import java.io.PrintWriter; import java.text.DecimalFormat; import java.util.ArrayList; import java.util.Arrays; import java.util.Random; import java.util.Scanner; public class CupCounterbalancing { static PrintWriter out; static void solve(Scanner fs, PrintWriter out, int tt) throws InterruptedException { int n = fs.nextInt(), r = fs.nextInt(), max = fs.nextInt(); Vec[] points = new Vec[n]; for (int i = 0; i < n; i++) { points[i] = new Vec(fs.nextInt(), fs.nextInt()); } double ans = solveNaive1e8(points, r, max); out.println("Case #" + tt + ": " + ans); out.flush(); } static double solveNaive1e8(Vec[] points, int r, int max) { long time = System.currentTimeMillis(); Vec test = new Vec(1, 0).rotate(random.nextDouble()).scale(max / Math.sqrt(1e8)); double res = solveNaive(points, r, max, test); return res; } static double solveNaive1e8Multithreaded( Vec[] points, int r, int max, boolean print, int nThreads) throws InterruptedException { long time = System.currentTimeMillis(); Thread[] threads = new Thread[nThreads]; final double[] answers = new double[nThreads]; for (int i = 0; i < threads.length; i++) { final int ii = i; threads[i] = new Thread(null, null, "T") { public void run() { answers[ii] = solveNaive1e8(points, r, max); } }; threads[i].start(); } for (Thread t : threads) { t.join(); } double average = 0; for (double d : answers) { average += d; } return average / nThreads; } static Random random = new Random(); static double solveNaive(Vec[] points, int r, int max, Vec test) { test.x = Math.abs(test.x); test.y = Math.abs(test.y); Vec start = new Vec(0, 0); int row = 0; Vec rotated = test.rot90(); Seg[] segs = new Seg[points.length]; for (int i = 0; i < points.length; i++) { segs[i] = new Seg(points[i], points[(i + 1) % points.length]); } int worksCount = 0; int allCount = 0; while (true) { Vec rowStart = start.add(rotated.scale(row)); if (rowStart.y > max) break; Vec at = rowStart; while (at.x <= max && at.y <= max) { if (at.x >= 0 && at.y >= 0) { if (worksNaive(segs, at, r)) { worksCount++; } allCount++; } at = at.add(test); } row++; } row = -1; while (true) { Vec rowStart = start.add(rotated.scale(row)); Vec at = rowStart; boolean hit = false; while (at.x <= max && at.y <= max) { if (at.x >= 0 && at.y >= 0) { hit = true; if (worksNaive(segs, at, r)) { worksCount++; } allCount++; } at = at.add(test); } row--; if (!hit) { break; } } return worksCount / (double) allCount; } static boolean worksNaive(Seg[] segs, Vec query, long r) { ArrayList intersections = new ArrayList<>(); Circle c = new Circle(query, r); for (Seg s : segs) { Vec[] ans = c.intersectSeg(s); if (ans != null) { for (Vec v : ans) { intersections.add(v); } } } double[] angles = new double[intersections.size()]; for (int i = 0; i < angles.length; i++) angles[i] = intersections.get(i).sub(query).angle(); Arrays.sort(angles); if (angles.length <= 2) { return false; } for (int i = 1; i < angles.length; i++) { if (angles[i] - angles[i - 1] >= Math.PI) { return false; } } if (angles[0] - (angles[angles.length - 1] - Math.PI * 2) > Math.PI) { return false; } return true; } static class Vec { static final double EPS = 1e-6; double x, y; public Vec(double x, double y) { this.x = x; this.y = y; } public Vec add(Vec o) { return new Vec(x + o.x, y + o.y); } public Vec sub(Vec o) { return new Vec(x - o.x, y - o.y); } public Vec scale(double s) { return new Vec(x * s, y * s); } public double dot(Vec o) { return x * o.x + y * o.y; } public