Orientation (geometry)
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- This article deals with orientation of reference frames. For orientation of a space see Orientation (mathematics).
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. Usually an orientation is defined by a rotation from the initial system. Several tools to describe a three dimensional (sometimes extensible to more dimensions) rotations, and therefore orientations, have been developed.
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[edit] Euler angles
The first attempt to represent an orientation was owed to Euler. He did imagine three reference frames that could rotate one around the other. He realized that starting with a fixed reference frame and performing three rotations he could get any other reference frame in the space. (Two rotations to fix the vertical axis and other to fix the other two axes). This three rotations are called Euler angles.
[edit] Euler orientation vector
Euler also realized that the composition of two rotations is also a new rotation, with a new axis. Therefore the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore any orientation can be represented by a rotation vector that leads to it from the reference frame.
[edit] Orientation matrix
With the introduction of matrices the Euler theorems were rewritten. The rotations were described by a matrix of which the Euler vector is the eigenvector (a rotation matrix has a unique real eigenvalue). The product of two matrix is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe.
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 quaternion
Another way to describe rotations are quaternions. They are equivalent to rotation matrices, removing the redundant information. They are better than vectors because they can be easily converted to and from matrices.
[edit] Navigation angles
These are the three angles known as Yaw, pitch and roll, also known as Tait-Bryan angles angles or Cardan angles. In aerospace engineering they are usually referred to as Euler angles, creating confusion with the mathematical euler angles, which are different.
[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.
