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Then rotate the given axis and the point such that the axis is aligned with one of the two coordinate axes for that particular coordinate plane (x, y or z) Use one of the fundamental rotation matrices to rotate the point depending on the coordinate axis with which the rotation axis is aligned.
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
The rotation has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω 1 and ω 2 then all points not in the planes rotate through an angle between ω 1 and ω 2. Rotations in four dimensions about a fixed point have six degrees of freedom.
The engineering and robotics communities typically use 3-1-3 Euler angles. Notice that after composing the independent rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body.
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), producing the rotated vector . Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k,
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...