結果
| 問題 |
No.2564 衝突予測
|
| コンテスト | |
| ユーザー |
 Kemty
|
| 提出日時 | 2023-12-02 15:15:06 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 40,181 bytes |
| コンパイル時間 | 303 ms |
| コンパイル使用メモリ | 86,400 KB |
| 実行使用メモリ | 93,652 KB |
| 最終ジャッジ日時 | 2024-09-26 18:20:06 |
| 合計ジャッジ時間 | 12,220 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 6 WA * 3 |
ソースコード
from collections import defaultdict, deque, Counter
import copy
from itertools import combinations, permutations, product, accumulate, groupby, chain
from heapq import heapify, heappop, heappush
import math
import bisect
from pprint import pprint
from random import randint
import sys
# sys.setrecursionlimit(200000)
input = lambda: sys.stdin.readline().rstrip('\n')
inf = float('inf')
mod1 = 10**9+7
mod2 = 998244353
def ceil_div(x, y): return -(-x//y)
#################################################
from functools import cmp_to_key
from math import atan2, cos, sin, acos, asin, sqrt, degrees, pi
inf = float('inf')
EPS = 1e-10
def sgn(a):
return -1 if a < -EPS else (1 if a > EPS else 0)
class Point:
def __init__(self, *point) -> None:
if len(point) == 0:
self.x = 0
self.y = 0
elif len(point) == 1:
self.x = point[0][0]
self.y = point[0][1]
else:
self.x = point[0]
self.y = point[1]
def __getitem__(self, i):
if i == 0: return self.x
elif i == 1: return self.y
raise IndexError
def __setitem__(self, i, value):
if i == 0: self.x = value
elif i == 1: self.y = value
raise IndexError
def __neg__(self):
return Point(-self.x, -self.y)
def __add__(self, other):
return Point(self.x+other.x, self.y+other.y)
def __sub__(self, other):
return Point(self.x-other.x, self.y-other.y)
def __mul__(self, other):
if type(other) is Point: # 複素数としての積
return Point(self.x*other.x - self.y*other.y, self.x*other.y + self.y*other.x)
return Point(self.x*other, self.y*other) # スカラー倍
def __rmul__(self, other):
if type(other) is Point: # 複素数としての積
return Point(other.x*self.x - other.y*self.y, other.x*self.y + other.y*self.x)
return Point(self.x*other, self.y*other) # スカラー倍
def __truediv__(self, other):
if type(other) is Point: # 複素数としての除算
return self*other.conjugation()/other.length_sq()
return Point(self.x/other, self.y/other) # スカラー除算
def __floordiv__(self, other):
if type(other) is Point: # 複素数としての除算、otherのノルムで割らない
return self*other.conjugation()
return Point(self.x//other, self.y//other) # スカラー除算
def __iadd__(self, other):
self.x += other.x
self.y += other.y
return self
def __isub__(self, other):
self.x -= other.x
self.y -= other.y
return self
def __imul__(self, other):
if type(other) is Point: # 複素数としての積
a, b, c, d = self.x, self.y, other.x, other.y
self.x = a*c - b*d
self.y = a*d + b*c
else: # スカラー倍
self.x *= other
self.y *= other
return self
def __itruediv__(self, other):
if type(other) is Point: # 複素数としての除算
self *= other.conjugation()/other.length_sq()
else: # スカラー除算
self.x /= other
self.y /= other
return self
def __ifloordiv__(self, other):
if type(other) is Point: # 複素数としての除算、otherのノルム^2で割らない
self *= other.conjugation()
else: # スカラー除算
self.x //= other
self.y //= other
return self
def __eq__(self, other) -> bool:
return sgn(self.x-other.x) == sgn(self.y-other.y) == 0
def __lt__(self, other) -> bool:
if sgn(self.x-other.x) != 0:
return sgn(self.x-other.x) < 0
return sgn(self.y-other.y) < 0
def __repr__(self) -> str:
return (self.x, self.y).__repr__()
def __abs__(self):
return self.length()
