:Math by such orthogonal matrix transformations have the same trace: the trace is a class function. This matrix transformation is clearly an equivalence relation, that is, all such equivalent matrices form an equivalence class.
In fact, all proper rotation Шаблон:Nowrap rotation matrices form a group, 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 class share their rotation angle, but all rotations are around different axes. If Шаблон:Math is an eigenvector of Шаблон:Math with eigenvalue 1, then Шаблон:Math is also an eigenvector of Шаблон:MathT, also with eigenvalue 1. Unless Шаблон:Math, Шаблон:Math and Шаблон:Math are different.
Applications
Generators of rotations
Suppose we specify an axis of rotation by a unit vector Шаблон:Math, and suppose we have an infinitely small rotation of angle Шаблон:Math about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Шаблон:Math is represented as:
- <math>
\Delta R =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} +
\begin{bmatrix}
0 & z & -y \\
-z & 0 & x \\
y & -x & 0
\end{bmatrix}\,\Delta \theta =
\mathbf{I} + \mathbf{A}\,\Delta \theta.
</math>
A finite rotation through angle Шаблон:Mvar about this axis may be seen as a succession of small rotations about the same axis. Approximating Шаблон:Math as Шаблон:Math where Шаблон:Math is a large number, a rotation of Шаблон:Mvar about the axis may be represented as:
- <math>R = \left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^N \approx e^{\mathbf{A}\theta}.</math>
It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product Шаблон:Math is the "generator" of the particular rotation, being the vector Шаблон:Math associated with the matrix Шаблон:Math. This shows that the rotation matrix and the axis–angle format are related by the exponential function.
One can derive a simple expression for the generator Шаблон:Math. One starts with an arbitrary plane (in Euclidean space) defined by a pair of perpendicular unit vectors Шаблон:Math and Шаблон:Math. In this plane one can choose an arbitrary vector Шаблон:Math with perpendicular Шаблон:Math. One then solves for Шаблон:Math in terms of Шаблон:Math and substituting into an expression for a rotation in a plane yields the rotation matrix Шаблон:Math which includes the generator Шаблон:Nowrap.
- <math>\begin{align}
\mathbf{x} &= \mathbf{a}\cos\alpha + \mathbf{b}\sin\alpha \\
\mathbf{y} &= -\mathbf{a}\sin\alpha + \mathbf{b}\cos\alpha \\[8pt]
\cos\alpha &= \mathbf{a}^\mathsf{T}\mathbf{x} \\
\sin\alpha &= \mathbf{b}^\mathsf{T}\mathbf{x} \\[8px]
\mathbf{y} &= -\mathbf{ab}^\mathsf{T}\mathbf{x} + \mathbf{ba}^\mathsf{T}\mathbf{x}
= \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\mathbf{x} \\[8px]
\mathbf{x}' &= \mathbf{x}\cos\beta + \mathbf{y}\sin\beta \\
&= \left( \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right) \sin\beta \right)\mathbf{x} \\[8px]
\mathbf{R} &= \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\sin\beta \\
&= \mathbf{I}\cos\beta + \mathbf{G}\sin\beta \\[8px]
\mathbf{G} &= \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T}
\end{align}</math>
To include vectors outside the plane in the rotation one needs to modify the above expression for Шаблон:Math by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an exponential function.
- <math>\begin{align}
\mathbf{P_{ab}} &= -\mathbf{G}^2 \\
\mathbf{R} &= \mathbf{I} - \mathbf{P_{ab}} + \left( \mathbf{I} \cos \beta + \mathbf{G} \sin \beta \right)\mathbf{P_{ab}} = e^{\mathbf{G}\beta }
\end{align}</math>
Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.
Quaternions
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 a quaternion.
While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers.
Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.
Generalizations
In higher dimensions, any rigid motion that preserves a point in dimension Шаблон:Math or Шаблон:Math is a composition of at most Шаблон:Mvar rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any rigid motion that preserves Шаблон:Math linearly independent points, which span an Шаблон:Math-dimensional body in dimension Шаблон:Math or Шаблон:Math, is a single plane of rotation. To put it another way, if two rigid bodies, with identical geometry, share at least Шаблон:Math points of 'identical' locations within themselves, the convex hull of which is Шаблон:Math-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension Шаблон:Math or Шаблон:Math.
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; see screw axis. This gives rise to screw theory.
See also
- 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.
Notes
References
- ↑ Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478)
- 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 in Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478) and was reprinted in Theoria motus corporum rigidorum, ed. nova, 1790, pp. 449–460 (E478a) and later in his collected works Opera Omnia, Series 2, Volume 9, pp. 84–98.
- Шаблон:Cite journal
External links
- Euler's original treatise in The Euler Archive: entry on E478, 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)
