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All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of rigid motions is called the special Euclidean group, and denoted SE(n). In kinematics, rigid motions in a 3 ...
For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. [13] A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.
In physics, a rigid body, also known as a rigid object, [2] is a solid body in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.
In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave " handedness " unchanged.
Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.
One takes f(0) to be the identity transformation I of , which describes the initial position of the body. The position and orientation of the body at any later time t will be described by the transformation f(t). Since f(0) = I is in E + (3), the same must be true of f(t) for any later time. For that reason, the direct Euclidean isometries are ...
In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations. Category theory , which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different ...
An example of rotation. Each part of the worm drive—both the worm and the worm gear—is rotating on its own axis. A rigid body is an object of a finite extent in which all the distances between the component particles are constant. No truly rigid body exists; external forces can deform any solid.