#include #include #include #include #include #include using namespace std; struct Point2D { double x, y; }; struct Circle { double x, y, r; }; struct Point3D { double x, y, z; int id; Point3D operator-(const Point3D& p) const { return {x - p.x, y - p.y, z - p.z, id}; } }; struct Face { int i, j, k; bool visible = true; vector pts; }; vector clip_polygon_halfplane(const vector& poly, double a, double b, double c) { vector res; int n = poly.size(); if (n == 0) return res; for (int i = 0; i < n; i++) { Point2D p1 = poly[i]; Point2D p2 = poly[(i + 1) % n]; double d1 = a * p1.x + b * p1.y + c; double d2 = a * p2.x + b * p2.y + c; if (d1 >= -1e-9) { res.push_back(p1); if (d2 < -1e-9) { double t = d1 / (d1 - d2); t = max(0.0, min(1.0, t)); res.push_back({p1.x + t * (p2.x - p1.x), p1.y + t * (p2.y - p1.y)}); } } else if (d2 >= -1e-9) { double t = d1 / (d1 - d2); t = max(0.0, min(1.0, t)); res.push_back({p1.x + t * (p2.x - p1.x), p1.y + t * (p2.y - p1.y)}); } } return res; } vector clip_to_rect(vector poly, double X, double Y) { poly = clip_polygon_halfplane(poly, 1.0, 0.0, 0.0); poly = clip_polygon_halfplane(poly, -1.0, 0.0, X); poly = clip_polygon_halfplane(poly, 0.0, 1.0, 0.0); poly = clip_polygon_halfplane(poly, 0.0, -1.0, Y); return poly; } double circle_polygon_area(double cx, double cy, double R, vector poly) { int n = poly.size(); if (n < 3) return 0.0; double area = 0.0; double R2 = R * R; for (int i = 0; i < n; i++) { Point2D p1 = poly[i]; Point2D p2 = poly[(i + 1) % n]; double x1 = p1.x - cx, y1 = p1.y - cy; double x2 = p2.x - cx, y2 = p2.y - cy; double dx = x2 - x1, dy = y2 - y1; double a = dx * dx + dy * dy; double b = 2.0 * (x1 * dx + y1 * dy); double c = x1 * x1 + y1 * y1 - R2; vector ts = {0.0, 1.0}; if (abs(a) > 1e-12) { double disc = b * b - 4 * a * c; if (disc > 0) { double sq = sqrt(disc); double t1 = (-b - sq) / (2 * a); double t2 = (-b + sq) / (2 * a); if (t1 > 0 && t1 < 1) ts.push_back(t1); if (t2 > 0 && t2 < 1) ts.push_back(t2); } } sort(ts.begin(), ts.end()); for (size_t j = 0; j < ts.size() - 1; j++) { double ta = ts[j], tb = ts[j + 1]; if (tb - ta < 1e-12) continue; Point2D sub_p1 = {x1 + ta * dx, y1 + ta * dy}; Point2D sub_p2 = {x1 + tb * dx, y1 + tb * dy}; double mx = (sub_p1.x + sub_p2.x) / 2.0; double my = (sub_p1.y + sub_p2.y) / 2.0; if (mx * mx + my * my <= R2 + 1e-9) { area += 0.5 * (sub_p1.x * sub_p2.y - sub_p1.y * sub_p2.x); } else { double ang1 = atan2(sub_p1.y, sub_p1.x); double ang2 = atan2(sub_p2.y, sub_p2.x); double dtheta = atan2(sin(ang2 - ang1), cos(ang2 - ang1)); area += 0.5 * R2 * dtheta; } } } return abs(area); } Point3D cross(const Point3D& a, const Point3D& b) { return { a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x }; } double dot(const Point3D& a, const Point3D& b) { return a.x * b.x + a.y * b.y + a.z * b.z; } double orient3d(const Point3D& a, const Point3D& b, const Point3D& c, const Point3D& d) { return dot(cross(b - a, c - a), d - a); } const double EPS = 1e-9; vector get_convex_hull_3d(vector pts) { int n = pts.size(); if (n < 4) return {}; int p0 = 0, p1 = 1, p2 = -1, p3 = -1; while (p1 < n && abs(pts[p1].x - pts[p0].x) < EPS && abs(pts[p1].y - pts[p0].y) < EPS && abs(pts[p1].z - pts[p0].z) < EPS) { p1++; } if (p1 == n) return {}; for (int i = p1 + 1; i < n; i++) { Point3D crs = cross(pts[p1] - pts[p0], pts[i] - pts[p0]); if (abs(crs.x) > EPS || abs(crs.y) > EPS || abs(crs.z) > EPS) { p2 = i; break; } } if (p2 == -1) return {}; for (int i = p2 + 1; i < n; i++) { if (abs(orient3d(pts[p0], pts[p1], pts[p2], pts[i])) > EPS) { p3 = i; break; } } if (p3 == -1) return {}; vector remaining; for (int i = 0; i < n; i++) { if (i != p0 && i != p1 && i != p2 && i != p3) { remaining.push_back(i); } } mt19937 rng(1337); shuffle(remaining.begin(), remaining.end(), rng); vector faces; auto add_face = [&](int a, int b, int c) { faces.push_back({a, b, c, true, {}}); }; if (orient3d(pts[p0], pts[p1], pts[p2], pts[p3]) > 0) { add_face(p0, p2, p1); add_face(p0, p1, p3); add_face(p1, p2, p3); add_face(p2, p0, p3); } else { add_face(p0, p1, p2); add_face(p0, p3, p1); add_face(p1, p3, p2); add_face(p2, p3, p0); } for (int p_idx : remaining) { for (auto& f : faces) { if (orient3d(pts[f.i], pts[f.j], pts[f.k], pts[p_idx]) > EPS) { f.pts.push_back(p_idx); break; } } } for (int p_idx : remaining) { vector vis_faces; for (int i = 0; i < (int)faces.size(); i++) { if (faces[i].visible && orient3d(pts[faces[i].i], pts[faces[i].j], pts[faces[i].k], pts[p_idx]) > EPS) { vis_faces.push_back(i); } } if (vis_faces.empty()) continue; map, int> edge_cnt; for (int f_idx : vis_faces) { const auto& f = faces[f_idx]; edge_cnt[{f.i, f.j}]++; edge_cnt[{f.j, f.k}]++; edge_cnt[{f.k, f.i}]++; } vector> horizon; for (int f_idx : vis_faces) { const auto& f = faces[f_idx]; pair edges[3] = {{f.i, f.j}, {f.j, f.k}, {f.k, f.i}}; for (auto& e : edges) { if (edge_cnt.find({e.second, e.first}) == edge_cnt.end()) { horizon.push_back(e); } } } vector orphaned; for (int f_idx : vis_faces) { faces[f_idx].visible = false; for (int pt_i : faces[f_idx].pts) { if (pt_i != p_idx) { orphaned.push_back(pt_i); } } } sort(orphaned.begin(), orphaned.end()); orphaned.erase(unique(orphaned.begin(), orphaned.end()), orphaned.end()); int new_start_idx = faces.size(); for (auto& e : horizon) { add_face(e.first, e.second, p_idx); } for (int pt_i : orphaned) { for (int i = new_start_idx; i < (int)faces.size(); i++) { if (orient3d(pts[faces[i].i], pts[faces[i].j], pts[faces[i].k], pts[pt_i]) > EPS) { faces[i].pts.push_back(pt_i); break; } } } } vector result; for (const auto& f : faces) { if (f.visible) { result.push_back(f); } } return result; } vector get_lower_hull(const vector& pts) { auto all_faces = get_convex_hull_3d(pts); vector lower_faces; for (const auto& f : all_faces) { Point3D norm = cross(pts[f.j] - pts[f.i], pts[f.k] - pts[f.i]); if (norm.z < -EPS) { lower_faces.push_back(f); } } return