rotations in the complex plane / trigonometry
definition of exponential function:
exponential of imaginary value:
$exp(i \theta) = \Sigma_{m=0}^\infty \frac{(i \theta)^m}{m!}$
$= \Sigma_{m=0}^\infty (-1)^m \frac{1}{(2 m)!} \theta^{(2 m)}
+ \Sigma_{m=0}^\infty (-1)^m \frac{1}{(2 m + 1)!} \theta^{(2 m + 1)}$
$= cos(\theta) + i sin(\theta)$
This is called Moore's law, or DeMoore's law, or something like that.
Inverse of imaginary exponential map / rotation formula:
$exp(-i \theta) = cos(-\theta) + i sin(-\theta)$
$ = cos(\theta) - i sin(\theta)$
Therefore
$cos(\theta) = \frac{1}{2}( exp(i \theta) + exp(-i \theta))$
$sin(\theta) = \frac{1}{2 i}( exp(i \theta) - exp(-i \theta))$
Complex rotations:
$(x' + i y') = exp(i \theta) (x + i y)$
rotations in 2D using a matrix in place of 'i'
Let $A = \left[ \begin{matrix}
0 & -\theta \\
\theta & 0
\end{matrix} \right]$
$A^2 = \left[ \begin{matrix}
-\theta^2 & 0 \\
0 & -\theta^2
\end{matrix} \right] = -\theta^2 I$
$A^3 = \left[ \begin{matrix}
0 & \theta^3 \\
-\theta^3 & 0
\end{matrix} \right] = -\theta^2 A$
$A^4 = \left[ \begin{matrix}
\theta^4 & 0 \\
0 & \theta^4
\end{matrix} \right] = \theta^4 I = \theta^4 A^0$
The exponential map of A is calculated as:
$exp(A) = \Sigma_{n=0}^\infty \frac{1}{n!} A^n$
$exp(A) =
\left[ \begin{matrix}
1 & 0 \\
0 & 1
\end{matrix} \right]
+
\left[ \begin{matrix}
0 & -\theta \\
\theta & 0
\end{matrix} \right]
+
\frac{1}{2}
\left[ \begin{matrix}
-\theta^2 & 0 \\
0 & -\theta^2
\end{matrix} \right]
+
\frac{1}{3!}
\left[ \begin{matrix}
0 & \theta^3 \\
-\theta^3 & 0
\end{matrix} \right]
+
\frac{1}{4!}
\left[ \begin{matrix}
\theta^4 & 0 \\
0 & \theta^4
\end{matrix} \right]
+
\frac{1}{5!}
\left[ \begin{matrix}
0 & -\theta^5 \\
\theta^5 & 0
\end{matrix} \right]
+
...
$
$exp(A) =
\left[ \begin{matrix}
1 & 0 \\
0 & 1
\end{matrix} \right]
+
\theta
\left[ \begin{matrix}
0 & -1 \\
1 & 0
\end{matrix} \right]
+
\frac{1}{2} \theta^2
\left[ \begin{matrix}
-1 & 0 \\
0 & -1
\end{matrix} \right]
+
\frac{1}{3!} \theta^3
\left[ \begin{matrix}
0 & 1 \\
-1 & 0
\end{matrix} \right]
+
\frac{1}{4!} \theta^4
\left[ \begin{matrix}
1 & 0 \\
0 & 1
\end{matrix} \right]
+
\frac{1}{5!} \theta^5
\left[ \begin{matrix}
0 & -1 \\
1 & 0
\end{matrix} \right]
+
...
$
$exp(A) = \left[ \begin{matrix}
\frac{1}{0!} \theta^0 - \frac{1}{2!} \theta^2 + \frac{1}{4!} \theta^4 - ... &
-\frac{1}{1!} \theta + \frac{1}{3!} \theta^3 - \frac{1}{5!} \theta^5 +... \\
\frac{1}{1!} \theta - \frac{1}{3!} \theta^3 + \frac{1}{5!} \theta^5 - ... &
\frac{1}{0!} \theta^0 - \frac{1}{2!} \theta^2 + \frac{1}{4!} \theta^4 - ...
