![]() ![]() This process exhibits the Euclidean group as the semi-direct product. The orientation preserving maps are those of determinant +1, forming the special orthogonal group SO( n). Hence they may be represented by the orthogonal matrix group O( n). If we fix a particular point O in the Euclidean space and consider the rigid motions which fix this point, we find that these must be linear maps of the underlying vector space which preserve distance. ![]() All translations are orientation-preserving. If we regard Euclidean space of n dimensions as an affine space built on a real vector space R n then the translations are the maps of the formįor a particular a in R n. It is a matter of convention whether the orientation-reversing maps such as reflections are considered "proper" rigid motions.Īn important subclass of rigid motions are the translations or displacements. Rigid motions are invertible functions, whose inverse functions are also rigid motions, and hence form a group, the Euclidean group.Īn important distinction is between those motions which preserve orientation or "handedness" and those which do not (for example, those three-dimensional motions which would transform a right-handed into a left-handed glove). Since Euclidean properties may be defined in terms of distance, the rigid motions are the distance-preserving mappings or isometries. In Euclidean geometry, a rigid motion is a transformation which preserves the geometrical properties of the Euclidean space. ![]()
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