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Svg arc, how to define sweeps and large arc flags given by the start end and through the point

In svg, I want to build a function that returns all the parameters for the "arc element" in the path-d attribute. Given the start point, end point and through point. (3 points on a fuzzy line correspond to the definition in a circle). I'm only interested in circle-arcs (rx == ry).

I can calculate the center and radius quite easily. But I'm struggling with two flags, is there a clear definition of how to set these flags by comparing the topology of the three points? how are angles or distances to each other?)

I know the meaning of the flags smallest vs largest arc, clockwise counterclockwise for the sweep flag.

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, SVG, , (S), (V) (E) .

:

  • | & angle; SVE | > & pi;/2, = 0
  • | & angle; SVE | < & pi;/2, = 1
    • , , V, .
    • "" (.. ), .

  • & angle; ESV > 0, = 0
  • & angle; ESV < 0, = 1
    • , S-to-V V.
    • S E, V , .

:

  • , : S-V-E E-S-V .
  • - & pi; & pi;, -180 & deg; 180 & deg;.
  • , .

- . , OP, :

const lgArcFl = (S,V,E) => Math.abs(angle(S,V,E)) > pi/2 ? 0 : 1;
const sweepFl = (S,V,E) =>          angle(E,S,V)  > 0    ? 0 : 1;

, " ", 3x , (S), (V) (E) . 4 , 0,0, 0,1, 1,0 1,1. , . . 3 , .. , 3 - , .

, (5 2016 ) Chrome, Opera Safari, Firefox. ( Explorer Edge, .) , , ES6/ES2015. " BabelJS/ES2015" , . , , , .

const lgArcFl = (S,V,E) => Math.abs(angle(S,V,E)) > pi/2 ? 0 : 1;
const sweepFl = (S,V,E) =>          angle(E,S,V)  > 0    ? 0 : 1;

const angle = ([a,b],[c,d],[e,f]) => (Math.atan2(f-d,e-c)-Math.atan2(b-d,a-c)+3*pi)%(2*pi)-pi;
const qs = sel => document.querySelector(sel), pi = Math.PI, pts = [];
const radius = ([a,b],[c,d],[e,f]) => {
  const g=c-a,h=2*(c-e)/g,i=d-b,j=c*c+d*d,k=j-a*a-b*b,l=(j-e*e-f*f-h*k/2)/(2*(d-f)-h*i);
  return Math.hypot(a+(i*l-k/2)/g,b-l);
};
const mkArc = (arc, [sx, sy], [ex, ey], r, lg, sw) => arc.setAttribute('d',
  `M ${sx} ${sy} A ${r} ${r} 0 ${lg} ${sw} ${ex} ${ey}`);
const calcArcs = (S,V,E) => {
  const args = [S, E, radius(S,V,E)];
  [[0,0],[0,1],[1,0],[1,1]].forEach(([lg,sw]) => mkArc(qs(`#arc${lg}${sw}`), ...args, lg, sw));
  mkArc(qs(`#arc`), ...args, lgArcFl(S,V,E), sweepFl(S,V,E));
};
let ptNum = 0;
qs('svg').addEventListener('click', evt => {
  const x = evt.x - 10, y = evt.y - 10;
  pts[ptNum] = [x, y];
  qs('#pt' + ptNum).setAttribute('transform', `translate(${x},${y})`);
  if (ptNum++ === 2) {
    calcArcs(...pts);
    ptNum = 0;
  }
});
text {
  font-family: courier;
  font-size: 18px;
  fill: black;
  stroke: none;
  transform: translate(-5px,5px);
}
#arc00, #arc01 {
  stroke: red;
}
#arc10, #arc11 {
  stroke: blue;
}
#arc00, #arc10 {
  stroke-width: 4;
  stroke-dasharray: 5,5;
}
#arc01, #arc11 {
  stroke-width: 2;
}
rect {
  fill: none;
  stroke: black;
  height: 200px;
  width: 600px;
}
<svg height="200" width="600">
  <defs>
    <path id="arc" />
    <circle id="circ" cx="0" cy="0" r="10" />
  </defs>
  <g fill="none">
    <use xlink:href="#arc" stroke="black" stroke-width="12" />
    <use xlink:href="#arc" stroke="#ff8"  stroke-width="10" />
    <path id="arc00" />
    <path id="arc01" />
    <path id="arc10" />
    <path id="arc11" />
  </g>
  <g fill="#ff8" stroke="black">
    <g id="pt0" transform="translate(-99,0)"><use xlink:href="#circ" /><text>S</text></g>
    <g id="pt1" transform="translate(-99,0)"><use xlink:href="#circ" /><text>V</text></g>
    <g id="pt2" transform="translate(-99,0)"><use xlink:href="#circ" /><text>E</text></g>
  </g>
  <rect />
</svg>
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svg d. .

//---x1,y1 and x2,y2 are the two end points---
    function polarToCartesian(centerX, centerY,radius, angleInDegrees)
    {
        var angleInRadians = (angleInDegrees) * Math.PI / 180.0;

        return {
        x: centerX + (radius * Math.cos(angleInRadians)),
        y: centerY + (radius * Math.sin(angleInRadians))
        };
    }
    var startAngle = 180/Math.PI*Math.atan2(y1-cy, x1-cx);
    var endAngle =  180/Math.PI*Math.atan2(y2-cy, x2-cx);

    StartPnt = polarToCartesian(cx, cy, radius, startAngle);
    EndPnt = polarToCartesian(cx, cy,  radius, endAngle);
    ArcSweep = endAngle - startAngle <= 180 ? "0" : "1";

    var d = [
    "M", StartPnt.x, StartPnt.y,
    "A", radius, radius, 0, ArcSweep, 0, EndPnt.x, EndPnt.y
    ].join(" ");
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