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  2. Translation (geometry) - Wikipedia

    en.wikipedia.org/wiki/Translation_(geometry)

    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.

  3. Screw theory - Wikipedia

    en.wikipedia.org/wiki/Screw_theory

    Now for any quaternion vector p, p* = −p, let q = 1 + pε ∈ F, where the required rotation and translation are effected. Evidently the group of units of the ring of dual quaternions is a Lie group. A subgroup has Lie algebra generated by the parameters a r and b s, where a, b ∈ R, and r, s ∈ H. These six parameters generate a subgroup ...

  4. Rigid transformation - Wikipedia

    en.wikipedia.org/wiki/Rigid_transformation

    A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form T ( v ) = R v + t where R T = R −1 (i.e., R is an orthogonal transformation ), and t is a vector giving the translation of the origin.

  5. Glide reflection - Wikipedia

    en.wikipedia.org/wiki/Glide_reflection

    Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it.

  6. Translation of axes - Wikipedia

    en.wikipedia.org/wiki/Translation_of_axes

    In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away.

  7. Vectorization (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Vectorization_(mathematics)

    For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...

  8. Vector (mathematics and physics) - Wikipedia

    en.wikipedia.org/wiki/Vector_(mathematics_and...

    [13] [14] [12] A simpler example of a bound vector is the translation vector from an initial point to an end point; in this case, the bound vector is an ordered pair of points in the same position space, with all coordinates having the same quantity dimension and unit (length an meters).

  9. Parallel transport - Wikipedia

    en.wikipedia.org/wiki/Parallel_transport

    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.

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