def length_sq(self):
return self.x*self.x + self.y*self.y
def length(self):
return sqrt(self.length_sq())
def dot(self, other):
return self.x*other.x + self.y*other.y
def cross(self, other):
return self.x*other.y - self.y*other.x
def is_zero(self):
return sgn(self.x) == sgn(self.y) == 0
def normalized(self):
return self/self.length()
def normal_vector(self):
return Point(-self.y, self.x)
def normal_unit_vector(self):
tmp = self.normalized()
return Point(-tmp.y, tmp.x)
def rotate(self, arg):
cs = cos(arg); sn = sin(arg)
return Point(self.x*cs - self.y*sn, self.x*sn + self.y*cs)
def angle(self):
return atan2(self.y, self.x)
def is_same_angle(self, other): # selfとotherの偏角は等しいか
return Point.ccw(Point(), self, other) in (0, 2)
def is_same_slope(self, other): # selfとotherの((0, 0)と結んだ直線での)傾きは等しいか
return sgn(self.cross(other)) == 0
def conjugation(self): # 共役な複素数
return Point(self.x, -self.y)
def print(self):
print(f"{self.x: .9f} {self.y: .9f}")
def distance_from_Point_sq(self, other):
if type(other) is not Point:
raise ValueError
return (self.x-other.x)*(self.x-other.x) + (self.y-other.y)*(self.y-other.y)
def distance_from_Point(self, other):
if type(other) is not Point:
raise ValueError
return sqrt(self.distance_from_Point_sq(other))
def distance_from_Line(self, other):
if type(other) is not Line:
raise ValueError
return abs(other.vec().cross(self-other.begin))/other.vec().length()
def distance_from_Ray(self, other):
if type(other) is not Ray:
raise ValueError
if Point.angle_type(self, other.begin, other.end) == 2:
return self.distance_from_Point(other.begin)
return abs(other.vec().cross(self-other.begin))/other.vec().length()
def distance_from_Segment(self, other):
if type(other) is not Segment:
raise ValueError
if Point.angle_type(self, other.begin, other.end) == 2:
return self.distance_from_Point(other.begin)
if Point.angle_type(self, other.end, other.begin) == 2:
return self.distance_from_Point(other.end)
return abs(other.vec().cross(self-other.begin))/other.vec().length()
def projection(self, line):
return line.begin+line.vec().normalized()*(self-line.begin).dot(line.vec())/line.vec().length()
def reflection(self, line):
return self+2*(self.projection(line)-self)
# 3点A, B, Cの位置関係を返す関数 A, Bがすべて異なった点であるのが前提
# ABから見てBCは左に曲がるのなら +1
# ABから見てBCは右に曲がるのなら -1
# ABC(CBA)の順番で一直線上に並ぶなら +2
# ACB(BCA)の順番で一直線上に並ぶなら 0
# BAC(CAB)の順番で一直線上に並ぶなら -2
# a ≠ b を仮定(a = b ならValueError)
def ccw(a, b, c):
if a == b:
return ValueError
flg = sgn((b-a).cross(c-a))
if flg == 1:
return 1
elif flg == -1:
return -1
# ABC(CBA)
if sgn((b-a).dot(c-b)) > 0:
return 2
# BAC(CAB)
elif (sgn((a-b).dot(c-a)) > 0):
return -2
# ACB(BCA) または cがa, bいずれかと一致
return 0
# 角ABCが鋭角なら0、直角なら1、鈍角なら2を返す。
# cc: ABから反時計回りに見るBCの偏角(0 ≤ Θ ≤ 2π)で考え、π < Θ < 2πなら3を返す。
def angle_type(a, b, c, cc=True):
if cc and sgn((a-b).cross(c-b)) < 0:
return 3
v = (a-b).dot(c-b)
if sgn(v) > 0:
return 0
elif sgn(v) == 0:
return 1
return 2
# Line, Ray, Segment は相互にキャスト可能
# ただし、Rayにキャストする際は向きに注意
class Line:
begin = None
end = None
def __init__(self, *args) -> None:
# args: begin, end
# begin = end なら ValueError
if len(args) == 2:
self.begin = Point(args[0])
self.end = Point(args[1])
# args: a, b, c (ax+by+c = 0)
# a = b = 0 なら ValueError
elif len(args) == 3:
a, b, c = args
if sgn(a) == sgn(b) == 0:
raise ValueError
if sgn(b) == 0:
self.begin = Point(-c/a, 0)
self.end = Point(-c/a, 1)
else:
