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.

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

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.

See also:

• Roll, pitch and yaw
• Spherical coordinate system