ロドリゲスの回転公式
s = r + ( sin  θ ) ( n × r ) + ( 1 − cos  θ ) { n × ( n × r ) } = r + ( sin  θ ) K ( n ) r + ( 1 − cos  θ ) K 2 ( n ) r = { E + ( sin  θ ) K ( n ) + ( 1 − cos  θ ) K 2 ( n ) } r {\displaystyle {\begin{aligned}{\boldsymbol {s}}&={\boldsymbol {r}}+(\sin \theta )({\boldsymbol {n}}\times {\boldsymbol {r}})+(1-\cos \theta )\{{\boldsymbol {n}}\times ({\boldsymbol {n}}\times {\boldsymbol {r}})\}\\&={\boldsymbol {r}}+(\sin \theta )K({\boldsymbol {n}}){\boldsymbol {r}}+(1-\cos \theta )K^{2}({\boldsymbol {n}}){\boldsymbol {r}}\\&=\{E+(\sin \theta )K({\boldsymbol {n}})+(1-\cos \theta )K^{2}({\boldsymbol {n}})\}{\boldsymbol {r}}\end{aligned}}}
R 上の回転軸を表す単位ベクトル n = [nx ny nz], 右手の法則に基づく回転角度 θ に対して、ロドリゲスの回転公式は次の様に与えられる。
R n ( θ ) = e θ K ( n ) = E + ( sin  θ ) K ( n ) + ( 1 − cos  θ ) K 2 ( n ) = [ n x 2 ( 1 − cos  θ ) + cos  θ n x n y ( 1 − cos  θ ) − n z sin  θ n z n x ( 1 − cos  θ ) + n y sin  θ n x n y ( 1 − cos  θ ) + n z sin  θ n y 2 ( 1 − cos  θ ) + cos  θ n y n z ( 1 − cos  θ ) − n x sin  θ n z n x ( 1 − cos  θ ) − n y sin  θ n y n z ( 1 − cos  θ ) + n x sin  θ n z 2 ( 1 − cos  θ ) + cos  θ ] {\displaystyle {\begin{aligned}R_{\boldsymbol {n}}(\theta )&=e^{\theta K({\boldsymbol {n}})}\\&=E+(\sin \theta )K({\boldsymbol {n}})+(1-\cos \theta )K^{2}({\boldsymbol {n}})\\&={\begin{bmatrix}n_{x}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{x}n_{y}\left(1-\cos \theta \right)-n_{z}\sin \theta &n_{z}n_{x}\left(1-\cos \theta \right)+n_{y}\sin \theta \\n_{x}n_{y}\left(1-\cos \theta \right)+n_{z}\sin \theta &n_{y}^{2}\left(1-\cos \theta \right)+\cos \theta &n_{y}n_{z}\left(1-\cos \theta \right)-n_{x}\sin \theta \\n_{z}n_{x}\left(1-\cos \theta \right)-n_{y}\sin \theta &n_{y}n_{z}\left(1-\cos \theta \right)+n_{x}\sin \theta &n_{z}^{2}\left(1-\cos \theta \right)+\cos \theta \\\end{bmatrix}}\end{aligned}}}