circle-parameters.md

Circle Parameters

Parameter	Formula	Getting R from Parameter	Getting $\alpha$ from Parameter
circumference	$2{\pi}R$	$\frac{x}{2\pi}$	-
area	${\pi}R^2$	$\sqrt{\frac{x}{\pi}}$	-
arc length	${\alpha}R$	$\frac{x}{\alpha}$	$\frac{x}{R}$
chord length	$2R \sin(\frac{\alpha}{2})$	$\frac{x}{2 \sin(\frac{\alpha}{2})}$	$2 \arcsin(\frac{x}{2R})$
sector area	$\frac{\alpha}{2} R^2$	$\sqrt{\frac{2x}{\alpha}}$	$2 \frac{x}{R^2}$
segment area	$\frac{R^2}{2}(\alpha - \sin(\alpha))$	$\sqrt{2\frac{x}{\alpha - \sin(\alpha)}}$	brute force method
perpendicular distance from circle center to chord	$R \cos(\frac{\alpha}{2})$	$\frac{x}{\cos(\frac{\alpha}{2})}$	$2 \arccos(\frac{x}{R})$
perpendicular distance from circle edge to chord	$R - R \cdot \cos\left(\frac{\alpha}{2}\right)$	$-x \cdot \frac{1}{\cos\left(\frac{\alpha}{2}\right)-1}$	$2 \arccos\left(\frac{-x+R}{R}\right)$

$\alpha - \sin(\alpha)$

Operation $\alpha - \sin(\alpha)$ cannot be simply reversed, some technique must be employeed.
The function below uses the Newton-Raphson method:

$x = \alpha - \sin(\alpha) \implies \text{calcAlpha}(x) \approx \alpha$

function calcAlpha(x) {
  const f = alpha => alpha - Math.sin(alpha) - x;
  const df = alpha => 1 - Math.cos(alpha);
  
  const tolerance = 1e-10;
  let alpha = x;
  let error = Infinity;

  while (error > tolerance) {
    const nextAlpha = alpha - f(alpha) / df(alpha);
    error = Math.abs(nextAlpha - alpha);
    alpha = nextAlpha;
  }

  return alpha;
}