Rodrigues' rotation formula

In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from so(3), the Lie algebra of SO(3), to SO(3) without actually computing the full matrix exponential. v r o t = v cos ⁡ θ + ( k × v ) sin ⁡ θ + k   ( k ⋅ v ) ( 1 − cos ⁡ θ ) . {displaystyle mathbf {v} _{mathrm {rot} }=mathbf {v} cos heta +(mathbf {k} imes mathbf {v} )sin heta +mathbf {k} ~(mathbf {k} cdot mathbf {v} )(1-cos heta ),.} R = I + ( sin ⁡ θ ) K + ( 1 − cos ⁡ θ ) K 2 {displaystyle mathbf {R} =mathbf {I} +(sin heta )mathbf {K} +(1-cos heta )mathbf {K} ^{2}} In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from so(3), the Lie algebra of SO(3), to SO(3) without actually computing the full matrix exponential. If v is a vector in ℝ3 and k is a unit vector describing an axis of rotation about which v rotates by an angle θ according to the right hand rule, the Rodrigues formula for the rotated vector vrot is An alternative statement is to write the axis vector as a cross product a × b of any two nonzero vectors a and b which define the plane of rotation, and the sense of the angle θ is measured away from a and towards b. Letting α denote the angle between these vectors, the two angles θ and α are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written This form may be more useful when two vectors defining a plane are involved. An example in physics is the Thomas precession which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane. Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ (right hand rule, anticlockwise in the figure). Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k, where the component parallel to k is called the vector projection of v on k, and the component perpendicular to k is called the vector rejection of v from k.