Euler angles (original) (raw)

Euler angles one way of representing rotations in 3-dimensional Euclidean space as a product of three successive 2D coordinate rotations θ, φ, ψ about the x-, y- and z-axes. Most often, the rotations are carried out first around Z axis, then around the X axis and finally around the Z axis again, but other definitions are possible.

As a representation of rotations, they have several flaws:

a single rotation can be represented by several different sets of Euler angles, allowing a phenomenon known as gimbal lock
it is difficult to compute the combination of sets of rotations within the Euler angle framework
they are difficult to interpolate smoothly, or in a coordinate-independent way

Quaternions are a far better representation of 3D rotations for many purposes, alleviating all of the above issues.

The Euler angles form a chart on SO(3), the mathematical group of rotations in 3D space. See Charts on SO(3) for a fuller treatment.

Some naming systems for Euler angles include:

"NASA standard aerospace" convention: "precession, nutation and spin";
"heading, attitude and bank"

References:

G. J. Minkler, Aerospace Coordinate Systems and Transformations. Magellan Book Company, Balitmore, MD, 1990.