baseline
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Horldsence
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"""问题一独立参考解：大框 Sutherland–Hodgman 半平面裁剪 + 精确直径/MEC。
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与生产实现 ``mcm26b.geometry.localization.solve`` 的独立性：
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- 生产实现枚举"半平面边界直线两两交点"再筛可行点；
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- 本参考解从一个大方框出发，逐条半平面做 Sutherland–Hodgman 裁剪，
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是另一条构造路径，且这里的半平面系数、直径、最小包围圆都在本文件内
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独立编码（不复用生产实现的 ``halfplanes`` / ``diameter`` / ``mec``）。
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只要区域有界，大框（±1e7 m）裁剪得到的顶点即精确多边形；顶点触到大框
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边界则判无界。最小包围圆用"枚举所有 1/2/3 点支撑圆"的暴力算法，直径用
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顶点对暴力枚举，二者都是 O(m²)/O(m³) 的朴素精确算法，m 为多边形顶点数。
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"""
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from __future__ import annotations
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import math
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from time import perf_counter
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from project.common import Method
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METHOD = Method(
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key="reference",
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name="大框半平面裁剪 + 精确直径/MEC",
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kind="reference",
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note="独立于生产顶点枚举的精确参考解",
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)
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BOX_M = 1.0e7
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_PARALLEL_EPS = 1e-12
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def _planes(case: dict) -> list[tuple[float, float, float]]:
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"""独立重写 a*x+b*y<=c 的半平面构造（与题面角域定义一致）。"""
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planes = []
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for row in case["measurements"]:
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half = math.radians(row.get("half_angle_deg", 1.0))
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theta = math.radians(row["bearing_deg"] % 360.0)
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sx, sy = row["x"], row["y"]
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for angle, sign in ((theta - half, 1.0), (theta + half, -1.0)):
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a = sign * math.sin(angle)
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b = -sign * math.cos(angle)
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planes.append((a, b, a * sx + b * sy))
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# 前向约束：不允许把角域取成反向无限双楔
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a, b = -math.cos(theta), -math.sin(theta)
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planes.append((a, b, a * sx + b * sy))
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return planes
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def _clip(polygon, a, b, c, tol=1e-7):
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"""Sutherland–Hodgman：保留 a*x+b*y<=c 的一侧。"""
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if not polygon:
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return []
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output = []
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count = len(polygon)
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for index in range(count):
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p, q = polygon[index], polygon[(index + 1) % count]
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fp = a * p[0] + b * p[1] - c
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fq = a * q[0] + b * q[1] - c
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p_in, q_in = fp <= tol, fq <= tol
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if p_in:
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output.append(p)
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if p_in != q_in:
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ratio = fp / (fp - fq)
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output.append((p[0] + ratio * (q[0] - p[0]),
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p[1] + ratio * (q[1] - p[1])))
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deduped = []
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for point in output:
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if not deduped or math.dist(point, deduped[-1]) > 1e-9:
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deduped.append(point)
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if len(deduped) > 1 and math.dist(deduped[0], deduped[-1]) <= 1e-9:
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deduped.pop()
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return deduped
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def _brute_force_diameter(points):
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best = 0.0
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pair = (points[0], points[0])
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for i, pi in enumerate(points):
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for pj in points[i + 1:]:
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value = math.dist(pi, pj)
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if value > best:
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best, pair = value, (pi, pj)
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return best, pair
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def _circumcircle(a, b, c):
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ux, uy = b[0] - a[0], b[1] - a[1]
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vx, vy = c[0] - a[0], c[1] - a[1]
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det = 2.0 * (ux * vy - uy * vx)
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if abs(det) <= _PARALLEL_EPS * max(
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math.hypot(ux, uy) * math.hypot(vx, vy), 1e-300):
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return None
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uu, vv = ux * ux + uy * uy, vx * vx + vy * vy
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center = (a[0] + (uu * vy - vv * uy) / det,
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a[1] + (ux * vv - vx * uu) / det)
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return center, math.dist(center, a)
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def _brute_force_mec(points):
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"""枚举 1/2/3 点支撑圆的精确最小包围圆（朴素但无迭代）。"""
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xs = [p[0] for p in points]
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ys = [p[1] for p in points]
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best_c = ((min(xs) + max(xs)) / 2.0, (min(ys) + max(ys)) / 2.0)
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best_r = max(math.dist(best_c, p) for p in points)
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candidates: list[tuple[tuple[float, float], float]] = []
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for i, p in enumerate(points):
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for q in points[i + 1:]:
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candidates.append((((p[0] + q[0]) / 2.0, (p[1] + q[1]) / 2.0),
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math.dist(p, q) / 2.0))
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for i, p in enumerate(points):
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for j in range(i + 1, len(points)):
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for k in range(j + 1, len(points)):
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circle = _circumcircle(p, points[j], points[k])
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if circle is not None:
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candidates.append(circle)
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for center, radius in candidates:
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if radius >= best_r:
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continue
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if all(math.dist(center, p) <= radius + 1e-7 for p in points):
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best_c, best_r = center, radius
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return best_c, best_r
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def solve_case(case: dict) -> dict:
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start = perf_counter()
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planes = _planes(case)
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polygon = [(-BOX_M, -BOX_M), (BOX_M, -BOX_M), (BOX_M, BOX_M), (-BOX_M, BOX_M)]
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for a, b, c in planes:
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polygon = _clip(polygon, a, b, c)
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if not polygon:
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return _finish(_degenerate("empty"), start)
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boundary_tol = 1e-6 * BOX_M
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if any(abs(point[0]) >= BOX_M - boundary_tol
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or abs(point[1]) >= BOX_M - boundary_tol for point in polygon):
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return _finish(_degenerate("unbounded"), start)
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if len(polygon) == 1:
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summary = {"status": "point", "diameter_m": 0.0, "covers": True,
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"mec_radius_m": 0.0, "max_excess_m": 0.0,
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"vertex_count": 1}
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else:
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value, (a, b) = _brute_force_diameter(polygon)
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if len(polygon) == 2:
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summary = {"status": "segment", "diameter_m": value, "covers": True,
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"mec_radius_m": value / 2.0, "max_excess_m": 0.0,
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"vertex_count": 2}
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else:
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_mec_center, radius = _brute_force_mec(polygon)
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diameter_center = ((a[0] + b[0]) / 2.0, (a[1] + b[1]) / 2.0)
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max_distance = max(math.dist(diameter_center, p) for p in polygon)
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summary = {
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"status": "polygon",
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"diameter_m": value,
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"covers": bool(radius <= value / 2.0 + 1e-7),
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"mec_radius_m": radius,
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"max_excess_m": round(max_distance - value / 2.0, 9),
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"vertex_count": len(polygon),
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}
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return _finish(summary, start)
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def _degenerate(status: str) -> dict:
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return {"status": status, "diameter_m": None, "covers": None,
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"mec_radius_m": None, "max_excess_m": None, "vertex_count": 0}
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def _finish(summary: dict, start: float) -> dict:
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result = {
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"method": METHOD.key,
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"algorithm": METHOD.name,
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"resolution_m": 0.0,
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"diameter_infinite": summary["status"] == "unbounded",
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"outside_vertex_count": None,
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"runtime_ms": round((perf_counter() - start) * 1000.0, 4),
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}
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result.update(summary)
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return result
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