self.begin = Point(0, -c/b)
self.end = Point(1, -(a+c)/b)
elif len(args) == 1 and isinstance(args[0], Line):
self.begin = Point(args[0].begin)
self.end = Point(args[0].end)
if self.begin == self.end:
raise ValueError
def __eq__(self, other): # 同一の直線か
if type(other) is not Line:
return False
return self.is_parallel(other) and self.is_intersect_with_Point(other.begin)
def __repr__(self) -> str:
return f"{type(self)}({self.begin}, {self.end})"
def vec(self):
return self.end-self.begin
def rvec(self):
return self.begin-self.end
def is_orthogonal(self, other):
if not isinstance(other, Line):
raise ValueError
return sgn(self.vec().dot(other.vec())) == 0
def is_parallel(self, other):
if not isinstance(other, Line):
raise ValueError
return self.vec().is_same_slope(other.vec())
def is_intersect_with_Point(self, other) -> bool:
if type(other) is not Point:
raise ValueError
return Point.ccw(self.begin, self.end, other) in (-2, 0, 2)
def is_intersect_with_Line(self, other) -> bool:
if type(other) is not Line:
raise ValueError
return not (self.is_parallel(other) and not self.is_intersect_with_Point(other.begin))
def is_intersect_with_Ray(self, other) -> bool:
if type(other) is not Ray:
return ValueError
c = Point.ccw(self.begin, self.end, other.begin)
if c in (-2, 0, 2): return True
elif c == 1:
return Point.ccw(other.begin, other.begin+self.vec(), other.end) == -1
else:
return Point.ccw(other.begin, other.begin+self.vec(), other.end) == 1
def is_intersect_with_Segment(self, other) -> bool:
if type(other) is not Segment:
return ValueError
return sgn(self.vec().cross(other.begin-self.begin))*sgn(self.vec().cross(other.end-self.begin)) <= 0
def intersection_with_Line(self, other):
if type(other) is not Line:
raise ValueError
if not self.is_intersect_with_Line(other):
return None
if self == other:
return Line(self)
return Point(self.begin+self.vec()*(other.end-self.begin).cross(other.vec())/self.vec().cross(other.vec()))
def intersection_with_Ray(self, other):
if type(other) is not Ray:
raise ValueError
if not self.is_intersect_with_Ray(other):
return None
if self == Line(other):
return Ray(other)
return self.intersection_with_Line(Line(other))
def intersection_with_Segment(self, other):
if type(other) is not Segment:
raise ValueError
if not self.is_intersect_with_Segment(other):
return None
if self == Line(other):
return Segment(other)
return self.intersection_with_Line(Line(other))
def distance_from_Point(self, other):
if type(other) is not Point:
raise ValueError
return other.distance_from_Line(self)
def distance_from_Line(self, other):
if type(other) is not Line:
raise ValueError
if self.is_intersect_with_Line(other):
return 0
return self.begin.distance_from_Line(other)
def distance_from_Ray(self, other):
if type(other) is not Ray:
raise ValueError
if self.is_intersect_with_Ray(other):
return 0
return other.begin.distance_from_Line(self)
def distance_from_segment(self, other):
if type(other) is not Segment:
raise ValueError
if self.is_intersect_with_Line:
return 0
return min(other.begin.distance_from_Line(self), other.end.distance_from_Line(self))
class Ray(Line):
def __init__(self, *args) -> None:
super().__init__(*args)
def __eq__(self, other) -> bool:
if type(other) is not Ray:
return False
return sgn(self.begin-other.begin) == 0 and self.vec().is_same_angle(other.vec())
def reverse(self):
self.begin, self.end = self.end, self.begin
