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Lecture

Rotations: Isometries and Axes

Description

This lecture covers the concept of rotations as isometries with a single fixed point, which can be expressed as compositions of reflections. It explains that an isometry is a rotation if it is the composition of two reflections with a common axis. The lecture also discusses the choice of axes for rotations, highlighting the flexibility in selecting axes when the rotation is the identity.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Rotation

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis.

Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.

Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is which gives rise to the sequence of iterated function applications which is hoped to converge to a point . If is continuous, then one can prove that the obtained is a fixed point of , i.e., More generally, the function can be defined on any metric space with values in that same space.

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.

Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space. Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space.

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