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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
Translation is done by shearing parallel to the xy plane, and rotation is performed around the z axis. To represent affine transformations with matrices, we can use homogeneous coordinates. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. Using this system, translation can be expressed with ...
Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. In spaces with dimension higher than 1, there may be multiple translational symmetries. For each set of k independent translation vectors, the symmetry group is isomorphic with Z k. In particular, the ...
Parallel transport of a vector around a closed loop (from A to N to B and back to A) on the sphere. The angle by which it twists, , is proportional to the area inside the loop. In differential geometry, parallel transport (or parallel translation [a]) is a way of transporting geometrical data along smooth curves in a manifold.
[15] [16] A sliding vector is the combination of an ordinary vector quantity and a line of application or line of action, over which the vector quantity can be translated (without rotations). A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement ...
For each Bravais lattice vector we define a translation operator ^ which, when operating on any function () shifts the argument by : ^ = (+) Since all translations form an Abelian group, the result of applying two successive translations does not depend on the order in which they are applied, i.e. ^ ^ = ^ ^ = ^ + In addition, as the Hamiltonian ...
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple ...
The algorithm only computes the rotation matrix, but it also requires the computation of a translation vector. When both the translation and rotation are actually performed, the algorithm is sometimes called partial Procrustes superimposition (see also orthogonal Procrustes problem ).