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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.
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference ...
Important theorems of screw theory include: the transfer principle proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws; [1] Chasles' theorem proves that any change between two rigid object poses can be performed by a single screw; Poinsot's theorem ...
Euler's theorem and its proof are contained in paragraphs 24–26 of the appendix (Additamentum. pp. 201–203) of L. Eulero (Leonhard Euler), Formulae generales pro translatione quacunque corporum rigidorum (General formulas for the translation of arbitrary rigid bodies), presented to the St. Petersburg Academy on October 9, 1775, and first ...
The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain, around a loop, and back to the base to obtain the loop equation, [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] …
Euler angles – Description of the orientation of a rigid body; Geometric terms of location – Directions or positions relative to the shape and position of an object; Ship motions – Terms connected to the six degrees of freedom of motion; Aircraft principal axes – Principal directions in aviation
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is