01
Finite rotations, 2D case
Consider a vector v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 v 0 v v v φ \varphi φ z z z v \htmlClass{fea-math-bold-italic}{\boldsymbol v} v
Figure 1. Finite rotation in the plane.We can write successively:
v 0 = { v c o s α v s i n α } \htmlClass{fea-math-bold-italic}{\boldsymbol v}_0=\begin{Bmatrix}v\mathop{\mathit{cos}}\alpha\\v\mathop{\mathit{sin}}\alpha\end{Bmatrix} v 0 = { v cos α v sin α } v = { v c o s ( α + φ ) v s i n ( α + φ ) } = { v ( c o s α c o s φ − s i n α s i n φ ) v ( c o s α s i n φ + s i n α c o s φ ) } = [ c o s φ − s i n φ s i n φ c o s φ ] ⏟ R { v c o s α v s i n α } ⏟ v 0 . \begin{aligned}\htmlClass{fea-math-bold-italic}{\boldsymbol v}&=\begin{Bmatrix}v\mathop{\mathit{cos}}(\alpha+\varphi)\\v\mathop{\mathit{sin}}(\alpha+\varphi)\end{Bmatrix}\\&=\begin{Bmatrix}v(\mathop{\mathit{cos}}\alpha\mathop{\mathit{cos}}\varphi-\mathop{\mathit{sin}}\alpha\mathop{\mathit{sin}}\varphi)\\v(\mathop{\mathit{cos}}\alpha\mathop{\mathit{sin}}\varphi+\mathop{\mathit{sin}}\alpha\mathop{\mathit{cos}}\varphi)\end{Bmatrix}\\&=\underbrace{\begin{bmatrix}\mathop{\mathit{cos}}\varphi&-\mathop{\mathit{sin}}\varphi\\\mathop{\mathit{sin}}\varphi&\mathop{\mathit{cos}}\varphi\end{bmatrix}}_{\htmlClass{fea-math-bold-italic}{\boldsymbol R}}\underbrace{\begin{Bmatrix}v\mathop{\mathit{cos}}\alpha\\v\mathop{\mathit{sin}}\alpha\end{Bmatrix}}_{\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0}.\end{aligned} v = { v cos ( α + φ ) v sin ( α + φ ) } = { v ( cos α cos φ − sin α sin φ ) v ( cos α sin φ + sin α cos φ ) } = R [ cos φ sin φ − sin φ cos φ ] v 0 { v cos α v sin α } . And finally:
v = R v 0 , R = [ c o s φ − s i n φ s i n φ c o s φ ] . \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol R}\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0,\qquad\htmlClass{fea-math-bold-italic}{\boldsymbol R}=\begin{bmatrix}\mathop{\mathit{cos}}\varphi&-\mathop{\mathit{sin}}\varphi\\\mathop{\mathit{sin}}\varphi&\mathop{\mathit{cos}}\varphi\end{bmatrix}. v = R v 0 , R = [ cos φ sin φ − sin φ cos φ ] . If the angle φ \varphi φ c o s φ ≈ 1 \mathop{\mathit{cos}}\varphi\approx1 cos φ ≈ 1 s i n φ ≈ φ \mathop{\mathit{sin}}\varphi\approx\varphi sin φ ≈ φ
R ≈ [ 1 − φ φ 1 ] . \htmlClass{fea-math-bold-italic}{\boldsymbol R}\approx\begin{bmatrix}1&-\varphi\\\varphi&1\end{bmatrix}. R ≈ [ 1 φ − φ 1 ] . 02
Infinitesimal rotations
