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2. Rotation (mathematics) - Wikipedia

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

   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. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn ...

3. Rotations in 4-dimensional Euclidean space - Wikipedia

   en.wikipedia.org/wiki/Rotations_in_4-dimensional...

   In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO (4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment [0, π] except ...

4. Rotational symmetry - Wikipedia

   en.wikipedia.org/wiki/Rotational_symmetry

   Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of ⁠ ⁠ (180°, 120°, 90°, 72°, 60°, 51 3⁄7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...

5. Rotor (mathematics) - Wikipedia

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

   In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin (V). We define this group below. Let V be a vector space equipped with a positive definite quadratic form q, and let Cl (V) be the geometric algebra associated to V. The algebra Cl (V) is the quotient of the tensor algebra ...

6. Quaternions and spatial rotation - Wikipedia

   en.wikipedia.org/wiki/Quaternions_and_spatial...

   A rotation can be represented by a unit-length quaternion q = (w, r →) with scalar (real) part w and vector (imaginary) part r →. The rotation can be applied to a 3D vector v → via the formula = + (+). This requires only 15 multiplications and 15 additions to evaluate (or 18 multiplications and 12 additions if the factor of 2 is done via ...

7. Rotations and reflections in two dimensions - Wikipedia

   en.wikipedia.org/wiki/Rotations_and_reflections...

   A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1. Then reflect P′ to its image P′′ on the other side of line L2. If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the ...