def reversed(ray):
return Ray(ray.end, ray.begin)
def is_intersect_with_Point(self, other) -> bool:
if type(other) is not Point:
raise ValueError
return Point.ccw(self.begin, self.end, other) in (0, 2)
def is_intersect_with_Line(self, other) -> bool:
if type(other) is not Line:
raise ValueError
return other.is_intersect_with_Ray(self)
def is_intersect_with_Ray(self, other) -> bool:
if type(other) is not Ray:
raise ValueError
if not Line(self).is_intersect_with_Ray(other):
return False
if Point.ccw(self.begin, self.end, other.begin) in (0, 2) or Point.ccw(self.begin, self.end, other.end) in (0, 2):
return True
if sgn(self.vec().cross(other.begin-self.begin)) > 0:
return sgn(other.vec().cross(self.begin-other.begin)) <= 0
if sgn(self.vec().cross(other.begin-self.begin)) < 0:
return sgn(other.vec().cross(self.begin-other.begin)) >= 0
return False
def is_intersect_with_Segment(self, other) -> bool:
if type(other) is not Segment:
raise ValueError
return self.is_intersect_with_Ray(Ray(other.begin, other.end)) and self.is_intersect_with_Ray(Ray(other.end, other.begin))
def intersection_with_Line(self, other):
return other.intersection_with_Ray(self)
def intersection_with_Ray(self, other):
if type(other) is not Ray:
raise ValueError
if not self.is_intersect_with_Ray(other):
return None
c = Point.ccw(self.begin, self.end, other.begin)
if c in (0, 2):
if self.vec().is_same_angle(other.vec()):
return Ray(other)
if self.vec().is_same_slope(other.vec()):
if self.begin == other.end:
return Point(self.begin)
return Segment(self.begin, other.end)
return Point(other.begin)
if c == -2 and self.is_parallel(other):
return Ray(self)
return super().intersection_with_Ray(other)
def intersection_with_Segment(self, other):
if type(other) is not Segment:
raise ValueError
if not self.is_intersect_with_Segment(other):
return None
if self.is_parallel(other):
if Point.ccw(self.begin, self.end, other.begin) == -2:
return Segment(self.begin, other.end)
if Point.ccw(self.begin, self.end, other.end) == -2:
return Segment(self.begin, other.begin)
return super().intersection_with_Segment(other)
def distance_from_Point(self, other):
if type(other) is not Point:
raise ValueError
return other.distance_from_Ray(self)
def distance_from_Line(self, other):
if type(other) is not Line:
raise ValueError
return other.distance_from_Ray(self)
def distance_from_Ray(self, other):
if type(other) is not Ray:
raise ValueError
if self.is_intersect_with_Ray(other):
return 0
return min(self.begin.distance_from_Ray(other), other.begin.distance_from_Ray(self))
def distance_from_segment(self, other):
if type(other) is not Segment:
raise ValueError
if self.is_intersect_with_Segment(other):
return 0
return min(self.begin.distance_from_Segment(other), other.begin.distance_from_Ray(self), other.end.distance_from_Ray(self))
class Segment(Line):
def __init__(self, *args) -> None:
super().__init__(*args)
def __eq__(self, other) -> bool:
if type(other) is not Segment:
return False
return (self.begin == other.begin and self.end == other.end) or (self.begin == other.end and self.end == other.begin)
def length_sq(self):
return self.vec().length_sq()
def length(self):
return self.vec().length()
def is_intersect_with_Point(self, other) -> bool:
if type(other) is not Point:
return ValueError
return Point.ccw(self.begin, self.end, other) == 0
