![]() In one-dimensional space, there are only trivial rotations. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. In the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. The " improper rotation" term refers to isometries that reverse (flip) the orientation. But a (proper) rotation also has to preserve the orientation structure. See the article below for details.ĭefinitions and representations In Euclidean geometry įurther information: Euclidean space § Rotations and reflections, and Special orthogonal group A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation.Ī motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation. The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations. Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. This meaning is somehow inverse to the meaning in the group theory. The axis (where present) and the plane of a rotation are orthogonal.Ī representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. Unlike the axis, its points are not fixed themselves.
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