Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body
(TO DO: fix math, expand, organize, cross-reference, illustrate.)
Description
To describe such an orientation in 3-dimensional Euclidean space three parameters are required.
They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others.
Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as [[Alpha|Template:Math]], [[Beta|Template:Math]], [[Gamma|Template:Math]], or [[Phi|Template:Math]], [[Theta|Template:Math]], [[Psi (letter)|Template:Math]].
Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system.
For instance, a first rotation about Template:Math by an angle Template:Math, a second rotation about Template:Math by an angle Template:Math, and a last rotation again about Template:Math, by an angle Template:Math. These rotations start from a known standard orientation.
In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.
Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations).
The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system.
See also
External links
- Euler angles @ Wikipedia