def is_intersect_with_Line(self, other) -> bool:
if type(other) is not Line:
return ValueError
return other.is_intersect_with_Segment(self)
def is_intersect_with_Ray(self, other) -> bool:
if type(other) is not Ray:
return ValueError
return other.is_intersect_with_Segment(self)
def is_intersect_with_Segment(self, other) -> bool:
if type(other) is not Segment:
return ValueError
ray = Ray(self)
return ray.is_intersect_with_Segment(other) and Ray.reversed(ray).is_intersect_with_Segment(other)
def intersection_with_Line(self, other):
if type(other) is not Line:
raise ValueError
return other.intersection_with_Segment(self)
def intersection_with_Ray(self, other):
if type(other) is not Ray:
raise ValueError
return other.intersection_with_Ray(self)
def intersection_with_Segment(self, other):
if type(other) is not Segment:
raise ValueError
if not self.is_intersect_with_Segment(other):
return None
if not self.is_parallel(other):
return super().intersection_with_Segment(other)
cb = Point.ccw(self.begin, self.end, other.begin)
ce = Point.ccw(self.begin, self.end, other.end)
if cb == 0:
if ce == 0:
return Segment(other)
if ce == -2:
if other.begin == self.begin:
return Point(other.begin)
return Segment(self.begin, other.begin)
if ce == 2:
if other.begin == self.end:
return Point(other.begin)
return Segment(self.end, other.begin)
if ce == 0:
if cb == -2:
if other.end == self.begin:
return Point(other.end)
return Segment(self.begin, other.end)
if cb == 2:
if other.end == self.end:
return Point(other.end)
return Segment(self.end, other.end)
return Segment(self)
def distance_from_Point(self, other):
if type(other) is not Point:
raise ValueError
return other.distance_from_Segment(self)
def distance_from_Line(self, other):
if type(other) is not Line:
raise ValueError
return other.distance_from_Segment(self)
def distance_from_Ray(self, other):
if type(other) is not Ray:
raise ValueError
return other.distance_from_Segment(self)
def distance_from_Segment(self, other):
if type(other) is not Segment:
raise ValueError
if self.is_intersect_with_Segment(other):
return 0
return min(self.distance_from_Point(other.begin), self.distance_from_Point(other.end),
other.distance_from_Point(self.begin), other.distance_from_Point(self.end))
class Polygon:
def __init__(self, points) -> None:
self.points = [Point(p) for p in points]
if len(points) < 3:
return ValueError
def __getitem__(self, i):
return self.points[i]
def __setitem__(self, i, x):
self.points[i] = Point(x)
def __iter__(self):
i = 0
for i in range(len(self.points)):
yield self.points[i]
def __len__(self):
return len(self.points)
def __repr__(self) -> str:
return f"{type(self)}({self.points})"
def area(self):
ret = 0
for i in range(len(self.points)-1):
ret += self.points[i].cross(self.points[i+1])
ret += self.points[-1].cross(self.points[0])
return ret/2
def is_convex(self):
size = len(self.points)
for i in range(size):
if Point.ccw(self.points[i-1], self.points[i], self.points[(i+1)%size]) == -1:
return False
return True
# 内部なら2, 境界上なら1, 外部なら0
def is_contain_Point(self, other):
if type(other) is not Point:
raise ValueError
ray = Ray(Point(), Point((1, 0)))
ret = 0
for i in range(len(self.points)):
a, b = self.points[i]-other, self.points[(i+1)%len(self.points)]-other
if Point.ccw(a, b, Point()) == 0:
return 1
if not ray.is_intersect_with_Segment(Segment(a, b)):
continue
if sgn(a.y-b.y) < 0:
ret ^= (sgn(b.y) != 0)
