Rotation (mathematics)
A geometric motion preserving at least one point.
In geometry, rotation is a type of motion where at least one point in a space stays fixed. For instance, it describes how a rigid body moves around a stationary point. A rotation can carry a sign, similar to an angle: turning clockwise gives a negative value, while turning counterclockwise gives a positive one.
Rotations differ from other motions. Translations have no fixed points at all. Reflections (across a hyperplane) keep an entire (n − 1)-dimensional flat of points fixed in an n-dimensional space.
Mathematically, a rotation is a map. All rotations around a single fixed point form a group under composition, known as the rotation group of that space. In physics and mechanics, however, rotation is often treated as a coordinate transformation—specifically, a transformation of an orthonormal basis. This is because any motion of a body has an inverse transformation; applying that inverse to the frame of reference leaves the body at the same coordinates. For example, in two dimensions, rotating a body clockwise around a point while keeping the axes fixed is equivalent to rotating the axes counterclockwise around the same point while keeping the body fixed. These two views are called active and passive transformations.
**Related definitions and terminology**
The rotation group is a Lie group of rotations about a fixed point. That common fixed point is called the center of rotation and is usually taken as the origin. The rotation group acts as a point stabilizer within the larger group of orientation-preserving motions.
For any specific rotation: - The **axis of rotation** is the line of points that remain fixed. Such an axis exists only in three dimensions (n = 3). - The **plane of rotation** is a plane that stays invariant under the rotation, though its individual points are not fixed. Where an axis exists, it is orthogonal to the plane of rotation. - A **representation of rotations** is a particular algebraic or geometric formalism used to parameterize a rotation map. This meaning is somewhat opposite to how the term is used in group theory.
Rotations of affine spaces of points and rotations of vector spaces are not always clearly distinguished. The former are sometimes called affine rotations (though the term can be misleading), while the latter are vector rotations.
**Definitions and representations**
*In Euclidean geometry*
A motion of a
- field
- Mathematics, Geometry
- known_for
- Concept of rotation as a motion preserving at least one point; distinction between active and passive transformations; rotation group as a Lie group
Lore & Background
Rotation is a concept originating in geometry, defined as a motion of a space that preserves at least one point. It can describe the motion of a rigid body around a fixed point. Rotation has a sign: clockwise rotation is negative, counterclockwise positive. Rotation differs from translations, which have no fixed points, and from hyperplane reflections, which have an entire (n−1)-dimensional flat of fixed points in n-dimensional space.
Reader's Guide
Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group. In mechanics and physics, this concept is frequently understood as a coordinate transformation, because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations. The rotation group is a Lie group of rotations about a fixed point. This common fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of orientation-preserving motions.
Did You Know?
- In one-dimensional space, there are only trivial rotations.
- In two dimensions, only a single angle is needed to specify a rotation about the origin.
- Rotations in three dimensions are generally not commutative, so the order in which rotations are applied is important even about the same point.
- A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation.
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