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Mathematically, a four-dimensional space is a space that needs four parameters to specify a point in it. For example, a general point might have position vector a, equal to. This can be written in terms of the four standard basis vectors (e1, e2, e3, e4), given by. so the general vector a is. Vectors add, subtract and scale as in three ...
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 ...
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 ...
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 ...
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 ...
Therefore the set of rotations has a group structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê. Its product by the rotation angle is known as an axis-angle vector.
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