A rotation is represented by a magnitude, the angle of rotation, and a direction, the axis of rotation. Let’s consider the magnitude φ \varphi φ n \htmlClass{fea-math-bold-italic}{\boldsymbol n} n
For an infinitesimal rotation, the rotation can be represented by a vector:
φ = φ n , o r φ = { φ x φ y φ z } . \htmlClass{fea-math-bold-italic}{\boldsymbol{\varphi}}=\varphi\htmlClass{fea-math-bold-italic}{\boldsymbol n},\qquad\mathit{or}\qquad\htmlClass{fea-math-bold-italic}{\boldsymbol{\varphi}}=\begin{Bmatrix}\varphi_x\\\varphi_y\\\varphi_z\end{Bmatrix}. φ = φ n , or φ = ⎩ ⎨ ⎧ φ x φ y φ z ⎭ ⎬ ⎫ . And:
φ = ∥ φ ∥ = φ x 2 + φ y 2 + φ z 2 . \varphi=\lVert\htmlClass{fea-math-bold-italic}{\boldsymbol{\varphi}}\rVert=\sqrt{\varphi_x^2+\varphi_y^2+\varphi_z^2}. φ = ∥ φ ∥ = φ x 2 + φ y 2 + φ z 2 . The components of rotation, φ x \varphi_x φ x φ y \varphi_y φ y φ z \varphi_z φ z
R = R z R y R x = [ 1 − φ z 0 φ z 1 0 0 0 1 ] [ 1 0 φ y 0 1 0 − φ y 0 1 ] [ 1 0 0 0 1 − φ x 0 φ x 1 ] ⟶ R = [ 1 − φ z φ y φ z 1 − φ x − φ y φ x 1 ] . \begin{aligned}\htmlClass{fea-math-bold-italic}{\boldsymbol R}&=\htmlClass{fea-math-bold-italic}{\boldsymbol R}_z\htmlClass{fea-math-bold-italic}{\boldsymbol R}_y\htmlClass{fea-math-bold-italic}{\boldsymbol R}_x\\&=\begin{bmatrix}1&-\varphi_z&0\\\varphi_z&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}1&0&\varphi_y\\0&1&0\\-\varphi_y&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\0&1&-\varphi_x\\0&\varphi_x&1\end{bmatrix}\\&\longrightarrow\quad\htmlClass{fea-math-bold-italic}{\boldsymbol R}=\begin{bmatrix}1&-\varphi_z&\varphi_y\\\varphi_z&1&-\varphi_x\\-\varphi_y&\varphi_x&1\end{bmatrix}.\end{aligned} R = R z R y R x = 1 φ z 0 − φ z 1 0 0 0 1 1 0 − φ y 0 1 0 φ y 0 1 1 0 0 0 1 φ x 0 − φ x 1 ⟶ R = 1 φ z − φ y − φ z 1 φ x φ y − φ x 1 . The terms of the second and third degrees were neglected, and it was assumed that:
s i n φ ≃ φ , c o s φ ≃ 1. \mathop{\mathit{sin}}\varphi\simeq\varphi,\qquad\mathop{\mathit{cos}}\varphi\simeq1. sin φ ≃ φ , cos φ ≃ 1. In fact, the rotation φ \htmlClass{fea-math-bold-italic}{\boldsymbol{\varphi}} φ
φ = { φ x φ y φ z } = { φ x 0 0 } + { 0 φ y 0 } + { 0 0 φ z } . \htmlClass{fea-math-bold-italic}{\boldsymbol{\varphi}}=\begin{Bmatrix}\varphi_x\\\varphi_y\\\varphi_z\end{Bmatrix}=\begin{Bmatrix}\varphi_x\\0\\0\end{Bmatrix}+\begin{Bmatrix}0\\\varphi_y\\0\end{Bmatrix}+\begin{Bmatrix}0\\0\\\varphi_z\end{Bmatrix}. φ = ⎩ ⎨ ⎧ φ x φ y φ z ⎭ ⎬ ⎫ = ⎩ ⎨ ⎧ φ x 0 0 ⎭ ⎬ ⎫ + ⎩ ⎨ ⎧ 0 φ y 0 ⎭ ⎬ ⎫ + ⎩ ⎨ ⎧ 0 0 φ z ⎭ ⎬ ⎫ . Composition of small (infinitesimal) rotations is commutative!
