Orientation (rigid body)
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The orientation of an object in space is the choice of positioning it with one point held in a fixed position. Since the object may still be rotated around its fixed point, position of the fixed point is not enough to completely describe the object. Thus the configuration space of a non-symmetrical object in n-dimensional space is SO(n) × Rn. Orientation may be visualized by attaching a basis of tangent vectors to an object. The direction in which each vector points determines its orientation.
[edit] Orientation of a rigid body
The orientation of a rigid body in the three dimensional space changes by rotation. In the case of rotation about an axis through the center of the body, only the orientation changes, otherwise also position. If the rigid body has any rotational symmetry, not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation.
In two dimensions the situation is similar. In one dimension a "rigid body" can not move (continuously change) from one orientation to the other.
This meaning of orientation should not be confused with the other meaning, where a different orientation means a change to the mirror image by an improper rotation, which includes a reflection, see orientation (mathematics).
Formally, for any dimension, the orientation of the image of an object under a direct isometry with respect to that object is the linear part of that isometry. Thus it is an element of SO(n), or, put differently, the corresponding coset in E+(n) / T, where T is the translation group.
