User talk:Paul Wormer/scratchbook1

Rotations in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3}

Consider a real 3×3 matrix R with columns r₁, r₂, r₃, i.e.,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R} = \left(\mathbf{r}_1,\, \mathbf{r}_2,\, \mathbf{r}_3 \right) } .

The matrix R is orthogonal if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_i \cdot \mathbf{r}_j = \delta_{ij}, \quad i,j = 1,2,3 . }

The matrix R is a proper rotation matrix, if it is orthogonal and if r₁, r₂, r₃ form a right-handed set, i.e.,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_i \times \mathbf{r}_j = \sum_{k=1}^3 \, \varepsilon_{ijk} \mathbf{r}_k . }

Here the symbol × indicates a cross product and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{ijk}} is the antisymmetric Levi-Civita symbol,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \varepsilon_{123} =&\; \varepsilon_{312} = \varepsilon_{231} = 1 \\ \varepsilon_{213} =&\; \varepsilon_{321} = \varepsilon_{132} = -1 \end{align} }

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{ijk} = 0} if two or more indices are equal.

The matrix R is an improper rotation matrix if its column vectors form a left-handed set, i.e.,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_i \times \mathbf{r}_j = - \sum_{k=1}^3 \, \varepsilon_{ijk} \mathbf{r}_k \; . }

The last two equations can be condensed into one equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_i \times \mathbf{r}_j = \det(\mathbf{R}) \sum_{k=1}^3 \; \varepsilon_{ijk} \mathbf{r}_k }

by virtue of the the fact that the determinant of a proper rotation matrix is 1 and of an improper rotation −1. This can be proved as follows: The determinant of a 3×3 matrix with column vectors a, b, and c can be written as scalar triple product

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(\mathbf{a},\,\mathbf{b},\, \mathbf{c}\right) = \mathbf{a} \cdot (\mathbf{b}\times\mathbf{c}) } .

It was just shown that for a proper rotation the columns of R are orthonormal and satisfy,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_1 \cdot (\mathbf{r}_2 \times \mathbf{r}_3 ) = \mathbf{r}_1 \cdot\left(\sum_{k=1}^3 \, \varepsilon_{23k} \, \mathbf{r}_k \right) = \varepsilon_{231} = 1 . }

Likewise the determinant is −1 for an improper rotation.

Theorem

A proper rotation matrix R can be factorized thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R} = \mathbf{R}_z (\omega_3 ) \; \mathbf{R}_y (\omega_2 ) \; \mathbf{R}_x (\omega_1 ) }

which is referred to as the Euler z-y-x parametrization, or also as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R} = \mathbf{R}_z (\alpha) \; \mathbf{R}_y (\beta ) \; \mathbf{R}_z (\gamma ) \quad }

the Euler z-y-z parametrization.

Here the matrices representing rotations around the z, y, and x axis, respectively, over arbitrary angle φ, are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}_z (\varphi ) \equiv \begin{pmatrix} \cos \varphi & -\sin \varphi & 0 \\ \sin \varphi & \cos \varphi & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}, \quad \mathbf{R}_y (\varphi ) \equiv \begin{pmatrix} \cos \varphi & 0 & \sin \varphi \\ 0 & 1 & 0 \\ -\sin \varphi& 0 & \cos \varphi \\ \end{pmatrix}, \quad \mathbf{R}_x (\varphi ) \equiv \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \varphi & -\sin \varphi \\ 0 & \sin \varphi & \cos \varphi \\ \end{pmatrix} . }

Proof

First the Euler z-y-x-parametrization will be proved by describing an algorithm for the factorization of R. Consider to that end the matrix product

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}_z (\omega_3 ) \, \mathbf{R}_y (\omega_2 ) = \begin{pmatrix} \cos \omega_3 \cos \omega_2 & -\sin \omega_3 & \cos \omega_3 \sin \omega_2 \\ \sin \omega_3 \cos \omega_2 & \cos \omega_3 & \sin \omega_3 \sin \omega_2 \\ -\sin \omega_2 & 0 & \cos \omega_2 \\ \end{pmatrix} \equiv (\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 ) . }