03
Finite rotations
Let’s find the rotation matrix R \htmlClass{fea-math-bold-italic}{\boldsymbol R} R
The problem: consider a vector v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 v 0 φ \varphi φ n \htmlClass{fea-math-bold-italic}{\boldsymbol n} n v \htmlClass{fea-math-bold-italic}{\boldsymbol v} v R \htmlClass{fea-math-bold-italic}{\boldsymbol R} R [1] , [2] , [3] , [4] :
v = R ( φ ) v 0 . \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol R}(\varphi)\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0. v = R ( φ ) v 0 . This is called an active rotation : the vector is rotated while the reference frame remains fixed.
Figure 2. Rotation about the axis n \htmlClass{fea-math-bold-italic}{\boldsymbol n} n Vector sum:
v = v 0 + Δ v . \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0+\mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol v}. v = v 0 + Δ v . Δ v = P D ‾ + D Q ‾ ⟶ v = v 0 + P D ‾ + D Q ‾ . \mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol v}=\overline{PD}+\overline{DQ}\quad\longrightarrow\quad\htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0+\overline{PD}+\overline{DQ}. Δ v = P D + D Q ⟶ v = v 0 + P D + D Q . Vector D Q ‾ \overline{DQ} D Q n × v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 n × v 0 D Q ‾ \overline{DQ} D Q ( n , v 0 ) (\htmlClass{fea-math-bold-italic}{\boldsymbol n},\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0) ( n , v 0 )
The modulus of n × v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 n × v 0
∥ n × v 0 ∥ = ∥ v ∥ s i n α ⟶ ∥ n × v 0 ∥ = a . \lVert\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\rVert=\lVert\htmlClass{fea-math-bold-italic}{\boldsymbol v}\rVert\mathop{\mathit{sin}}\alpha\quad\longrightarrow\quad\lVert\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\rVert=a. ∥ n × v 0 ∥ = ∥ v ∥ sin α ⟶ ∥ n × v 0 ∥ = a . As:
D Q = a s i n φ , DQ=a\mathop{\mathit{sin}}\varphi, D Q = a sin φ , we get:
D Q ‾ = ( n × v 0 ) s i n φ . \overline{DQ}=(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\mathop{\mathit{sin}}\varphi. D Q = ( n × v 0 ) sin φ . Vector P D ‾ \overline{PD} P D
P D = a − a c o s φ = 2 a s i n 2 φ 2 . PD=a-a\mathop{\mathit{cos}}\varphi=2a\mathop{\mathit{sin}}^2\frac{\varphi}{2}. P D = a − a cos φ = 2 a sin 2 2 φ . Obviously, n \htmlClass{fea-math-bold-italic}{\boldsymbol n} n n × v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 n × v 0 n × ( n × v 0 ) \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0) n × ( n × v 0 ) P D ‾ \overline{PD} P D
∥ n × ( n × v 0 ) ∥ = ∥ n × v 0 ∥ = a , \lVert\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\rVert=\lVert\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\rVert=a, ∥ n × ( n × v 0 )∥ = ∥ n × v 0 ∥ = a , therefore:
P D ‾ = 2 ( n × ( n × v 0 ) ) s i n 2 φ 2 . \overline{PD}=2\bigl(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\bigr)\mathop{\mathit{sin}}^2\frac{\varphi}{2}. P D = 2 ( n × ( n × v 0 ) ) sin 2 2 φ . The relation between v \htmlClass{fea-math-bold-italic}{\boldsymbol v} v v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 v 0