\end{matrix} \right]$
$exp(A) = \left[ \begin{matrix}
cos(\theta) & -sin(\theta) \\
sin(\theta) & cos(\theta)
\end{matrix} \right]$
2D rotation:
$\left[ \begin{matrix} x' \\ y' \end{matrix} \right] = \left[ \begin{matrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta
\end{matrix} \right] \left[ \begin{matrix}
x \\ y
\end{matrix} \right]$
rotations in $\textbf{R}^3$
3D (and higher?) rotations
rotation around an axis $\vec{n}$ by some angle $\theta$:
for unit vector $\vec{n} = n^i e_i$
for initial vector $\vec{x}$ and result vector $\vec{x}'$
$\vec{x}' = R(\vec{n}, \theta) \vec{x}$
Let $\vec{\omega} = \omega_i e^i = \theta n_i e^i$
Exponential map:
in 3D:
$exp( {\epsilon_{ij}}^k \omega_k ) = \Sigma_{m=0}^\infty ( {\epsilon_{ij}}^k \omega_k)^m$
Let $A_{ij} = {\epsilon_{ij}}^k \omega_k$
$A = \left[ \begin{matrix}
0 & \omega_z & -\omega_y \\
-\omega_z & 0 & \omega_x \\
\omega_y & -\omega_x & 0
\end{matrix} \right] = \star \vec{\omega}$
Notice that $\star (a \wedge b) = a^\sharp \lrcorner \star b = a \times b$
Therefore $(\star \vec{\omega}) \cdot \vec{\omega} = \vec{\omega} \times \vec{\omega} = 0$
$A^2 = \left[ \begin{matrix}
-((\omega_y)^2 + (\omega_z)^2) & \omega_x \omega_y & \omega_x \omega_z \\
\omega_x \omega_y & -((\omega_x)^2 + (\omega_z)^2) & \omega_y \omega_z \\
\omega_x \omega_z & \omega_y \omega_z & -((\omega_x)^2 + (\omega_y)^2)
\end{matrix} \right]
= \vec{\omega} \otimes \vec{\omega} - I |\omega|^2$
$A^3 = \left[ \begin{matrix}
0 & -\omega_z |\omega|^2 & \omega_y |\omega|^2 \\
\omega_z |\omega|^2 & 0 & -\omega_x |\omega|^2 \\
-\omega_y |\omega|^2 & \omega_x |\omega|^2 & 0
\end{matrix} \right]
= (\star \omega) \cdot (\vec{\omega} \otimes \vec{\omega} - I |\omega|^2)
= -(\star \omega) |\omega|^2
= -|\omega|^2 A$
$A^4 = \left[ \begin{matrix}
( (\omega_y)^2 + (\omega_z)^2) |\omega|^2 &
-\omega_x \omega_y |\omega|^2 &
-\omega_x \omega_z |\omega|^2 \\
-\omega_x \omega_y |\omega|^2 &
( (\omega_x)^2 + (\omega_z)^2) |\omega|^2 &
-\omega_y \omega_z |\omega|^2 \\
-\omega_x \omega_z |\omega|^2 &
-\omega_y \omega_z |\omega|^2 &
( (\omega_x)^2 + (\omega_y)^2) |\omega|^2
\end{matrix} \right]
= -(\vec{\omega} \otimes \vec{\omega} - I |\omega|^2) |\omega|^2
= -|\omega|^2 A^2
$
$A^5 = \left[ \begin{matrix}
0 & \omega_z |\omega|^4 & -\omega_y |\omega|^4 \\
-\omega_z |\omega|^4 & 0 & \omega_x |\omega|^4 \\
\omega_y |\omega|^4 & -\omega_x |\omega|^4 & 0
\end{matrix} \right]
= (\star \vec{\omega}) |\omega|^4
= A |\omega|^4$
$exp(A) = \Sigma_{n=0}^\infty \frac{1}{n!} A^n$
$exp(A) = \frac{1}{0!} A^0 + \frac{1}{1!} A^1 + \frac{1}{2!} A^2 + \frac{1}{3!} A^3 + \frac{1}{4!} A^4 + \frac{1}{5!} A^5 + ...$
$exp(A) = \frac{1}{0!} I
+ \frac{1}{1!} (\star \vec{\omega})
+ \frac{1}{2!} (\vec{\omega} \otimes \vec{\omega} - I |\omega|^2)
+ \frac{1}{3!} (-(\star \omega) |\omega|^2)
+ \frac{1}{4!} (-(\vec{\omega} \otimes \vec{\omega} - I |\omega|^2) |\omega|^2)
+ \frac{1}{5!} ((\star \vec{\omega}) |\omega|^4)
+ ...
$
$exp(A) =
\frac{1}{0!} I
- \frac{1}{2!} |\omega|^2 I
+ \frac{1}{4!} |\omega|^4 I
+ \frac{1}{1!} \star \vec{\omega}
- \frac{1}{3!} \star \vec{\omega} |\omega|^2
+ \frac{1}{5!} \star \vec{\omega} |\omega|^4
+ \frac{1}{2!} \vec{\omega} \otimes \vec{\omega}
- \frac{1}{4!} |\omega|^2 \vec{\omega} \otimes \vec{\omega}
+ ...
