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Any object will keep the same shape and size after a proper rigid transformation. 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 ...
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.
The rigid-edge and elastic-edge cuboctahedron transformations differ only in having reciprocal parameters: in the elastic-edge transformation the Jessen's icosahedron's short edges stretch and its long edges are rigid, and in the rigid-edge transformation its long edges compress and its short edges are rigid.
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 ...
These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.
More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object.
For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur , that is, its motion relative to a ( local ) coordinate system which moves together with the femur, rather ...
The terms on the left corresponding to the columns of a vertex in () yield the entry in that is the net force applied to by the stresses on edges incident to . Hence, the domain of the dual linear transformation is the set of stresses on edges and the image is the set of net forces on vertices.