if sgn(a.y-b.y) > 0:
ret ^= (sgn(a.y) != 0)
return 2 if ret else 0
class Convex(Polygon):
def __init__(self, points) -> None:
super().__init__(points)
# if not self.is_convex(): # 安全性を考慮するならやるべき
# raise ValueError
def convex_full(points, on_line=True):
if type(points[0]) is not Point:
Points.cast_to_Point(points)
sp = sorted(points)
ret = [sp[0], sp[1]]
if on_line:
for i in range(2, len(sp)):
while len(ret) >= 2 and Point.ccw(ret[-2], ret[-1], sp[i]) < 0:
ret.pop()
ret.append(sp[i])
t = len(ret)+1
for i in range(len(sp)-2, -1, -1):
while len(ret) >= t and Point.ccw(ret[-2], ret[-1], sp[i]) < 0:
ret.pop()
ret.append(sp[i])
else:
for i in range(2, len(sp)):
while len(ret) >= 2 and Point.ccw(ret[-2], ret[-1], sp[i]) <= 0:
ret.pop()
ret.append(sp[i])
t = len(ret)+1
for i in range(len(sp)-2, -1, -1):
while len(ret) >= t and Point.ccw(ret[-2], ret[-1], sp[i]) <= 0:
ret.pop()
ret.append(sp[i])
ret.pop()
return Convex(ret)
def cut(self, line):
if not isinstance(line, Line):
raise ValueError
points = self.points
l = None
r = None
for i in range(len(points)):
c = Point.ccw(line.begin, line.end, points[i])
if c == 1:
l = i
elif c == -1:
r = i
if r is None:
return Convex(self), None
if l is None:
return None, Convex(self)
left = [points[l]]
right = []
i = (l+1)%len(points)
t = 0
for _ in range(len(points)-1):
p = line.intersection_with_Segment(Segment(points[i], points[(i-1)%len(points)]))
if p is None or p == points[(i-1)%len(points)]:
if t == 0:
left.append(points[i])
else:
right.append(points[i])
else:
if t == 0:
left.append(p)
right.append(p)
if p != points[i]:
right.append(points[i])
t ^= 1
else:
right.append(p)
left.append(p)
if p != points[i]:
left.append(points[i])
t ^= 1
i = (i+1)%len(points)
if t:
p = line.intersection_with_Segment(Segment(points[i], points[(i-1)%len(points)]))
right.append(p)
left.append(p)
return Convex(left), Convex(right)
def common_with_Convex(self, other):
if type(other) is not Convex:
raise ValueError
ret = Convex(other)
for i in range(len(self.points)):
line = Line(self.points[i], self.points[(i+1)%len(self.points)])
ret = ret.cut(line)[0]
if ret is None:
return None
return ret
def common_area_with_Convex(self, other):
if type(other) is not Convex:
raise ValueError
c = self.common_with_Convex(other)
if c is None:
return 0
return c.area()
# 内部なら2, 境界上なら1, 外部なら0
def is_contain_Point(self, other):
if type(other) is not Point:
raise ValueError
if len(self.points) == 3:
for i in range(len(self.points)):
a, b = self.points[i], self.points[(i+1)%len(self.points)]
c = Point.ccw(a, b, other)
if c == 0:
return 1
if c < 0:
return 0
return 2
ok, ng = 0, len(self.points)-1
while(abs(ok-ng)>1):
mid = (ok+ng)//2
a = self.points[mid]-self.points[0]
b = other-self.points[0]
if sgn(a.cross(b)) >= 0:
ok = mid
else:
ng = mid
if ok == 0:
return 0
x = Convex([self.points[0], self.points[ok], self.points[ng]]).is_contain_Point(other)
if x != 1:
return x
if sgn(Point.ccw(self.points[ok], self.points[ng], other)) == 0:
return 1
if ok == 1 and sgn(Point.ccw(self.points[0], self.points[1], other)) == 0:
return 1
if ng == len(self.points)-1 and sgn(Point.ccw(self.points[0], self.points[-1], other)) == 0:
return 1
return 2
class Points:
def cast_to_Point(points):
for i in range(len(points)):
points[i] = Point(points[i])
# 最近点対
def closest_pair(points):
if type(points[0]) is not Point:
Points.cast_to_Point(points)