$
$exp(A) =
I (
\frac{1}{0!} |\omega|^0
- \frac{1}{2!} |\omega|^2
+ \frac{1}{4!} |\omega|^4
- ...
)
+ \vec{\omega} \otimes \vec{\omega} (
\frac{1}{2!}
- \frac{1}{4!} |\omega|^2
+ ...
)
+ \star \vec{\omega} (
\frac{1}{1!}
- \frac{1}{3!} |\omega|^2
+ \frac{1}{5!} |\omega|^4
- ...
)
$
$exp(A) =
I cos|\omega|
+ \vec\omega \otimes \vec\omega (1 - cos|\omega|)
+ \star \vec{\omega} sin|\omega|
$
$exp(A) =
(I - \vec\omega \otimes \vec\omega) cos|\omega|
+ \star \vec{\omega} sin|\omega|
+ \vec\omega \otimes \vec\omega
$
aka Rodrigues rotation formula, with angle $|\omega|$ and axis $\frac{\vec\omega}{|\omega|}$.
rotations in $\textbf{R}^n$
Cartesian basis: $e_i$
inner product: $e_i \cdot e_j = \delta_{ij}$ for $\delta_{ij} =$ Kronecker delta.
exponential map:
$exp(x) = \Sigma_{m=0}^\infty \frac{x^m}{m!}$
... TODO some work...
$ = \delta_{ij} cos(\theta) + n_i n_j (1 - cos(\theta)) + \epsilon_{ijk} n^k sin(\theta)$
(aka Rodrigues rotation formula)
In any dimension:
Let A be an antisymmetric matrix. Spoilers that this can be represented as the components of $a \wedge b$ in some basis.
TODO ...
hyperbolic trigonometry
$cosh(\alpha) = \frac{1}{2} (exp(\alpha) + exp(-\alpha))$
$sinh(\alpha) = \frac{1}{2} (exp(\alpha) - exp(-\alpha))$
Taylor expansion:
$cosh(\alpha) = \Sigma_{m=0}^\infty \frac{1}{(2 m)!} \alpha^{(2 m)}$
$sinh(\alpha) = \Sigma_{m=0}^\infty \frac{1}{(2 m + 1)!} \alpha^{(2 m + 1)}$
hyperbolic rotations in 2D
Let $A = \left[ \begin{matrix}
0 & \alpha \\
\alpha & 0
\end{matrix} \right]$
$A^2 = \left[ \begin{matrix}
\alpha^2 & 0 \\
0 & \alpha^2
\end{matrix} \right]$
$A^3 = \left[ \begin{matrix}
0 & \alpha^3 \\
\alpha^3 & 0
\end{matrix} \right]$
$A^4 = \left[ \begin{matrix}
\alpha^4 & 0 \\
0 & \alpha^4
\end{matrix} \right]$
Let $\hat{A} = \frac{1}{\alpha} A$
$exp(A) = I + \alpha \hat{A} + \frac{1}{2} \alpha^2 \hat{A}^2 + ...$
$exp(A) = I (1 + \frac{1}{2!} \alpha^2 + \frac{1}{4!} \alpha^4 + ...)
+ \hat{A} (\alpha + \frac{1}{3!} \alpha^3 + \frac{1}{5!} \alpha^5 + ....)$
$exp(A) = cosh(\alpha) I + sinh(\alpha) \hat{A}$
$exp(A) = \left[ \begin{matrix}
cosh(\alpha) & sinh(\alpha) \\
sinh(\alpha) & cosh(\alpha)
\end{matrix} \right]$
hyperbolic rotation in 3D
Let $A = \left[ \begin{matrix}
0 & \alpha_3 & \alpha_2 \\
\alpha_3 & 0 & \alpha_1 \\
\alpha_2 & \alpha_1 & 0
\end{matrix} \right]$
Let $\hat{A} = \frac{1}{|\alpha|} A$
$A^2 = \left[ \begin{matrix}
|\alpha|^2 - (\alpha_1)^2 & \alpha_1 \alpha_2 & \alpha_1 \alpha_3 \\
\alpha_1 \alpha_2 & |\alpha|^2 - (\alpha_2)^2 & \alpha_2 \alpha_3 \\
\alpha_1 \alpha_3 & \alpha_2 \alpha_3 & |\alpha|^2 - (\alpha_3)^2
\end{matrix} \right]$
Let $v = 2 \alpha_1 \alpha_2 \alpha_3$
$A^3 = \left[ \begin{matrix}
