Rotation formalisms in three dimensions

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, there exist various rotation formalisms to express a <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> in three <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimensions</a> as a mathematical <a href="/facts/Transformation_(geometry)/qRjrasJf">transformation</a>. In physics, this concept is applied to <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a> where rotational (or angular) <a href="/facts/Kinematics/o7mSyeJA">kinematics</a> is the science of <a href="/facts/Quantity/E2jCKetq">quantitative</a> description of a purely rotational <a href="/facts/Motion_(physics)/dCnP7MlB">motion</a>. The <a href="/facts/Orientation_(geometry)/Tj24mTJ8">orientation</a> of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.
According to <a href="/facts/Euler%2527s_rotation_theorem/t0moTljA">Euler's rotation theorem</a>, the rotation of a <a href="/facts/Rigid_body/THezBlq2">rigid body</a> (or three-dimensional <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> with a fixed <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three <a href="/facts/Real_number/R02gw5Pb">real</a> parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three <a href="/facts/Degrees_of_freedom_(mechanics)/2BqcdnLA">degrees of freedom</a>.
An example where rotation representation is used is in <a href="/facts/Computer_vision/Tl2Yyk66">computer vision</a>, where an <a href="/facts/Automation/3YlF2K5q">automated</a> observer needs to track a target. Consider a rigid body, with three <a href="/facts/Orthonormal_basis/pzfKHlJG">orthogonal unit vectors</a> fixed to its body (representing the three axes of the object's local <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">coordinate system</a>). The basic problem is to specify the orientation of these three <a href="/facts/Unit_vector/GyaXneDn">unit vectors</a>, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.

Rotation formalisms in three dimensions open-in-new

Rotation formalisms in three dimensions