n = len(points)
return Points._cp_rec(sorted(points), 0, n)[:2]
def _cp_rec(points, i, n):
if n <= 1:
return inf, None, [points[i]]
m = n//2
x = points[i+m][0]
dl, pairl, pointsl = Points._cp_rec(points, i, m)
dr, pairr, pointsr = Points._cp_rec(points, i+m, n-m)
if dl <= dr:
d = dl
pair = pairl
else:
d = dr
pair = pairr
sorted_points = [None]*n
l = r = idx = 0
while l < m and r < n-m:
if pointsl[l].y < pointsr[r].y:
sorted_points[idx] = pointsl[l]
l += 1
else:
sorted_points[idx] = pointsr[r]
r += 1
idx += 1
while l < m:
sorted_points[idx] = pointsl[l]
l += 1
idx += 1
while r < n-m:
sorted_points[idx] = pointsr[r]
r += 1
idx += 1
b = []
for i in range(n):
p = sorted_points[i]
if abs(p.x-x) >= d:
continue
for j in range(len(b)-1, -1, -1):
if p.y-b[j].y >= d: break
nd = p.distance_from_Point(b[j])
if nd < d:
d = nd
pair = (p, b[j])
b.append(p)
return d, pair, sorted_points
# 最遠点対
def farthest_pair(points):
if type(points[0]) is not Point:
Points.cast_to_Point(points)
if len(points) == 2:
return points[0].distance_from_Point(points[1]), (points[0], points[1])
convex = Convex.convex_full(points, False)
if len(convex) > len(points):
p, q = min(points), max(points)
return p.distance_from_Point(q), (p, q)
j = 1
n = len(convex)
d_sq = 0
pair = None
for i in range(n):
while j == i or sgn((convex[(i+1)%n]-convex[i]).dot(convex[(j+1)%n]-convex[i])) > 0:
j += 1
j %= n
nd_sq = convex[i].distance_from_Point_sq(convex[j])
if d_sq < nd_sq:
d_sq = nd_sq
pair = (convex[i], convex[j])
nd_sq = convex[i].distance_from_Point_sq(convex[(j+1)%n])
if d_sq < nd_sq:
d_sq = nd_sq
pair = (convex[i], convex[(j+1)%n])
return sqrt(d_sq), pair
def argsort(points, base=Point(), by_cross=True):
if by_cross:
U, D = [], []
for p in points:
p = p-base
if p.is_zero(): continue
if sgn(p.y) == 0:
if sgn(p.x) > 0:
U.append(p)
else:
D.append(p)
else:
if sgn(p.y) > 0:
U.append(p)
else:
D.append(p)
U.sort(key=cmp_to_key(lambda p, q: sgn(q.cross(p))))
D.sort(key=cmp_to_key(lambda p, q: sgn(q.cross(p))))
return list(map(lambda p: p+base, U))+list(map(lambda p: p+base, D))
else:
args = [(degrees((p-base).angle())%360, p) for p in points if p != base]
args.sort(key=lambda x: x[0])
return args
class Circle:
def __init__(self, centre, radius) -> None:
if type(centre) is not Point:
raise ValueError
self.centre = centre
self.radius = radius
def __eq__(self, other):
if type(other) is not Circle:
return False
return self.centre == other.centre and sgn(self.radius-other.radius) == 0
def __repr__(self) -> str:
return f"{type(self)}({self.centre}, {self.radius})"
def area(self):
return pi*self.radius**2
def common_area_with_Circle(self, other):
if type(other) is not Circle:
raise ValueError
if self == other:
return self.area()
tn = self.tangent_num_with_Circle(other)
if tn == 0 or tn == 1:
return min(self.area(), other.area())
elif tn == 2:
a, b = self.intersection_with_Circle(other)
arg1 = ((a-self.centre).angle()-(other.centre-self.centre).angle())%(2*pi)
t1 = Convex([self.centre, b, a]).area()
s1 = self.area()/pi*arg1
arg2 = ((b-other.centre).angle()-(self.centre-other.centre).angle())%(2*pi)
t2 = Convex([other.centre, a, b]).area()
s2 = other.area()/pi*arg2
return s1+s2-t1-t2
return 0
def inscribed_circle(triangle): # 内接円
if not isinstance(triangle, Polygon):
raise ValueError
if len(triangle) != 3:
raise ValueError
if not triangle.is_convex():
