Rotation

Lore & Background

Rotation is a rigid body movement that, unlike a translation, keeps at least one point fixed. This definition applies to rotations in two dimensions, where exactly one point is kept fixed, and in three dimensions, where additional points may be kept fixed, as in rotation around a fixed axis. All rigid body movements are rotations, translations, or combinations of the two. A rotation is simply a progressive radial orientation to a common point, which lies within the axis of that motion, and the axis is perpendicular to the plane of the motion. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results, and the reverse of a rotation is also a rotation, forming a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, such as a translation. Rotations around the x, y, and z axes are called principal rotations, and any spatial rotation can be decomposed into a combination of these principal rotations.

Reader's Guide

The concept of rotation is fundamental to understanding motion in physics and mathematics. It distinguishes between spin, where the axis passes through the body's own center of mass, and revolution, where the axis is external to the moving body, as in planetary orbits. This distinction is crucial for analyzing angular velocity and angular momentum, which are key to mechanics. Mathematically, rotations form a group, and their representation through matrices allows for precise calculations in two and three dimensions. In three dimensions, every proper rotation has an axis corresponding to an eigenvector with eigenvalue 1, and the rotation angle can be derived from the trace of the rotation matrix. The ability to decompose any spatial rotation into principal rotations around the x, y, and z axes is essential for computer graphics, robotics, and physics simulations. The concept also extends to higher dimensions, where rotations are described in planes rather than around axes.

Did You Know?

• A rotation with an internal axis through the body's own center of mass is called a spin or autorotation.
• A rotation around an axis external to the moving body is called a revolution or orbit.
• In two dimensions, a rotation has no axis of rotation, only a point about which the rotation occurs.
• Every proper rotation in 3D space has an axis of rotation, which is an eigenvector of the rotation matrix with eigenvalue 1.