v = v 0 + ( n × v 0 ) s i n φ + 2 ( n × ( n × v 0 ) ) s i n 2 φ 2 , \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0+(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\mathop{\mathit{sin}}\varphi+2\bigl(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\bigr)\mathop{\mathit{sin}}^2\frac{\varphi}{2}, v = v 0 + ( n × v 0 ) sin φ + 2 ( n × ( n × v 0 ) ) sin 2 2 φ , or:
v = v 0 + ( n × v 0 ) s i n φ + ( n × ( n × v 0 ) ) ( 1 − c o s φ ) . (*) \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0+(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\mathop{\mathit{sin}}\varphi+\bigl(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\bigr)(1-\mathop{\mathit{cos}}\varphi).\tag{*} v = v 0 + ( n × v 0 ) sin φ + ( n × ( n × v 0 ) ) ( 1 − cos φ ) . ( * ) To convert this equation into matrix form, we introduce the notations:
n = { n x n y n z } , n x 2 + n y 2 + n z 2 = 1. \htmlClass{fea-math-bold-italic}{\boldsymbol n}=\begin{Bmatrix}n_x\\n_y\\n_z\end{Bmatrix},\qquad n_x^2+n_y^2+n_z^2=1. n = ⎩ ⎨ ⎧ n x n y n z ⎭ ⎬ ⎫ , n x 2 + n y 2 + n z 2 = 1. n × v 0 ⟶ [ 0 − n z n y n z 0 − n x − n y n x 0 ] { v 0 x v 0 y v 0 z } . \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\ \longrightarrow\ \begin{bmatrix}0&-n_z&n_y\\n_z&0&-n_x\\-n_y&n_x&0\end{bmatrix}\begin{Bmatrix}v_{0x}\\v_{0y}\\v_{0z}\end{Bmatrix}. n × v 0 ⟶ 0 n z − n y − n z 0 n x n y − n x 0 ⎩ ⎨ ⎧ v 0 x v 0 y v 0 z ⎭ ⎬ ⎫ . Or:
n × v 0 ⟶ n ~ v 0 , \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\ \longrightarrow\ \widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0, n × v 0 ⟶ n v 0 , with:
n ~ = [ 0 − n z n y n z 0 − n x − n y n x 0 ] . \widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}=\begin{bmatrix}0&-n_z&n_y\\n_z&0&-n_x\\-n_y&n_x&0\end{bmatrix}. n = 0 n z − n y − n z 0 n x n y − n x 0 . n × ( n × v 0 ) ⟶ n ~ 2 v 0 , \htmlClass{fea-math-bold-italic}{\boldsymbol n}\times(\htmlClass{fea-math-bold-italic}{\boldsymbol n}\times\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0)\ \longrightarrow\ \widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}^{\,2}\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0, n × ( n × v 0 ) ⟶ n 2 v 0 , where:
n ~ 2 = n ~ n ~ = [ − ( n y 2 + n z 2 ) n x n y n z n x n x n y − ( n z 2 + n x 2 ) n y n z n z n x n y n z − ( n x 2 + n y 2 ) ] . \widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}^{\,2}=\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}=\begin{bmatrix}-(n_y^2+n_z^2)&n_xn_y&n_zn_x\\n_xn_y&-(n_z^2+n_x^2)&n_yn_z\\n_zn_x&n_yn_z&-(n_x^2+n_y^2)\end{bmatrix}. n 2 = n n = − ( n y 2 + n z 2 ) n x n y n z n x n x n y − ( n z 2 + n x 2 ) n y n z n z n x n y n z − ( n x 2 + n y 2 ) . This gives:
v = ( E + s i n φ n ~ + ( 1 − c o s φ ) n ~ 2 ) v 0 ⟶ v = R v 0 . \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\bigl(\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\mathop{\mathit{sin}}\varphi\,\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}+(1-\mathop{\mathit{cos}}\varphi)\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}^{\,2}\bigr)\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0\quad\longrightarrow\quad\htmlClass{fea-math-bold-italic}{\boldsymbol v}=\htmlClass{fea-math-bold-italic}{\boldsymbol R}\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0. v = ( E + sin φ n + ( 1 − cos φ ) n 2 ) v 0 ⟶ v = R v 0 . E \htmlClass{fea-math-bold-italic}{\boldsymbol E} E
R = E + s i n φ n ~ + ( 1 − c o s φ ) n ~ 2 . (**) \htmlClass{fea-math-bold-italic}{\boldsymbol R}=\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\mathop{\mathit{sin}}\varphi\,\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}+(1-\mathop{\mathit{cos}}\varphi)\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}^{\,2}.\tag{**} R = E + sin φ n + ( 1 − cos φ ) n 2 . ( ** ) Finally:
R = [ c o s φ + n x 2 ( 1 − c o s φ ) n x n y ( 1 − c o s φ ) − n z s i n φ n z n x ( 1 − c o s φ ) + n y s i n φ n x n y ( 1 − c o s φ ) + n z s i n φ c o s φ + n y 2 ( 1 − c o s φ ) n y n z ( 1 − c o s φ ) − n x s i n φ n z n x ( 1 − c o s φ ) − n y s i n φ n y n z ( 1 − c o s φ ) + n x s i n φ c o s φ + n z 2 ( 1 − c o s φ ) ] . \htmlClass{fea-math-bold-italic}{\boldsymbol R}=\begin{bmatrix}\mathop{\mathit{cos}}\varphi+n_x^2(1-\mathop{\mathit{cos}}\varphi)&n_xn_y(1-\mathop{\mathit{cos}}\varphi)-n_z\mathop{\mathit{sin}}\varphi&n_zn_x(1-\mathop{\mathit{cos}}\varphi)+n_y\mathop{\mathit{sin}}\varphi\\n_xn_y(1-\mathop{\mathit{cos}}\varphi)+n_z\mathop{\mathit{sin}}\varphi&\mathop{\mathit{cos}}\varphi+n_y^2(1-\mathop{\mathit{cos}}\varphi)&n_yn_z(1-\mathop{\mathit{cos}}\varphi)-n_x\mathop{\mathit{sin}}\varphi\\n_zn_x(1-\mathop{\mathit{cos}}\varphi)-n_y\mathop{\mathit{sin}}\varphi&n_yn_z(1-\mathop{\mathit{cos}}\varphi)+n_x\mathop{\mathit{sin}}\varphi&\mathop{\mathit{cos}}\varphi+n_z^2(1-\mathop{\mathit{cos}}\varphi)\end{bmatrix}. R = cos φ + n x 2 ( 1 − cos φ ) n x n y ( 1 − cos φ ) + n z sin φ n z n x ( 1 − cos φ ) − n y sin φ n x n y ( 1 − cos φ ) − n z sin φ cos φ + n y 2 ( 1 − cos φ ) n y n z ( 1 − cos φ ) + n x sin φ n z n x ( 1 − cos φ ) + n y sin φ n y n z ( 1 − cos φ ) − n x sin φ cos φ + n z 2 ( 1 − cos φ ) . 04
Another form of the rotation matrix
Now we introduce this vector:
ρ = φ n = { φ x φ y φ z } , \htmlClass{fea-math-bold-italic}{\boldsymbol{\rho}}=\varphi\htmlClass{fea-math-bold-italic}{\boldsymbol n}=\begin{Bmatrix}\varphi_x\\\varphi_y\\\varphi_z\end{Bmatrix}, ρ = φ n = ⎩ ⎨ ⎧ φ x φ y φ z ⎭ ⎬ ⎫ , and:
φ = φ x 2 + φ y 2 + φ z 2 . \varphi=\sqrt{\varphi_x^2+\varphi_y^2+\varphi_z^2}. φ = φ x 2 + φ y 2 + φ z 2 . ρ \htmlClass{fea-math-bold-italic}{\boldsymbol{\rho}} ρ v \htmlClass{fea-math-bold-italic}{\boldsymbol v} v v 0 \htmlClass{fea-math-bold-italic}{\boldsymbol v}_0 v 0
