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  2. Euclidean plane isometry - Wikipedia

    en.wikipedia.org/wiki/Euclidean_plane_isometry

    Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a ...

  3. Conjugation of isometries in Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Conjugation_of_isometries...

    the conjugation of a translation by a rotation is a translation by a rotated translation vector; the conjugation of a translation by a reflection is a translation by a reflected translation vector; Thus the conjugacy class within the Euclidean group E(n) of a translation is the set of all translations by the same distance.

  4. Conformal geometric algebra - Wikipedia

    en.wikipedia.org/wiki/Conformal_geometric_algebra

    This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.

  5. 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. In a Euclidean space, any translation is ...

  6. Rotation (mathematics) - Wikipedia

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

    Rotation of an object in two dimensions around a point O. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point.

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  8. Rotation of axes in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_of_axes_in_two...

    In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .

  9. Geometric algebra - Wikipedia

    en.wikipedia.org/wiki/Geometric_algebra

    The rotors in a space of dimension + have () / + degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an ⁠ ⁠-dimensional space. This is the case in Projective Geometric Algebra (PGA), which is used [ 33 ] [ 34 ] [ 35 ] to represent Euclidean isometries in Euclidean geometry (thereby ...

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