2 \alpha_1 \alpha_2 \alpha_3 & \alpha_3 |\alpha|^2 & \alpha_2 |\alpha|^2 \\
\alpha_3 |\alpha|^2 & 2 \alpha_1 \alpha_2 \alpha_3 & \alpha_1 |\alpha|^2 \\
\alpha_2 |\alpha|^2 & \alpha_1 |\alpha|^2 & 2 \alpha_1 \alpha_2 \alpha_3
\end{matrix} \right]
= |\alpha|^2 \left[ \begin{matrix}
\frac{v}{|\alpha|^2} & \alpha_3 & \alpha_2 \\
\alpha_3 & \frac{v}{|\alpha|^2} & \alpha_1 \\
\alpha_2 & \alpha_1 & \frac{v}{|\alpha|^2}
\end{matrix} \right]
= v I + |\alpha|^2 A
$
$A^4
= \left[ \begin{matrix}
|\alpha|^2 (|\alpha|^2 - (\alpha_1)^2) &
\alpha_1 \alpha_2 (|\alpha|^2 + 2 (\alpha_3)^2) &
\alpha_1 \alpha_3 (|\alpha|^2 + 2 (\alpha_2)^2) \\
\alpha_1 \alpha_2 (|\alpha|^2 + 2 (\alpha_3)^2) &
|\alpha|^2 (|\alpha|^2 - (\alpha_2)^2) &
\alpha_2 \alpha_3 (|\alpha|^2 + 2 (\alpha_1)^2) \\
\alpha_1 \alpha_3 (|\alpha|^2 + 2 (\alpha_2)^2) &
\alpha_2 \alpha_3 (|\alpha|^2 + 2 (\alpha_1)^2) &
|\alpha|^2 (|\alpha|^2 - (\alpha_3)^2)
\end{matrix} \right]
= |\alpha|^2 \left[ \begin{matrix}
|\alpha|^2 - (\alpha_1)^2 &
\alpha_1 \alpha_2 (1 + \frac{2 (\alpha_3)^2}{|\alpha|^2}) &
\alpha_1 \alpha_3 (1 + \frac{2 (\alpha_2)^2}{|\alpha|^2}) \\
\alpha_1 \alpha_2 (1 + \frac{2 (\alpha_3)^2}{|\alpha|^2}) &
|\alpha|^2 - (\alpha_2)^2 &
\alpha_2 \alpha_3 (1 + \frac{2 (\alpha_1)^2}{|\alpha|^2}) \\
\alpha_1 \alpha_3 (1 + \frac{2 (\alpha_2)^2}{|\alpha|^2}) &
\alpha_2 \alpha_3 (1 + \frac{2 (\alpha_1)^2}{|\alpha|^2}) &
|\alpha|^2 - (\alpha_3)^2
\end{matrix} \right]
= v A + |\alpha|^2 A^2
$
$A^5 = v A^2 + |\alpha|^2 A^3$
$A^5 = v |\alpha|^2 I
+ |\alpha|^4 A
+ v A^2
$
$A^6 = v |\alpha|^2 A
+ |\alpha|^4 A^2
+ v A^3$
$A^6 = v |\alpha|^2 A
+ |\alpha|^4 A^2
+ v (v I + |\alpha|^2 A)$
$A^6 =
(v)^2 I
+ 2 v |\alpha|^2 A
+ |\alpha|^4 A^2
$
$A^6 = (v I + |\alpha|^2 A)^2
$
$exp(A) = I + A + \frac{1}{2} A^2 + ....$
$exp(A) =
I
+ A
+ A^2
+ \frac{1}{3!} (v I + |\alpha|^2 A)
+ \frac{1}{4!} (v A + |\alpha|^2 A^2)
+ \frac{1}{5!} (
v |\alpha|^2 I
+ |\alpha|^4 A
+ v A^2
)
+ ...
$
$exp(A) =
I (
v \frac{1}{3!}
+ v |\alpha|^2 \frac{1}{5!}
+ (v)^2 \frac{1}{6!}
)
+ A(
1
+ |\alpha|^2 \frac{1}{3!}
+ v \frac{1}{4!}
+ |\alpha|^4 \frac{1}{5!}
+ 2 v |\alpha|^2 \frac{1}{6!}
)
+ A^2 (
1
+ |\alpha|^2 \frac{1}{4!}
+ v \frac{1}{5!}
+ |\alpha|^4 \frac{1}{6!}
)
+ ...
$
...bleh this seems to be going nowhere.
especially not until I find a relation between $|\alpha|^2$ and v.
And the fact that my powers are on a 3 point cycle isn't helpful for the fact that hyperbolic is 2-point rather than 4-point as non-hyperbolic exponent of imaginary expansion is.
I guess I'll just say this is the same as $exp(U A U^{-1}) = U exp(A) U^{-1})$ (does that work in general or only for diagonal A?)
And then define my new A as the old 2D A with an extra dimension tacked on the end.