Utente:Grasso Luigi/sandbox2/Formula di Eulero–Rodrigues

In matematica e meccanica, la formula di Eulero–Rodrigues descrive la rotazione di un vettore nello spazio. Si basa sulla formula di Rodrigues, ma con una parametrizzazione diversa.

La rotazione viene descritta dai quattro parametri di Eulero dovuti a Leonhard Euler. La formula di Rodrigues (che prende il nome da Olinde Rodrigues), un metodo per calcolare la posizione di un punto ruotato, viene utilizzata in applicazioni software, come nei simulatori di volo e giochi per PC.

Definizione

A rotation about the origin is represented by four real numbers, Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar such that

$a^{2}+b^{2}+c^{2}+d^{2}=1.$

When the rotation is applied, a point at position Template:Math rotates to its new position

${\vec {x}}'={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}{\vec {x}}.$

Formulazione vettoriale

The parameter Template:Mvar may be called the scalar parameter, while Template:Math the vector parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form Template:Equation box 1

Simmetria

The parameters Template:Math and Template:Math describe the same rotation. Apart from this symmetry, every set of four parameters describes a unique rotation in three-dimensional space.

Composizione di rotazioni

The composition of two rotations is itself a rotation. Let Template:Math and Template:Math be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:

${\begin{aligned}a&=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2};\\b&=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2};\\c&=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2};\\d&=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}.\end{aligned}}$

It is straightforward, though tedious, to check that Template:Math. (This is essentially Euler's four-square identity, also used by Rodrigues.)

Angolo e asse di rotazione

Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a unit vector Template:Math) and the rotation angle Template:Math. The Euler parameters for this rotation are calculated as follows:

${\begin{aligned}a&=\cos {\frac {\varphi }{2}};\\b&=k_{x}\sin {\frac {\varphi }{2}};\\c&=k_{y}\sin {\frac {\varphi }{2}};\\d&=k_{z}\sin {\frac {\varphi }{2}}.\end{aligned}}$

Note that if Template:Math is increased by a full rotation of 360 degrees, the arguments of sine and cosine only increase by 180 degrees. The resulting parameters are the opposite of the original values, Template:Math; they represent the same rotation.

In particular, the identity transformation (null rotation, Template:Math) corresponds to parameter values Template:Math. Rotations of 180 degrees about any axis result in Template:Math.

Connection with quaternions

The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter Template:Mvar is the real part, the vector parameters Template:Mvar, Template:Mvar, Template:Mvar are the imaginary parts. Thus we have the quaternion

$q=a+bi+cj+dk,$

which is a quaternion of unit length (or versor) since

$\left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.$

Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with composition.

Connection with SU(2) spin matrices

The Lie group SU(2) can be used to represent three-dimensional rotations in 2 × 2 matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is

$U={\begin{pmatrix}\ \ \,a+di&b+ci\\-b+ci&a-di\end{pmatrix}}.$

Alternatively, this can be written as the sum

${\begin{aligned}U&=a\ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b\ {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}+c\ {\begin{pmatrix}0&i\\i&0\end{pmatrix}}+d\ {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\\&=a\,I+ic\,\sigma _{x}+ib\,\sigma _{y}+id\,\sigma _{z},\end{aligned}}$

where the Template:Mvar are the Pauli spin matrices. Thus, the Euler parameters are the coefficients for the representation of a three-dimensional rotation in SU(2).

Voci correlate

• Rotation formalisms in three dimensions
• Quaternions and spatial rotation
• Versor
• Spinors in three dimensions
• SO(4)
• 3D rotation group

Bibliografia

• Élie Cartan, The Theory of Spinors, Dover, 1981, ISBN 0-486-64070-1.
• W. R. Hamilton, Elements of Quaternions, Cambridge University Press, 1899.
• E.J. Haug, Computer-Aided Analysis and Optimization of Mechanical Systems Dynamics., Springer-Verlag, 1984.
• (ES) Eduardo Garza, Benjamin Olinde Rodrigues, matemático y filántropo, y su influencia en la Física Mexicana (PDF), in Revista Mexicana de Física, June 2011, pp. 109–113.
• (EN) Shuster, A Survey of Attitude Representations (PDF), in Journal of the Astronautical Sciences, vol. 41, n. 4, 1993, pp. 439–517.