Synchrotron Geometry

A synchrotron’s beam trajectory can be characterized by the perimeter of a rounded polygon with $n$ bends of radius $\rho$ and drifts of length $L$:

Rounded n-gon

Our perimeter $C$ is defined by a characteristic radius $R$:

\[C = 2\pi R = 2\pi \rho + nL\]

The drift length $L$ is therefore:

\[L = 2\pi\frac{(R-\rho)}{n}\]

Our sector unit is defined by angle $\alpha$:

\[\alpha = \frac{2\pi}{n}\]

The rounded corner can be defined by a right kite of sides $\rho$ and $b$:

Corner Details

The interior angle of our n-gon is define by $\pi-\alpha$ so using law of cosines and law of sines respectively:

\[s^2 = 2\rho^2(1-\cos\alpha)\] \[b = s\frac{\sin\alpha/2}{\sin\alpha}\]

We can then define a minor and major spoke length $h$ and $g$:

Bisector

\[h = \frac{b+\frac{L}{2}}{\tan\frac{\alpha}{2}}\]

The subsector drift angle is accordingly defined by:

\[\gamma = 2\arctan\left(\frac{L}{2h}\right)\]

And the subsector bend angle is defined by:

\[\beta = \alpha - \gamma\]

Our relative sector coordinate $\psi$ is defined by the absolute polar angle $\phi$ given by:

\[\psi = \mod\left(\phi, \alpha\right)\]

Within the drift sector, our relative coordinate $\theta$ is defined by:

\[\theta = \psi-\frac{\gamma}{2}\]

Drift Polar Subsection

\[\boxed{r(\psi < \gamma) = \frac{h}{\cos\theta}}\]

For the bend sector we define a different offset by:

\[\varphi = \psi - \frac{\beta}{2}-\gamma\]

Bend Polar Subsection

Where:

\[\delta = \frac{\alpha - \beta}{2}\]

and:

\[k = \rho\frac{\sin\delta}{\sin\beta/2}\]

Therefore from law of cosines:

\[\boxed{r(\psi > \gamma) = k\cos\varphi \pm \sqrt{k^2\cos^2\varphi-k^2+\rho^2}}\]

Then we get:

Generalized Geometry