double cross(Vec o) { return x * o.y - y * o.x; } public double mag2() { return dot(this); } public double mag() { return Math.sqrt(mag2()); } public Vec unit() { return scale(1 / mag()); } public Vec rot90() { return new Vec(-y, x); } public Vec rot270() { return new Vec(y, -x); } public Vec rotate(double theta) { double PI = Math.PI; double newX = x * Math.cos(theta) + y * Math.cos(PI / 2 + theta); double newY = x * Math.sin(theta) + y * Math.sin(PI / 2 + theta); return new Vec(newX, newY); } // Angle between 0 and 2PI public double angle() { return (Math.atan2(y, x) + 2 * Math.PI) % (2 * Math.PI); } public String toString() { DecimalFormat df = new DecimalFormat("#.##"); return "(" + df.format(x) + ", " + df.format(y) + ")"; } static boolean eq(double a, double b) { return Math.abs(a - b) < EPS; } static boolean leq(double a, double b) { return a - EPS < b; } static boolean geq(double a, double b) { return a + EPS > b; } public boolean equals(Object oo) { Vec o = (Vec) oo; return eq(x, o.x) && eq(y, o.y); } } static class Seg { Vec from, to, dir; public Seg(Vec from, Vec to) { this.from = from; this.to = to; dir = to.sub(from); } // Line-line intersection public Vec lineIntersect(Seg o) { double det = o.dir.x * dir.y - dir.x * o.dir.y; if (Vec.eq(det, 0)) return null; double dist = (o.dir.x * (o.from.y - from.y) - o.dir.y * (o.from.x - from.x)) / det; return from.add(dir.scale(dist)); } public boolean containsPoint(Vec o) { double distFromLine = dir.unit().cross(o.sub(from)); if (!Vec.eq(distFromLine, 0)) { return false; } return Vec.eq(dir.mag(), from.sub(o).mag() + to.sub(o).mag()); } // Seg-seg intersection public Vec segIntersection(Seg o) { Vec intersect = lineIntersect(o); if (intersect == null) { return null; } return containsPoint(intersect) && o.containsPoint(intersect) ? intersect : null; } // Returns 1 if above, 0 if on, -1 if below. public int side(Vec o) { Vec oDir = o.sub(from); double distFromLine = dir.unit().cross(oDir); return Vec.eq(distFromLine, 0) ? 0 : (int)Math.signum(distFromLine); } public boolean intersects(Seg o) { return side(o.from) != side(o.to) && o.side(from) != o.side(to); } public Vec getClosestTo(Vec o) { double percentThere = o.sub(from).dot(dir) / dir.mag2(); return from.add(dir.scale(Math.max(0, Math.min(1, percentThere)))); } public Vec projectToLine(Vec o) { return dir.scale(o.sub(from).dot(dir) / dir.mag2()).add(from); } // Returns the shortest segment from the x-axis to the point to the y axis. // toContain must be a point with positive x and y coordinates. public static Seg getShortestSegFromAxesContainingQ1Point(Vec toContain) { double slope = -Math.pow(toContain.y / toContain.x, 1.0 / 3); double b = toContain.y - toContain.x * slope; double xInt = -b / slope; return new Seg(new Vec(0, b), new Vec(xInt, 0)); } public String toString() { return from + " -> " + to; } } static class Circle { Vec c; double r; public Circle(Vec c, double r) { this.c = c; this.r = r; } public boolean contains(Vec v) { return c.sub(v).mag2() - Vec.EPS * Vec.EPS <= r * r; } // When standing at this circle, returns right tangent, then left tangent. public Vec[] intersect(Circle o) { if (c.equals(o.c)) { return null; } Vec dir = o.c.sub(c); double d2 = dir.mag2(), d = Math.sqrt(d2); if (r + o.r < d || r + d < o.r || o.r + d < r) { return null; } if (Vec.eq(r + o.r, d) || Vec.eq(o.r + d, r)) { return new Vec[] {c.add(dir.scale(r / d))}; } if (Vec.eq(r + d, o.r)) { return