@antv/g2/src/data/utils/venn/circleintersection.ts

the Grammar of Graphics in Javascript

const SMALL = 1e-10;

/**
 * Returns the intersection area of a bunch of circles (where each circle
 * is an object having an x,y and radius property)
 */
export function intersectionArea(circles, stats?: any) {
  // Get all the intersection points of the circles
  const intersectionPoints = getIntersectionPoints(circles);

  // Filter out points that aren't included in all the circles
  const innerPoints = intersectionPoints.filter(function (p) {
    return containedInCircles(p, circles);
  });

  let arcArea = 0,
    polygonArea = 0,
    i;
  const arcs = [];
  // If we have intersection points that are within all the circles,
  // then figure out the area contained by them
  if (innerPoints.length > 1) {
    // Sort the points by angle from the center of the polygon, which lets
    // us just iterate over points to get the edges
    const center = getCenter(innerPoints);
    for (i = 0; i < innerPoints.length; ++i) {
      const p = innerPoints[i];
      p.angle = Math.atan2(p.x - center.x, p.y - center.y);
    }
    innerPoints.sort(function (a, b) {
      return b.angle - a.angle;
    });

    // Iterate over all points, get arc between the points
    // and update the areas
    let p2 = innerPoints[innerPoints.length - 1];
    for (i = 0; i < innerPoints.length; ++i) {
      const p1 = innerPoints[i];

      // Polygon area updates easily ...
      polygonArea += (p2.x + p1.x) * (p1.y - p2.y);

      // Updating the arc area is a little more involved
      const midPoint = { x: (p1.x + p2.x) / 2, y: (p1.y + p2.y) / 2 };
      let arc = null;

      for (let j = 0; j < p1.parentIndex.length; ++j) {
        if (p2.parentIndex.indexOf(p1.parentIndex[j]) > -1) {
          // Figure out the angle halfway between the two points
          // on the current circle
          const circle = circles[p1.parentIndex[j]],
            a1 = Math.atan2(p1.x - circle.x, p1.y - circle.y),
            a2 = Math.atan2(p2.x - circle.x, p2.y - circle.y);

          let angleDiff = a2 - a1;
          if (angleDiff < 0) {
            angleDiff += 2 * Math.PI;
          }

          // and use that angle to figure out the width of the
          // arc
          const a = a2 - angleDiff / 2;
          let width = distance(midPoint, {
            x: circle.x + circle.radius * Math.sin(a),
            y: circle.y + circle.radius * Math.cos(a),
          });

          // Clamp the width to the largest is can actually be
          // (sometimes slightly overflows because of FP errors)
          if (width > circle.radius * 2) {
            width = circle.radius * 2;
          }

          // Pick the circle whose arc has the smallest width
          if (arc === null || arc.width > width) {
            arc = { circle: circle, width: width, p1: p1, p2: p2 };
          }
        }
      }

      if (arc !== null) {
        arcs.push(arc);
        arcArea += circleArea(arc.circle.radius, arc.width);
        p2 = p1;
      }
    }
  } else {
    // No intersection points, is either disjoint - or is completely
    // overlapped. figure out which by examining the smallest circle
    let smallest = circles[0];
    for (i = 1; i < circles.length; ++i) {
      if (circles[i].radius < smallest.radius) {
        smallest = circles[i];
      }
    }

    // Make sure the smallest circle is completely contained in all
    // the other circles
    let disjoint = false;
    for (i = 0; i < circles.length; ++i) {
      if (
        distance(circles[i], smallest) >
        Math.abs(smallest.radius - circles[i].radius)
      ) {
        disjoint = true;
        break;
      }
    }

    if (disjoint) {
      arcArea = polygonArea = 0;
    } else {
      arcArea = smallest.radius * smallest.radius * Math.PI;
      arcs.push({
        circle: smallest,
        p1: { x: smallest.x, y: smallest.y + smallest.radius },
        p2: { x: smallest.x - SMALL, y: smallest.y + smallest.radius },
        width: smallest.radius * 2,
      });
    }
  }

  polygonArea /= 2;
  if (stats) {
    stats.area = arcArea + polygonArea;
    stats.arcArea = arcArea;
    stats.polygonArea = polygonArea;
    stats.arcs = arcs;
    stats.innerPoints = innerPoints;
    stats.intersectionPoints = intersectionPoints;
  }

  return arcArea + polygonArea;
}

/**
 * Returns whether a point is contained by all of a list of circles
 */
export function containedInCircles(point, circles) {
  for (let i = 0; i < circles.length; ++i) {
    if (distance(point, circles[i]) > circles[i].radius + SMALL) {
      return false;
    }
  }
  return true;
}

/** Gets all intersection points between a bunch of circles */
function getIntersectionPoints(circles) {
  const ret = [];
  for (let i = 0; i < circles.length; ++i) {
    for (let j = i + 1; j < circles.length; ++j) {
      const intersect = circleCircleIntersection(circles[i], circles[j]);
      for (let k = 0; k < intersect.length; ++k) {
        const p: any = intersect[k];
        p.parentIndex = [i, j];
        ret.push(p);
      }
    }
  }
  return ret;
}

/** Circular segment area calculation. See http://mathworld.wolfram.com/CircularSegment.html */
export function circleArea(r, width) {
  return (
    r * r * Math.acos(1 - width / r) -
    (r - width) * Math.sqrt(width * (2 * r - width))
  );
}

/** Euclidean distance between two points */
export function distance(p1, p2) {
  return Math.sqrt(
    (p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y),
  );
}

/** Returns the overlap area of two circles of radius r1 and r2 - that
have their centers separated by distance d. Simpler faster
circle intersection for only two circles */
export function circleOverlap(r1, r2, d) {
  // no overlap
  if (d >= r1 + r2) {
    return 0;
  }

  // Completely overlapped
  if (d <= Math.abs(r1 - r2)) {
    return Math.PI * Math.min(r1, r2) * Math.min(r1, r2);
  }

  const w1 = r1 - (d * d - r2 * r2 + r1 * r1) / (2 * d),
    w2 = r2 - (d * d - r1 * r1 + r2 * r2) / (2 * d);
  return circleArea(r1, w1) + circleArea(r2, w2);
}

/** Given two circles (containing a x/y/radius attributes),
returns the intersecting points if possible.
note: doesn't handle cases where there are infinitely many
intersection points (circles are equivalent):, or only one intersection point*/
export function circleCircleIntersection(p1, p2) {
  const d = distance(p1, p2),
    r1 = p1.radius,
    r2 = p2.radius;

  // If to far away, or self contained - can't be done
  if (d >= r1 + r2 || d <= Math.abs(r1 - r2)) {
    return [];
  }

  const a = (r1 * r1 - r2 * r2 + d * d) / (2 * d),
    h = Math.sqrt(r1 * r1 - a * a),
    x0 = p1.x + (a * (p2.x - p1.x)) / d,
    y0 = p1.y + (a * (p2.y - p1.y)) / d,
    rx = -(p2.y - p1.y) * (h / d),
    ry = -(p2.x - p1.x) * (h / d);

  return [
    { x: x0 + rx, y: y0 - ry },
    { x: x0 - rx, y: y0 + ry },
  ];
}

/** Returns the center of a bunch of points */
export function getCenter(points) {
  const center = { x: 0, y: 0 };
  for (let i = 0; i < points.length; ++i) {
    center.x += points[i].x;
    center.y += points[i].y;
  }
  center.x /= points.length;
  center.y /= points.length;
  return center;
}