v = ( E + s i n φ φ ρ ~ + 1 − c o s φ φ 2 ρ ~ 2 ) v 0 . \htmlClass{fea-math-bold-italic}{\boldsymbol v}=\left(\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\frac{\mathop{\mathit{sin}}\varphi}{\varphi}\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol{\rho}}}+\frac{1-\mathop{\mathit{cos}}\varphi}{\varphi^2}\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol{\rho}}}^{\,2}\right)\htmlClass{fea-math-bold-italic}{\boldsymbol v}_0. v = ( E + φ sin φ ρ + φ 2 1 − cos φ ρ 2 ) v 0 . And finally:
R = E + s i n φ φ [ 0 − φ z φ y φ z 0 − φ x − φ y φ x 0 ] + 1 − c o s φ φ 2 [ − ( φ y 2 + φ z 2 ) φ x φ y φ z φ x φ x φ y − ( φ z 2 + φ x 2 ) φ y φ z φ z φ x φ y φ z − ( φ x 2 + φ y 2 ) ] . \begin{aligned}\htmlClass{fea-math-bold-italic}{\boldsymbol R}={}&\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\frac{\mathop{\mathit{sin}}\varphi}{\varphi}\begin{bmatrix}0&-\varphi_z&\varphi_y\\\varphi_z&0&-\varphi_x\\-\varphi_y&\varphi_x&0\end{bmatrix}\\&+\frac{1-\mathop{\mathit{cos}}\varphi}{\varphi^2}\begin{bmatrix}-(\varphi_y^2+\varphi_z^2)&\varphi_x\varphi_y&\varphi_z\varphi_x\\\varphi_x\varphi_y&-(\varphi_z^2+\varphi_x^2)&\varphi_y\varphi_z\\\varphi_z\varphi_x&\varphi_y\varphi_z&-(\varphi_x^2+\varphi_y^2)\end{bmatrix}.\end{aligned} R = E + φ sin φ 0 φ z − φ y − φ z 0 φ x φ y − φ x 0 + φ 2 1 − cos φ − ( φ y 2 + φ z 2 ) φ x φ y φ z φ x φ x φ y − ( φ z 2 + φ x 2 ) φ y φ z φ z φ x φ y φ z − ( φ x 2 + φ y 2 ) . Or:
R = E + s i n φ [ 0 − n z n y n z 0 − n x − n y n x 0 ] + ( 1 − c o s φ ) [ − ( n y 2 + n z 2 ) n x n y n z n x n x n y − ( n z 2 + n x 2 ) n y n z n z n x n y n z − ( n x 2 + n y 2 ) ] . \begin{aligned}\htmlClass{fea-math-bold-italic}{\boldsymbol R}={}&\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\mathop{\mathit{sin}}\varphi\begin{bmatrix}0&-n_z&n_y\\n_z&0&-n_x\\-n_y&n_x&0\end{bmatrix}\\&+(1-\mathop{\mathit{cos}}\varphi)\begin{bmatrix}-(n_y^2+n_z^2)&n_xn_y&n_zn_x\\n_xn_y&-(n_z^2+n_x^2)&n_yn_z\\n_zn_x&n_yn_z&-(n_x^2+n_y^2)\end{bmatrix}.\end{aligned} R = E + sin φ 0 n z − n y − n z 0 n x n y − n x 0 + ( 1 − cos φ ) − ( n y 2 + n z 2 ) n x n y n z n x n x n y − ( n z 2 + n x 2 ) n y n z n z n x n y n z − ( n x 2 + n y 2 ) . 05
Small rotation angles
If the angle of rotation φ \varphi φ
s i n φ φ ≈ 1. \frac{\mathop{\mathit{sin}}\varphi}{\varphi}\approx1. φ sin φ ≈ 1. Neglecting second-degree terms:
R = [ 1 − φ z φ y φ z 1 − φ x − φ y φ x 1 ] . \htmlClass{fea-math-bold-italic}{\boldsymbol R}=\begin{bmatrix}1&-\varphi_z&\varphi_y\\\varphi_z&1&-\varphi_x\\-\varphi_y&\varphi_x&1\end{bmatrix}. R = 1 φ z − φ y − φ z 1 φ x φ y − φ x 1 . If the rotation angle is small and second-order terms are retained, the rotation angles are assumed to be small, but not infinitesimal; therefore, the second-order terms cannot be neglected. Since the rotation matrix contains quadratic terms in the rotation components, the resulting problem is geometrically nonlinear.
