Rules for geometry rotations3/28/2024 ![]() ![]() One can also understand "pure" rotations as linear maps in a vector space equipped with Euclidean structure, not as maps of points of a corresponding affine space. Whichever the order of their composition will be, the "pure" rotation component wouldn't change, uniquely determined by the complete motion. Any proper motion of the Euclidean space decomposes to a rotation around the origin and a translation. Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions. Rotation formalisms are focused on proper ( orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to. Main articles: Motion (geometry) and Rotation (mathematics) 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. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom.Īn example where rotation representation is used is in computer vision, where an automated observer needs to track a target. However, for various reasons, there are several ways to represent it. ![]() Such a rotation may be uniquely described by a minimum of three real parameters. 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.Īccording to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. 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. In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. For broader coverage of this topic, see Rotation group SO(3). ![]()
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