Euler angles represent a 3D rotation and are used in mathematics, engineering, and physics. A standard set of terms can help identify and plot angles. A coordinate system is needed to measure Euler angles, and a drawing can help clarify them. An algebraic matrix is often used to keep track of Euler angles.
An Euler angle is a term that represents a three-dimensional rotation and the three separate angles that make up the rotation. Euler angles can be applied to different aspects of mathematics, engineering and physics. They are used in the construction of devices such as airplanes and telescopes. Because of the mathematics involved, Euler angles are often represented algebraically.
Dealing with Euler angle terminology can be tricky due to widespread inconsistency in the field. One way to identify and plot angles is to use a standard set of terms for them. Traditionally, the Euler angle applied first is called the bow. The angle applied second is the trim, while the third and last angle applied is referred to as the bank.
To measure the object you also need a coordinate system for Euler angle coordinates and rotations. First, it’s important to establish the order in which the corners are combined. 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 can be measured with respect to the ground or with respect to the object being rotated. Considering this factor, the number of possible sequences doubles to 24. When the project requires a representation in absolute coordinates, it generally makes sense to measure relative to the ground. When the activity requires calculating the dynamics of the object, each Euler angle should be measured in terms of the coordinates of the rotating object.
An Euler angle is usually made clearer by a drawing. This can be an easy way to flesh out corners, but it can get tricky when a second spin is started. Now a second set of three Euler angles has to be measured, and they cannot simply be added to the first set because the order of the rotations is critical. Depending on the axis on which the pivot occurs, a rotation could of course cancel out.
To keep each Euler angle and its corresponding rotations straight, an algebraic matrix is often employed. A rotation around an axis is represented by a vector in a positive direction, if the rotation has taken place in a counterclockwise direction. Taking the point where x and y cross on the graph will rotate to another point, representing a new point using sine 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. Therefore, descriptions are often listed in a rotation matrix and can be represented by numbers, such as a, b, and c, to keep them straight.
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