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What is a Euler Angle?

By Rachel Burkot
Updated May 21, 2024
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A Euler angle is a term that represents a three dimensional rotation and the three separate angles that compose the rotation. Euler angles can be applied to several aspects of math, engineering and physics. They are used in the construction of appliances such as airplanes and telescopes. Due to the mathematics involved, Euler angles are often represented algebraically.

Tackling the terminology of Euler angles can be tricky because of widespread inconsistency in the field. One way to identify and track the angles is by using a standard set of terms for them. Traditionally, the Euler angle that is applied first is called the heading. The angle applied second is the attitude, while the third and final angle applied is referred to as the bank.

A coordinate system for the coordinates and rotations of the Euler angles is also necessary for measuring the object. First, the order of combining angles is important to establish. The order of 3-d rotations often uses an xyz representation, with each letter representing a plane. This allows for 12 different angle sequences.

Each Euler angle may be measured either relative to the ground or relative to the object being rotated. When this factor is considered, the number of possible sequences doubles to 24. When the project calls for a representation in absolute coordinates, it generally makes sense to measure relative to the ground. When the task requires calculating the object’s dynamics, each Euler angle should be measured in terms of the rotating object’s coordinates.

A Euler angle is generally made most clear by a drawing. This can be a simple way to flesh out the angles, but it can get complicated when a second rotation is set in motion. A second set of three Euler angles must now be measured, and they cannot simply be added to the first set because the order of rotations is critical. Depending on the axis on which the pivot occurs, a rotation might naturally cancel itself out.

To keep each Euler angle and its corresponding rotations straight, an algebraic matrix is often employed. A rotation about an axis is represented by a vector in a positive direction, if the rotation occurred counterclockwise. Taking the point where x and y cross one another on the chart will rotate to another point, representing a new point using sin and cosine.

In a matrix, each Euler angle is given a separate line. According to Euler’s rotation theorem, any rotation can be described in three angles. Thus, the descriptions are often listed in a rotation matrix and may be represented by numbers — such as a, b, and c — to keep them straight.

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