s i n φ φ ≈ 1 , 1 − c o s φ φ 2 ≈ 1 2 . \frac{\mathop{\mathit{sin}}\varphi}{\varphi}\approx1,\qquad\frac{1-\mathop{\mathit{cos}}\varphi}{\varphi^2}\approx\frac12. φ sin φ ≈ 1 , φ 2 1 − cos φ ≈ 2 1 . And:
R = [ 1 − 1 2 ( φ y 2 + φ z 2 ) − φ z + 1 2 φ x φ y φ y + 1 2 φ z φ x φ z + 1 2 φ x φ y 1 − 1 2 ( φ z 2 + φ x 2 ) − φ x + 1 2 φ y φ z − φ y + 1 2 φ z φ x φ x + 1 2 φ y φ z 1 − 1 2 ( φ x 2 + φ y 2 ) ] . \htmlClass{fea-math-bold-italic}{\boldsymbol R}=\begin{bmatrix}1-\frac12(\varphi_y^2+\varphi_z^2)&-\varphi_z+\frac12\varphi_x\varphi_y&\varphi_y+\frac12\varphi_z\varphi_x\\\varphi_z+\frac12\varphi_x\varphi_y&1-\frac12(\varphi_z^2+\varphi_x^2)&-\varphi_x+\frac12\varphi_y\varphi_z\\-\varphi_y+\frac12\varphi_z\varphi_x&\varphi_x+\frac12\varphi_y\varphi_z&1-\frac12(\varphi_x^2+\varphi_y^2)\end{bmatrix}. R = 1 − 2 1 ( φ y 2 + φ z 2 ) φ z + 2 1 φ x φ y − φ y + 2 1 φ z φ x − φ z + 2 1 φ x φ y 1 − 2 1 ( φ z 2 + φ x 2 ) φ x + 2 1 φ y φ z φ y + 2 1 φ z φ x − φ x + 2 1 φ y φ z 1 − 2 1 ( φ x 2 + φ y 2 ) . 06
Example
The 3D part in the figure below must be divided into finite elements. The part can be divided into 12 identical slices. Only one slice will be divided into finite elements, and then, by rotating, reflecting and copying this slice, the entire FE mesh will be obtained.
Figure 3. The 3D part and its finite element mesh.Figure 4. One slice and the construction of the complete mesh by rotation and reflection.One “slice” is depicted in the figure above, on the left. Nodal coordinates are known: x \htmlClass{fea-math-bold-italic}{\boldsymbol x} x n n n x , y , z x,y,z x , y , z n n n x \htmlClass{fea-math-bold-italic}{\boldsymbol x} x
Firstly, this slice is rotated around the axis computed using the normalized cross product:
n 1 = n 0 × n ∥ n 0 × n ∥ , \htmlClass{fea-math-bold-italic}{\boldsymbol n}_1=\frac{\htmlClass{fea-math-bold-italic}{\boldsymbol n}_0\times\htmlClass{fea-math-bold-italic}{\boldsymbol n}}{\lVert\htmlClass{fea-math-bold-italic}{\boldsymbol n}_0\times\htmlClass{fea-math-bold-italic}{\boldsymbol n}\rVert}, n 1 = ∥ n 0 × n ∥ n 0 × n , with the angle computed using a dot product:
φ = a r c c o s ( n 0 ⋅ n ) . \varphi=\mathop{\mathit{arccos}}(\htmlClass{fea-math-bold-italic}{\boldsymbol n}_0\cdot\htmlClass{fea-math-bold-italic}{\boldsymbol n}). φ = arccos ( n 0 ⋅ n ) . This gives the rotation matrix R 1 \htmlClass{fea-math-bold-italic}{\boldsymbol R}_1 R 1
R 1 = E + s i n φ n ~ 1 + ( 1 − c o s φ ) n ~ 1 2 . \htmlClass{fea-math-bold-italic}{\boldsymbol R}_1=\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\mathop{\mathit{sin}}\varphi\,\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}_1+(1-\mathop{\mathit{cos}}\varphi)\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}_1^{\,2}. R 1 = E + sin φ n 1 + ( 1 − cos φ ) n 1 2 . The new coordinates are:
