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var 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) {
// get all the intersection points of the circles
var intersectionPoints = getIntersectionPoints(circles);
// filter out points that aren't included in all the circles
var innerPoints = intersectionPoints.filter(function (p) {
return containedInCircles(p, circles);
});
var arcArea = 0, polygonArea = 0, arcs = [], i;
// 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
var center = getCenter(innerPoints);
for (i = 0; i < innerPoints.length; ++i ) {
var 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
var p2 = innerPoints[innerPoints.length - 1];
for (i = 0; i < innerPoints.length; ++i) {
var 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
var midPoint = {x : (p1.x + p2.x) / 2,
y : (p1.y + p2.y) / 2},
arc = null;
for (var 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
var 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);
var angleDiff = (a2 - a1);
if (angleDiff < 0) {
angleDiff += 2*Math.PI;
}
// and use that angle to figure out the width of the
// arc
var a = a2 - angleDiff/2,
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
var 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
var 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 (var 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) {
var ret = [];
for (var i = 0; i < circles.length; ++i) {
for (var j = i + 1; j < circles.length; ++j) {
var intersect = circleCircleIntersection(circles[i],
circles[j]);
for (var k = 0; k < intersect.length; ++k) {
var p = 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);
}
var 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) {
var 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 [];
}
var 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) {
var center = {x: 0, y: 0};
for (var 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;
}