Rotation
Rotational motion leaves at least one point unchanged.
Rotation is the motion of an object where at least one point stays fixed. In two dimensions, a flat shape turns around a central point, either clockwise or counterclockwise. In three dimensions, a solid object spins around an imaginary line known as an axis. When that axis runs through the object’s own center of mass, the motion is called a spin (or autorotation); the spot where the axis meets the surface is a pole—Earth’s rotation, for instance, defines its geographic poles. If the axis lies entirely outside the moving body, the motion is called a revolution (or orbit), like Earth’s path around the Sun; the ends of that external axis are the orbital poles. Both types involve their own angular velocity and angular momentum—spin angular velocity and orbital angular velocity, and corresponding angular momenta.
Mathematically, a rotation is a rigid-body movement that keeps at least one point fixed, unlike a translation. In two dimensions, exactly one point stays fixed; in three dimensions, an entire line (the axis) may remain fixed. Every rigid-body motion is either a rotation, a translation, or a combination of the two. A rotation is a progressive radial orientation toward a common point that lies on the motion’s axis, and that axis is perpendicular to the plane of motion. Performing two rotations around the same point or axis yields a third rotation, and the inverse of a rotation is also a rotation, so rotations around a given point or axis form a group. However, rotating around two different points or axes can produce something other than a rotation, such as a translation. Rotations around the x, y, and z axes are called principal rotations; any spatial rotation can be broken down into a sequence of these three.
In three dimensions, any sequence of rotations about a fixed point is equivalent to a single rotation about some axis (which lies perpendicular to the rotation plane). At any instant, the rotation rate is about an axis, though that axis may shift over time. In dimensions other than three, describing a rotation as around an axis becomes problematic because more than one axis may stay fixed; instead, simple rotations are described as occurring in a plane. In four or more dimensions, combining two or more rotations about different planes does not generally result in a rotation in a single plane.
Two-dimensional rotations have no axis—only a fixed point. This means no direction in the plane remains unchanged (except for the identity rotation). Mathematically, a 2D rotation about the origin by an angle θ counterclockwise is represented by the matrix [[cosθ, -sinθ], [sinθ, cosθ]]. Its eigenvalues are cosθ ± i sinθ; unless cosθ = ±1, there is no real eigenvalue, so no real vector stays fixed.
For a proper orthogonal 3×3 rotation matrix, the rotation angle α is found from the trace: α = arccos((trace(A) - 1)/2).
- field
- Physics, Mathematics
- known_for
- Defining rotational motion, spin, revolution, and the mathematical representation of rotations
- type
- Concept
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.
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