Orientation (geometry)
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- This article deals with orientation of reference axes or frames. For orientation of a space see Orientation (mathematics).
The orientation (or angular position) in space of an axis (straight line), segment of axis, directed axis, or segment of directed axis (vector) is defined by the angles it forms with the axes of a reference frame, or other equivalent methods, such as direction cosines.
The orientation of a plane is given by the orientation of a vector normal to that plane.
The orientation of a rigid body 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, the position of the latter is not enough to completely describe the position of the whole object. The position of a rigid body has two components: linear and angular position. The first component is represented by the position of a reference point fixed to the body, often coinciding with its center of mass or geometric center. The angular position, or orientation, is usually defined by a motion of rotation from a given reference orientation (which may not coincide with the initial orientation of the body). Several tools to describe three dimensional rigid body rotations, and therefore orientations, have been developed. Some of them are extensible to spaces with four or more dimensions.
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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). These three rotations are called Euler angles.
[edit] Euler orientation vector
Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed 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 orthogonal matrices referred to as rotation matrices or direction cosine matrices.
The Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue). The product of two rotation matrices 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 orientation quaternions, also called versors. They are equivalent to rotation matrices, removing the redundant information. With respect to orientation vectors, they can be more 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 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. All the points of the body change their position during a rotation about a fixed axis, except for those lying on the rotation axis. 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.
[edit] Orientation in mathematics
The above described geometrical meaning of the word orientation should not be confused with its meaning in the context of linear algebra, where a different orientation means a change to the mirror image by a reflection.
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.