x n = x R 1 T + Δ x , \htmlClass{fea-math-bold-italic}{\boldsymbol x}_n=\htmlClass{fea-math-bold-italic}{\boldsymbol x}\htmlClass{fea-math-bold-italic}{\boldsymbol R}_1^T+\mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol x}, x n = x R 1 T + Δ x , where Δ x \mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol x} Δ x [ 20 10 5 ] \begin{bmatrix}20&10&5\end{bmatrix} [ 20 10 5 ]
The remaining slices are obtained with the rotation matrix:
R t = R 2 R 1 ⟶ x n = x R t T + Δ x , \htmlClass{fea-math-bold-italic}{\boldsymbol R}_t=\htmlClass{fea-math-bold-italic}{\boldsymbol R}_2\htmlClass{fea-math-bold-italic}{\boldsymbol R}_1\quad\longrightarrow\quad\htmlClass{fea-math-bold-italic}{\boldsymbol x}_n=\htmlClass{fea-math-bold-italic}{\boldsymbol x}\htmlClass{fea-math-bold-italic}{\boldsymbol R}_t^T+\mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol x}, R t = R 2 R 1 ⟶ x n = x R t T + Δ x , where R 2 \htmlClass{fea-math-bold-italic}{\boldsymbol R}_2 R 2 n \htmlClass{fea-math-bold-italic}{\boldsymbol n} n
R 2 = E + s i n φ 0 n ~ + ( 1 − c o s φ 0 ) n ~ 2 . \htmlClass{fea-math-bold-italic}{\boldsymbol R}_2=\htmlClass{fea-math-bold-italic}{\boldsymbol E}+\mathop{\mathit{sin}}\varphi_0\,\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}+(1-\mathop{\mathit{cos}}\varphi_0)\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol n}}^{\,2}. R 2 = E + sin φ 0 n + ( 1 − cos φ 0 ) n 2 . Angle φ 0 \varphi_0 φ 0 π / 3 \pi/3 π /3 π / 3 \pi/3 π /3
R t = R 2 R 1 R a ⟶ x n = x R t T + Δ x , \htmlClass{fea-math-bold-italic}{\boldsymbol R}_t=\htmlClass{fea-math-bold-italic}{\boldsymbol R}_2\htmlClass{fea-math-bold-italic}{\boldsymbol R}_1\htmlClass{fea-math-bold-italic}{\boldsymbol R}_a\quad\longrightarrow\quad\htmlClass{fea-math-bold-italic}{\boldsymbol x}_n=\htmlClass{fea-math-bold-italic}{\boldsymbol x}\htmlClass{fea-math-bold-italic}{\boldsymbol R}_t^T+\mathit{\Delta}\htmlClass{fea-math-bold-italic}{\boldsymbol x}, R t = R 2 R 1 R a ⟶ x n = x R t T + Δ x , where:
R a = [ 1 0 0 0 1 0 0 0 − 1 ] . \htmlClass{fea-math-bold-italic}{\boldsymbol R}_a=\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}. R a = 1 0 0 0 1 0 0 0 − 1 . In fact, the matrix R a \htmlClass{fea-math-bold-italic}{\boldsymbol R}_a R a x y xy x y
Figure 5. Interactive 3D view of the example. Drag to rotate; switch Zoom ON to use wheel or pinch zoom. Reset view restores the initial camera position.A simple MATLAB program is available for download below.
MATLAB Finite rotations in three dimensions Four MATLAB files and README Download ZIP ↓ 07
References
Rotation Formula, Wolfram MathWorld. Finite rotations, Math Wiki. Carlos Felippa, NFEM, Nonlinear Finite Element Methods (ASEN 5107) , Appendix R. Enrico C. Da Lozzo, Geometrically exact three-dimensional beam theory: modeling and FEM implementation for statics and dynamics analysis , Istituto Universitario di Studi Superiori di Pavia, Master’s Degree in Earthquake Engineering, 2010. PREVIOUS CHAPTER ← 20. Buckling, plane stress state NEXT SECTION 21.2. Tait-Bryan angles → 