lower_faces; } double f(int X, int Y, vector C) { vector unique_C; for (auto c : C) { bool dup = false; for (auto& uc : unique_C) { if (abs(c.x - uc.x) < 1e-7 && abs(c.y - uc.y) < 1e-7) { uc.r = max(uc.r, c.r); dup = true; break; } } if (!dup) unique_C.push_back(c); } C = unique_C; int N = C.size(); if (N == 0) return 0.0; double min_x = 0, max_x = X, min_y = 0, max_y = Y; for (auto c : C) { min_x = min(min_x, c.x); max_x = max(max_x, c.x); min_y = min(min_y, c.y); max_y = max(max_y, c.y); } double CX = (min_x + max_x) / 2.0; double CY = (min_y + max_y) / 2.0; double M = max({max_x - min_x, max_y - min_y, (double)X, (double)Y, 100.0}) * 10.0; vector C_ext = C; C_ext.push_back({CX - M, CY - M, 0.0}); C_ext.push_back({CX + M, CY - M, 0.0}); C_ext.push_back({CX + M, CY + M, 0.0}); C_ext.push_back({CX - M, CY + M, 0.0}); vector pts; for (auto c : C_ext) { pts.push_back({c.x, c.y, c.x * c.x + c.y * c.y - c.r * c.r}); } vector lower = get_lower_hull(pts); vector> circle_to_centers(C_ext.size()); for (auto& face : lower) { Circle p_i = C_ext[face.i], p_j = C_ext[face.j], p_k = C_ext[face.k]; double A = 2 * (p_j.x - p_i.x); double B_coef = 2 * (p_j.y - p_i.y); double E = pts[face.j].z - pts[face.i].z; double C_coef = 2 * (p_k.x - p_i.x); double D_coef = 2 * (p_k.y - p_i.y); double F = pts[face.k].z - pts[face.i].z; double det = A * D_coef - B_coef * C_coef; if (abs(det) < 1e-12) continue; double cx = (E * D_coef - B_coef * F) / det; double cy = (A * F - E * C_coef) / det; Point2D center = {cx, cy}; circle_to_centers[face.i].push_back(center); circle_to_centers[face.j].push_back(center); circle_to_centers[face.k].push_back(center); } double area = 0.0; for (int i = 0; i < N; i++) { auto centers = circle_to_centers[i]; if (centers.size() < 3) continue; vector unique_centers; for (auto c : centers) { bool dup = false; for (auto u : unique_centers) { if (hypot(c.x - u.x, c.y - u.y) < 1e-6) { dup = true; break; } } if (!dup) unique_centers.push_back(c); } if (unique_centers.size() < 3) continue; double gx = 0, gy = 0; for (auto p : unique_centers) gx += p.x, gy += p.y; gx /= unique_centers.size(); gy /= unique_centers.size(); sort(unique_centers.begin(), unique_centers.end(), [&](const Point2D& a, const Point2D& b) { return atan2(a.y - gy, a.x - gx) < atan2(b.y - gy, b.x - gx); }); vector clipped_poly = clip_to_rect(unique_centers, X, Y); if (clipped_poly.size() >= 3) { area += circle_polygon_area(C[i].x, C[i].y, C[i].r, clipped_poly); } } return area; } int main() { int X, Y, N; if (!(cin >> X >> Y >> N)) return 0; vector v, h; mt19937 rnd(1337); uniform_real_distribution dist(-1e-9, 1e-9); for (int i = 0; i < N; i++) { double x, y, r; char d; cin >> x >> y >> r >> d; x += dist(rnd); y += dist(rnd); r += abs(dist(rnd)); (d == 'V' ? v : h).push_back({x, y, r}); } double ans = (double) X * Y; ans -= f(X, Y, v) * 0.5; ans -= f(X, Y, h) * 0.5; printf("%.20lf\n", ans); return 0; }