Quaternions and spatial rotation

Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more compact, more numerically stable, and may be more efficient. Quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics,orbital mechanics of satellites and crystallographic texture analysis.

When used to represent rotation, unit quaternions are also called rotation quaternions. When used to represent an orientation (rotation relative to a reference coordinate system), they are called orientation quaternions or attitude quaternions.

In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle θ about a fixed axis (called the Euler axis) that runs through the fixed point. The Euler axis is typically represented by a unit vector u→. Therefore, any rotation in three dimensions can be represented as a combination of a vector u→ and a scalar θ. Quaternions give a simple way to encode this axis–angle representation in four numbers, and can be used to apply the corresponding rotation to a position vector, representing a point relative to the origin in R³.

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