Infinitesimal rotation matrix

An infinitesimal rotation matrix or differential rotation matrix has the form:

I+A\,d\theta ,

where $dθ$ is vanishingly small and $A \in so (n)$ , for instance with $A = L x$ ,

dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.

While the matrices in the Lie algebra are not themselves rotations, the skew-symmetric matrices are derivatives, proportional differences of rotations.

The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.^[1] It turns out that the order in which infinitesimal rotations are applied is irrelevant.

Discussion

An infinitesimal rotation matrix is a skew-symmetric matrix where:

As any rotation matrix has a single real eigenvalue, which is equal to +1, the corresponding eigenvector defines the rotation axis.
Its module defines an infinitesimal angular displacement.

The shape of the matrix is as follows: $A={\begin{pmatrix}1&-d\phi _{z}(t)&d\phi _{y}(t)\\d\phi _{z}(t)&1&-d\phi _{x}(t)\\-d\phi _{y}(t)&d\phi _{x}(t)&1\\\end{pmatrix}}$

Associated quantities

We can introduce here the associated infinitesimal angular displacement tensor or rotation generator:

d\Phi (t)={\begin{pmatrix}0&-d\phi _{z}(t)&d\phi _{y}(t)\\d\phi _{z}(t)&0&-d\phi _{x}(t)\\-d\phi _{y}(t)&d\phi _{x}(t)&0\\\end{pmatrix}}

Such that its associated rotation matrix is $A=I+d\Phi (t)$ . When it is divided by the time, this will yield the angular velocity vector.

Order of rotations

These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals .^[2] To understand what this means, consider

dA_{\mathbf {x} }={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.

First, test the orthogonality condition, $Q T Q = I$ . The product is

dA_{\mathbf {x} }^{\textsf {T}}\,dA_{\mathbf {x} }={\begin{bmatrix}1&0&0\\0&1+d\theta ^{2}&0\\0&0&1+d\theta ^{2}\end{bmatrix}},

differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.

Next, examine the square of the matrix,

dA_{\mathbf {x} }^{2}={\begin{bmatrix}1&0&0\\0&1-d\theta ^{2}&-2d\theta \\0&2\,d\theta &1-d\theta ^{2}\end{bmatrix}}.

Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,

dA_{\mathbf {y} }={\begin{bmatrix}1&0&d\phi \\0&1&0\\-d\phi &0&1\end{bmatrix}}.

Compare the products $dA x dA y$ to $dA y dA x$ ,

{\begin{aligned}dA_{\mathbf {x} }\,dA_{\mathbf {y} }&={\begin{bmatrix}1&0&d\phi \\d\theta \,d\phi &1&-d\theta \\-d\phi &d\theta &1\end{bmatrix}}\\dA_{\mathbf {y} }\,dA_{\mathbf {x} }&={\begin{bmatrix}1&d\theta \,d\phi &d\phi \\0&1&-d\theta \\-d\phi &d\theta &1\end{bmatrix}}.\\\end{aligned}}

Since $d\theta \,d\phi$ is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,

dA_{\mathbf {x} }\,dA_{\mathbf {y} }=dA_{\mathbf {y} }\,dA_{\mathbf {x} },

again to first order. In other words, the order in which infinitesimal rotations are applied is irrelevant.

This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals one finds a bona fide vector space. Technically, this dismissal of any second order terms amounts to Group contraction.

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:

\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 =I+A\,\Delta \theta .

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:

R=\left(I+{\frac {A\theta }{N}}\right)^{N}\approx e^{A\theta }.

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^[3] 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 = ba^T − ab^T.

{\begin{aligned}x&=a\cos \left(\alpha \right)+b\sin \left(\alpha \right)\\y&=-a\sin \left(\alpha \right)+b\cos \left(\alpha \right)\\\cos \left(\alpha \right)&=a^{\mathrm {T} }x\\\sin \left(\alpha \right)&=b^{\mathrm {T} }x\\y&=-ab^{\mathrm {T} }x+ba^{\mathrm {T} }x=\left(ba^{\mathrm {T} }-ab^{\mathrm {T} }\right)x\\\\x'&=x\cos \left(\beta \right)+y\sin \left(\beta \right)\\&=\left[I\cos \left(\beta \right)+\left(ba^{\mathrm {T} }-ab^{\mathrm {T} }\right)\sin \left(\beta \right)\right]x\\\\R&=I\cos \left(\beta \right)+\left(ba^{\mathrm {T} }-ab^{\mathrm {T} }\right)\sin \left(\beta \right)\\&=I\cos \left(\beta \right)+G\sin \left(\beta \right)\\\\G&=ba^{\mathrm {T} }-ab^{\mathrm {T} }\\\end{aligned}}

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.

{\begin{aligned}P_{ab}&=-G^{2}\\R&=I-P_{ab}+\left[I\cos \left(\beta \right)+G\sin \left(\beta \right)\right]P_{ab}=e^{G\beta }\\\end{aligned}}

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.

Exponential map

Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for $e A$ ^[4] For any skew-symmetric matrix $A$ , $exp(A)$ is always a rotation matrix.^[a]

An important practical example is the $3 \times 3$ case. In rotation group SO(3), it is shown that one can identify every $A \in so (3)$ with an Euler vector $ω = θ u$ , where $u = (x, y, z)$ is a unit magnitude vector.

By the properties of the identification $su (2) ≅ R 3$ , $u$ is in the null space of $A$ . Thus, $u$ is left invariant by $exp(A)$ and is hence a rotation axis.

Using Rodrigues' rotation formula on matrix form with $θ = .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}θ⁄2 + θ⁄2$ , together with standard double angle formulae one obtains,

{\begin{aligned}\exp(A)&{}=\exp(\theta ({\boldsymbol {u\cdot L}}))=\exp \left(\left[{\begin{smallmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{smallmatrix}}\right]\right)={\boldsymbol {I}}+2\cos {\frac {\theta }{2}}\sin {\frac {\theta }{2}}~{\boldsymbol {u\cdot L}}+2\sin ^{2}{\frac {\theta }{2}}~({\boldsymbol {u\cdot L}})^{2},\end{aligned}}

This is the matrix for a rotation around axis $u$ by the angle $θ$ in half-angle form. For full detail, see exponential map SO(3).

Notice that for infinitesimal angles second order terms can be ignored and remains $exp(A) = I + A$

Notes

^ Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order, $e^{2A}-{\frac {I+A}{I-A}}=-{\frac {2}{3}}A^{3}+\mathrm {O} (A^{4})~.$
Conversely, a skew-symmetric matrix $A$ specifying a rotation matrix through the Cayley map specifies the same rotation matrix through the map $exp(2 artanh A)$ .

References

^ (Goldstein, Poole & Safko 2002, §4.8)
^ (Goldstein, Poole & Safko 2002, §4.8)
^ in Euclidean space
^ (Wedderburn 1934, §8.02)

[5] Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order, $e^{2A}-{\frac {I+A}{I-A}}=-{\frac {2}{3}}A^{3}+\mathrm {O} (A^{4})~.$
Conversely, a skew-symmetric matrix $A$ specifying a rotation matrix through the Cayley map specifies the same rotation matrix through the map $exp(2 artanh A)$ .

[1] (Goldstein, Poole & Safko 2002, §4.8)

[2] (Goldstein, Poole & Safko 2002, §4.8)

[3] Euclidean space

[4] (Wedderburn 1934, §8.02)

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