new Vec[] {c.sub(dir.scale(r / d))}; } double d1 = (r * r + d2 - o.r * o.r) / (2 * d); double h = Math.sqrt(r * r - d1 * d1); Vec unitDir = dir.unit(); Vec rInt = c.add(unitDir.scale(d1).add(unitDir.rot270().scale(h))); Vec lInt = c.add(unitDir.scale(d1).add(unitDir.rot90().scale(h))); return new Vec[] {rInt, lInt}; } public double intersectionArea(Circle o) { double d = o.c.sub(c).mag(); if (r + o.r < d) { return 0; } double minR = Math.min(r, o.r), maxR = Math.max(r, o.r), pi = Math.PI; if (Vec.leq(d + minR, maxR)) { return pi * minR * minR; } double theta1 = 2 * Math.acos((r * r + d * d - o.r * o.r) / (2 * r * d)); double theta2 = 2 * Math.acos((o.r * o.r + d * d - r * r) / (2 * o.r * d)); double part1Area = theta1 / 2 * r * r; double part2Area = theta2 / 2 * o.r * o.r; double tri1 = r * r * Math.sin(theta1) / 2; double tri2 = o.r * o.r * Math.sin(theta2) / 2; return part1Area + part2Area - tri1 - tri2; } // Returns right tangent, then left tangent from perspective of the point public Vec[] getTangentPoints(Vec p) { if (contains(p)) { return null; } double d2 = c.sub(p).mag2(); return new Circle(p, Math.sqrt(d2 - r * r)).intersect(this); } // Line going from my left to his right, then my right to his left lines go // from me to him public Seg[] internalTangentLines(Circle o) { Vec[] tangentPoints = new Circle(c, r + o.r).getTangentPoints(o.c); Vec offset1 = tangentPoints[0].sub(o.c).rot90().unit().scale(o.r); Vec offset2 = tangentPoints[1].sub(o.c).rot270().unit().scale(o.r); return new Seg[] { new Seg(tangentPoints[0].add(offset1), o.c.add(offset1)), new Seg(tangentPoints[1].add(offset2), o.c.add(offset2)) }; } // Right external tangent, then left external tangent, from my perspective // lines go from me to him public Seg[] externalTangentLines(Circle o) { if (o.r > r) { Seg[] oAnswer = o.externalTangentLines(this); return new Seg[] { new Seg(oAnswer[1].to, oAnswer[1].from), new Seg(oAnswer[0].to, oAnswer[0].from) }; } Vec[] tangentPoints = new Circle(c, r - o.r).getTangentPoints(o.c); Vec offset1 = tangentPoints[0].sub(o.c).rot270().unit().scale(o.r); Vec offset2 = tangentPoints[1].sub(o.c).rot90().unit().scale(o.r); return new Seg[] { new Seg(tangentPoints[1].add(offset2), o.c.add(offset2)), new Seg(tangentPoints[0].add(offset1), o.c.add(offset1)) }; } // Line (not line segment)-circle intersection in the order of line.dir public Vec[] intersectLine(Seg line) { Vec closest = line.projectToLine(c); double d2 = closest.sub(c).mag2(); if (d2 > r * r) { return null; } double l = Math.sqrt(r * r - d2); if (Vec.eq(l, 0)) { return new Vec[] {closest}; } Vec lVec = line.dir.unit().scale(l); return new Vec[] {closest.sub(lVec), closest.add(lVec)}; } // Line segment-circle intersection public Vec[] intersectSeg(Seg seg) { Vec[] lineIntersections = intersectLine(seg); if (lineIntersections == null) return null; ArrayList contained = new ArrayList<>(); for (Vec v : lineIntersections) { if (seg.containsPoint(v)) { contained.add(v); } } if (contained.isEmpty()) { return null; } return contained.toArray(new Vec[contained.size()]); } public String toString() { DecimalFormat df = new DecimalFormat(); return "center: " + c + ", r: " + df.format(r); } } public static void main(String[] args) throws InterruptedException { Scanner fs = new Scanner(System.in); out = new PrintWriter(System.out); int T = fs.nextInt(); for (int t = 1; t <= T; t++) { solve(fs, out, t); } fs.close(); out.close(); } }