The columns of the matrix product are for ease of reference designated by a₁, a₂, and a₃. Note that the multiplication by R_x(ω₁) on the right does not affect the first column, so that a₁ = r₁ (the first column of the matrix to be factorized). Solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3\;} from the first column of R,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}_1 = \begin{pmatrix} \cos \omega_3 \; \cos \omega_2 \\ \sin \omega_3 \; \cos \omega_2 \\ -\sin \omega_2 \\ \end{pmatrix} = \begin{pmatrix} R_{11} \\ R_{21} \\ R_{31} \\ \end{pmatrix} \equiv \mathbf{r}_1 . }

This is possible. First solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2\;} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\pi/2 \leq \omega_2 \leq \pi/2} from

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \omega_2 = - R_{31}. \, }

Then solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3\;} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq \omega_3 \leq 2 \pi} from the two equations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos \omega_3 =& {R_{11} \over \cos \omega_2} \\ \sin \omega_3 =& {R_{21} \over \cos \omega_2} . \end{align} }

Knowledge of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3\;} determines the vectors a₂ and a₃.

Since a₁, a₂ and a₃ are the columns of a proper rotation matrix they form an orthonormal right-handed system. The plane spanned by a₂ and a₃ is orthogonal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}_1 \equiv \mathbf{r}_1} and hence the plane contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_3} . Thus the latter two vectors are a linear combination of the first two,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( \mathbf{r}_2 , \mathbf{r}_3 ) = (\mathbf{a}_2 , \mathbf{a}_3 ) \begin{pmatrix} \cos \omega_1 & -\sin \omega_1 \\ \sin \omega_1 & \cos \omega_1 \\ \end{pmatrix} . }

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_2,\; \mathbf{a}_2,\; \mathbf{a}_3} are known unit vectors we can compute

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{a}_2 \cdot \mathbf{r}_2 =& \cos \omega_1 \\ \mathbf{a}_3 \cdot \mathbf{r}_2 =& \sin \omega_1. \end{align} }

These equations give Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_1\;} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq \omega_1 \leq 2 \pi} .

Augment the 2×2 matrix to the 3×3 matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}_x(\omega_1)} , then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{R} \equiv ( \mathbf{r}_1 , \mathbf{r}_2 , \mathbf{r}_3 ) = ( \mathbf{r}_1 , \mathbf{a}_2 , \mathbf{a}_3 ) \mathbf{R}_x (\omega_1 ) = (\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3)\mathbf{R}_x (\omega_1 ) = \mathbf{R}_z (\omega_3 ) \, \mathbf{R}_y (\omega_2 ) \, \mathbf{R}_x (\omega_1 ) . \end{align} }

This concludes the proof of the z-y-x parametrization.

The Euler z-y-z parametrization is obtained by a small modification of the previous proof. Solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3\;} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_3 = \mathbf{a}_3 } (the rightmost multiplication by R_z(ω₁) does not affect r₃) and then consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( \mathbf{r}_1, \; \mathbf{r}_2 ) = (\mathbf{a}_1, \; \mathbf{a}_2 ) \begin{pmatrix} \cos \omega_1 & -\sin \omega_1 \\ \sin \omega_1 & \cos \omega_1 \\ \end{pmatrix} }

or, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}_1 \cdot \mathbf{r}_1 = \cos \omega_1 \; , \quad \mathbf{a}_2 \cdot \mathbf{r}_1 = \sin \omega_1 . } The equation for R can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( \mathbf{r}_1 , \mathbf{r}_2 , \mathbf{r}_3 ) = ( \mathbf{a}_1 , \mathbf{a}_2 , \mathbf{r}_3 ) \, \mathbf{R}_z (\omega_1 ) = \mathbf{R}_z (\omega_3 ) \, \mathbf{R}_y (\omega_2 ) \, \mathbf{R}_z (\omega_1 ) \; , }

which proves the Euler z-y-z parametrization. It is common in this parametrization to write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3 = \alpha,\quad \omega_2 = \beta, \quad \omega_1 = \gamma. }