In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another. Euclid, detail from The School of Athens by Raphael. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... A parameter is a measurement or value on which something else depends. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...

The candidates include:

• Euler angles (θ,φ,ψ), representing a product of rotations about the z-, y- and z-axes;

• Tait-Bryan angles (θ,φ,ψ), representing a product of rotations about the x-, y- and z-axes;

• a pair (n, θ) of a unit vector representing an axis, and an angle of rotation about it (see coordinate rotation);

• Euler-Rodrigues parameters, a 4-vector v of length 1, an older name for the following;

• a quaternion q of length 1 (cf. quaternions and spatial rotation, 3-sphere);

• a 3×3 skew-symmetric matrix, via exponentiation;

• Cayley rational parameters, based on the Cayley transform, usable in all characteristics;

• fractional linear transformations, , acting on the Riemann sphere.

There are problems in using these as more than local charts, to do with their multiple-valued nature, and singularities. That is, one must be careful above all to work only with diffeomorphisms in the definition of chart. This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the quaternions in a 3-sphere. The uniqueness of the representation by Euler angles breaks down at some points (cf. gimbal lock), while the quaternion representation is always a double cover, with q and −q giving the same rotation. Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ... In geometry, Tait-Bryan angles are three angles used to describe a general rotation in three-dimensional Euclidean space by three successive rotations, once about the x-axis, once about the y-axis, and once about the z-axis. ... In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ... In mathematics, Euler-Rodrigues parameters, also called just Euler parameters, are four numbers a, b, c, d such that a2+b2+c2+d2=1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... The Wikipedia article on quaternions describes the history and purely mathematical properties of the algebra of quaternions. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... Cayley transform maps upper half plane to open unit disk In complex analysis, the Cayley transform is the map The Cayley transform is a linear fractional transformation. ... Möbius transformations should not be confused with the Möbius transform. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... A torus. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... In gyroscopic devices controlled by Euler mechanics or Euler angles, gimbal lock is caused by the alignment of two of the three gimbals together so that one of the rotation references (pitch/yaw/roll, often yaw) is cancelled. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

Looking more closely, the sixth representation gives parameters in R³. The third gives parameters in S²×S¹; if we replace the unit vector by the actual axis of rotation, so that n and -n give the same axis line, this becomes RP²×S¹, where RP² is the real projective plane. In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ...

That makes four or five manifolds that are used to try to give charts on SO(3). The truth about it, so to speak, is that it is diffeomorphic to real projective space RP³: the quaternion representation is precisely a two-to-one mapping from S³ to SO(3). This suggests that it has certain theoretical advantages; and also that conversions from other representations to it will encounter chart problems. In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...

One area in which these considerations, in some form, become inevitable, is the kinematics of a rigid body. One can take as definition the idea of a curve in the Euclidean group E(3) of three-dimensional Euclidean space, starting at the identity (initial position). The translation subgroup T of E(3) is a normal subgroup, with quotient SO(3) if we look at the subgroup E⁺(3) of direct isometries only (which is reasonable in kinematics). Therefore any rigid body movement leads directly to SO(3), when we factor out the translational part. In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. ... In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

See also

• atlas