Rotationsdelen

Rotationsdelen, literally the rotation part, is a concept in 3D geometry, robotics, computer graphics and related fields. It denotes the component of a rigid-body transformation that encodes only the object's orientation, independent of its position. In three-dimensional space a rigid-body pose is commonly represented by a homogeneous transformation matrix T = [R t; 0 1], where R is a 3x3 rotation matrix and t is a 3x1 translation vector. The rotation part R belongs to the special orthogonal group SO(3): it is orthogonal (R^T R = I) and has determinant 1. R transforms vectors from the object's local frame to the world frame and can be applied to rotate coordinates.

Rotations can be represented in several formats, including rotation matrices, Euler angles, axis-angle (rotation vector), and quaternions. Each representation has advantages and drawbacks: rotation matrices are robust to composition but require 9 numbers; Euler angles are compact but can suffer from gimbal lock; quaternions avoid singularities and support smooth interpolation.

In practice, the rotation part is manipulated by multiplying rotation matrices (composition of rotations) or by converting between representations. Interpolation between orientations is commonly done with spherical linear interpolation (slerp) for quaternions or similar methods for rotation matrices.

Applications include determining the orientation of robotic end effectors, camera poses in computer graphics, satellite attitude control, and animation. Key properties to verify include orthogonality and determinant 1, with numerical maintenance to prevent drift.

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