Скачать или смотреть Rotation representation (mathematics) | Wikipedia audio article

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https://en.wikipedia.org/wiki/Rotatio...


00:00:39 1 Rotations and motions
00:01:18 2 Formalism alternatives
00:01:38 2.1 Rotation matrix
00:03:55 2.2 Euler axis and angle (rotation vector)
00:04:34 2.3 Euler rotations
00:05:13 2.4 Quaternions
00:05:52 2.5 Rodrigues parameters and Gibbs representation
00:10:07 2.6 Cayley–Klein parameters
00:13:04 2.7 Higher-dimensional analogues
00:13:43 3 Conversion formulae between formalisms
00:14:03 3.1 Rotation matrix ↔ Euler angles
00:14:22 3.1.1 Rotation matrix → Euler angles (spaniz/i-ix/i-iz/i
00:15:01 3.1.2 Euler angles (spaniz/i-iy/i′-ix/i″
00:16:00 3.2 Rotation matrix ↔ Euler axis/angle
00:18:18 3.3 Rotation matrix ↔ quaternion
00:20:54 3.4 Euler angles ↔ quaternion
00:25:49 3.4.1 Euler angles (spaniz/i-ix/i-iz/i
00:26:08 3.4.2 Euler angles (spaniz/i-iy/i′-ix/i″
00:27:27 3.4.3 Quaternion → Euler angles (spaniz/i-ix/i-iz/i
00:28:45 3.4.4 Quaternion → Euler angles (spaniz/i-iy/i′-ix/i″
00:30:03 3.5 Euler axis–angle ↔ quaternion
00:31:22 4 Conversion formulae for derivatives
00:32:40 4.1 Rotation matrix ↔ angular velocities
00:33:00 4.2 Quaternion ↔ angular velocities
00:34:18 5 Rotors in a geometric algebra
00:35:17 6 See also
00:35:56 7 References
00:43:08 8 Further reading



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SUMMARY
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In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation 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 Euler's rotation theorem the rotation of a rigid body (or three-dimensional coordinate system with the fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real 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 degrees of freedom.
An example where rotation representation is used is in computer vision, where an automated observer needs to track a target. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.