Aksiskulmaesityksestä

Aksiskulmaesityksestä (Finnish for "about the axis‑angle representation") refers to a way of expressing the orientation of a rigid body using a single unit vector that defines an axis of rotation and a scalar angle that measures the amount of rotation about that axis. In this representation any rotation in three‑dimensional space can be described by a pair (𝑛,θ), where 𝑛 is a unit vector specifying the direction of the rotation axis and θ is the rotation angle measured in radians. The corresponding rotation matrix R can be constructed from (𝑛,θ) by Rodrigues’ rotation formula: R = I + sinθK + (1−cosθ)K², where K is the cross‑product matrix of 𝑛.

The axis‑angle representation is widely used in robotics, aerospace engineering, computer graphics, and biomechanics to describe the attitude of spacecraft, satellite, or an articulated limb. It forms a bridge between Euler angles, which describe orientation by successive rotations about coordinate axes, and quaternions, which provide a four‑parameter representation that avoids gimbal lock. Because an axis‑angle vector encapsulates both axis direction and rotation magnitude in a compact form, it is convenient for interpolation tasks such as spherical linear interpolation (slerp).

Advantages of the axis‑angle representation include its uniqueness for rotations smaller than 360°, intuitive geometric interpretation, and ease of converting to other representations. Limitations arise from its singularity at zero rotation, where the axis direction is undefined, and from the fact that it is less computationally efficient for repeated rotations compared to quaternion or matrix approaches.

In practice, the axis‑angle representation is often used in combination with quaternions: a quaternion can be converted to an axis‑angle pair, manipulated, and then reconverted for further processing. It is a foundational concept in rigid‑body dynamics and spatial visualization.