Euler's rotation theorem

Last updated
A rotation represented by an Euler axis and angle. Euler AxisAngle.png
A rotation represented by an Euler axis and angle.

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 .

Contents

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 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.

Figure 1: Blue great circle on sphere transforms into red great circle when rotated about diameter through O. Euler Rotation 1.JPG
Figure 1: Blue great circle on sphere transforms into red great circle when rotated about diameter through O.

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 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.)

Figure 2: Arcs connecting preimage a and image a of A with bisector AO of the angle at A. Euler Rotation 2.JPG
Figure 2: Arcs connecting preimage α and image a of A with bisector AO of the angle at A.

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.

Figure 3: O goes to O', but O' must coincide with O. Euler Rotation 3.JPG
Figure 3: O goes to O′, but O′ must coincide with O.

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 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 α, and connect these two points to A with arcs αA and Aa. These arcs have the same length. Construct the great circle that bisects αAa 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 αAa. (That is, the yellow region in Figure 3.) Then since αA = Aa and O is on the bisector of αAa, we also have αO = aO.

Proof of its invariance under the transformation

Now let us suppose that O′ is the image of O. Then we know αAO = ∠AaO′ and orientation is preserved, [lower-alpha 1] so O′ must be interior to αAa. Now AO is transformed to aO′, so AO = aO′. Since AO is also the same length as aO, then aO = aO′ and AaO = ∠aAO. But αAO = ∠aAO, so αAO = ∠AaO and AaO = ∠AaO′. 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.

Final notes about the construction

Euler's original drawing where ABC is the blue circle and ACc is the red circle Eulerrotation.svg
Euler's original drawing where ABC is the blue circle and ACc is the red circle

Euler also points out that O can be found by intersecting the perpendicular bisector of Aa 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 center C of the sphere), and
  • the symmetry plane of the arc Aa (which also passes through C).
Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for.
Proof. Let us call O either of the endpoints (there are two) of this diameter over the sphere surface. Since αA is mapped on Aa and the triangles have the same angles, it follows that the triangle OαA is transported onto the triangle OAa. Therefore the point O 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 α, A, a lie. The rotation axis is obviously orthogonal to this plane, and passes through the center C 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 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R, there is a nonzero vector n for which Rn = n; this is exactly the claim that n is an eigenvector of R associated with the eigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue of R; the rotation axis of R will be the line μn, where n is the eigenvector with eigenvalue 1.

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

where I is the 3 × 3 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,

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 R has at least one invariant vector n, i.e., Rn = n. Because this requires that (RI)n = 0, we see that the vector n must be an eigenvector of the matrix R with eigenvalue λ = 1. Thus, this is equivalent to showing that det(RI) = 0.

Use the two relations

for any 3 × 3 matrix A and

(since det(R) = 1) to compute

This shows that λ = 1 is a root (solution) of the characteristic equation, that is,

In other words, the matrix RI is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say n, for which

The line μn for real μ is invariant under R, i.e., μn 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:

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 R has more than one invariant vector then φ = 0 and R = I. Any vector is an invariant vector of I.

Excursion into matrix theory

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

An m × m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if AA = AA. [lower-alpha 2] This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

and U is unitary, that is,

The eigenvalues α1, ..., αm are roots of the characteristic equation. If the matrix A happens to be unitary (and note that unitary matrices are normal), then

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

Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an mth order polynomial in λ) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if α is a root then so is α. 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 R. It follows from its realness and orthogonality that we can find a U such that:

If a matrix U can be found that gives the above form, and there is only one purely real component and it is −1, then we define 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 3 × 3 matrix U will then be equal to the invariant vector n. Writing u1 and u2 for the first two columns of U, this equation gives

If u1 has eigenvalue 1, then φ = 0 and u2 has also eigenvalue 1, which implies that in that case R = I. In general, however, as implies that also holds, so can be chosen for . Similarly, can result in a with real entries only, for a proper rotation matrix . Finally, the matrix equation is transformed by means of a unitary matrix,

which gives

The columns of U are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that is the complex conjugate of and that is a vector with real-valued components. The third column is still n, the other two columns of U are perpendicular to n. 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 R corresponding to a proper rotation is equivalent to a rotation over an angle φ around an axis n.

Equivalence classes

The trace (sum of diagonal elements) of the real rotation matrix given above is 1 + 2 cos φ. Since a trace is invariant under an orthogonal matrix similarity transformation,

it follows that all matrices that are equivalent to R 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 3 × 3 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 n is an eigenvector of R with eigenvalue 1, then An is also an eigenvector of ARAT, also with eigenvalue 1. Unless A = I, n and An are different.

Applications

Generators of rotations

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

A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/N where N is a large number, a rotation of θ about the axis may be represented as:

It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product Aθ is the "generator" of the particular rotation, being the vector (x,y,z) associated with the matrix A. 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 G. One starts with an arbitrary plane (in Euclidean space) defined by a pair of perpendicular unit vectors a and b. In this plane one can choose an arbitrary vector x with perpendicular y. One then solves for y in terms of x and substituting into an expression for a rotation in a plane yields the rotation matrix R which includes the generator G = baTabT.

To include vectors outside the plane in the rotation one needs to modify the above expression for R by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an exponential 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 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 2n or 2n + 1 is a composition of at most n 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 n linearly independent points, which span an n-dimensional body in dimension 2n or 2n + 1, is a single plane of rotation. To put it another way, if two rigid bodies, with identical geometry, share at least n points of 'identical' locations within themselves, the convex hull of which is n-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension 2n or 2n + 1.

A screw motion. Pure screw.svg
A screw motion.

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

Notes

  1. Orientation is preserved in the sense that if αA is rotated about A counterclockwise to align with OA, then Aa must be rotated about a counterclockwise to align with O′a. Likewise if the rotations are clockwise.
  2. The dagger symbol stands for complex conjugation followed by transposition. For real matrices complex conjugation does nothing and daggering a real matrix is the same as transposing it.

Related Research Articles

Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.

<span class="mw-page-title-main">Pauli matrices</span> Matrices important in quantum mechanics and the study of spin

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

<span class="mw-page-title-main">Angular velocity</span> Pseudovector representing an objects change in orientation with respect to time

In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

In mathematics, a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

<span class="mw-page-title-main">Euler equations (fluid dynamics)</span> Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.

An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation.

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

<span class="mw-page-title-main">Bloch sphere</span> Geometrical representation of the pure state space of a two-level quantum mechanical system

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

<span class="mw-page-title-main">Dual quaternion</span> Eight-dimensional algebra over the real numbers

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices, functions of matrices, and the eigenvalues of matrices.

The quaternion estimator algorithm (QUEST) is an algorithm designed to solve Wahba's problem, that consists of finding a rotation matrix between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a quadratic form, using the Cayley–Hamilton theorem and the Newton–Raphson method to efficiently solve the eigenvalue problem and construct a numerically stable representation of the solution.

References

  1. Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478)
This article incorporates material from the Citizendium article "Euler's theorem (rotation)", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.