triangle[0], triangle[2] = triangle[2], triangle[0]
a, b, c = triangle
B = b-a
C = c-a
arg = (C.angle()-B.angle())/2
la = Line(a, B.rotate(arg)+a)
A = a-b
C = c-b
arg = (A.angle()-C.angle())/2
lb = Line(b, C.rotate(arg)+b)
c = la.intersection_with_Line(lb)
return Circle(c, c.distance_from_Line(Line(a, b)))
def circumscribed_circle(triangle): # 外接円
if not isinstance(triangle, Polygon):
raise ValueError
if len(triangle) != 3:
raise ValueError
A, B, C = triangle
AB = B-A
AC = C-A
a = AB.dot(AC)
b = A.distance_from_Point_sq(B)
c = A.distance_from_Point_sq(C)
t = b*(a-c)/(2*a**2-2*b*c)
s = 0.5-a*t/b
AO = Point((s*AB.x+t*AC.x, s*AB.y+t*AC.y))
O = A+AO
return Circle(O, O.distance_from_Point(A))
def tangent_num_with_Circle(self, other) -> int:
if type(other) is not Circle:
raise ValueError
d = self.centre.distance_from_Point_sq(other.centre)
r1 = self.radius
r2 = other.radius
if sgn(d-(r1+r2)**2) > 0: # 外部
return 4
elif sgn(d-(r1+r2)**2) == 0: # 外接
return 3
elif sgn((r1-r2)**2-d) == 0: # 内接
return 1
elif sgn((r1-r2)**2-d) > 0: # 内部
return 0
else: # 交差
return 2
def tangent_Point_from_Point(self, other):
if type(other) is not Point:
raise ValueError
icp = self.is_contain_Point(other)
if icp == 2:
return None, None
elif icp == 1:
return Point(other), None
d = self.centre-other
arg = asin(self.radius/d.length())
x = d.normalized()*sqrt(d.length_sq()-self.radius**2)
a = x.rotate(arg)+other
b = x.rotate(-arg)+other
return a, b
def tangent_Line_from_Point(self, other):
if type(other) is not Point:
raise ValueError
a, b = self.tangent_point_from_Point(other)
if a is None:
return None, None
elif b is None:
d = other-self.centre
return Line(other, other+d.normal_vector()), None
return Line(other, a), Line(other, b)
def tangent_Line_with_Circle(self, other):
if type(other) is not Circle:
raise ValueError
if self.centre == other.centre:
return []
tn = self.tangent_num_with_Circle(other)
if tn == 0:
return []
elif tn == 1:
p, _ = self.intersection_with_Circle(other)
n = (p-self.centre).normal_vector()
return [Line(p, p+n)]
elif tn == 2:
if self.radius < other.radius:
c1, c2 = self, other
elif self.radius > other.radius:
c2, c1 = self, other
else:
d = other.centre-self.centre
n = d.normal_unit_vector()*self.radius
a1, b1 = self.centre+n, self.centre-n
a2, b2 = other.centre+n, other.centre-n
return [Line(a1, a2), Line(b1, b2)]
q = c2.centre+(c1.centre-c2.centre)/(c2.radius-c1.radius)*c2.radius
a1, b1 = self.tangent_Point_from_Point(q)
a2, b2 = other.tangent_Point_from_Point(q)
return [Line(a1, a2), Line(b1, b2)]
elif tn == 3:
p, _ = self.intersection_with_Circle(other)
n = (p-self.centre).normal_vector()
if self.radius < other.radius:
c1, c2 = self, other
elif self.radius > other.radius:
c2, c1 = self, other
else:
d = other.centre-self.centre
n = d.normal_unit_vector()*self.radius
a1, b1 = self.centre+n, self.centre-n
a2, b2 = other.centre+n, other.centre-n
return [Line(p, p+n), Line(a1, a2), Line(b1, b2)]
q = c2.centre+(c1.centre-c2.centre)/(c2.radius-c1.radius)*c2.radius
a1, b1 = self.tangent_Point_from_Point(q)
a2, b2 = other.tangent_Point_from_Point(q)
return [Line(p, p+n), Line(a1, a2), Line(b1, b2)]
else:
p = self.centre+(other.centre-self.centre)/(self.radius+other.radius)*self.radius
a1, b1 = self.tangent_Point_from_Point(p)
a2, b2 = other.tangent_Point_from_Point(p)
la, lb = Line(a1, a2), Line(b1, b2)
if self.radius < other.radius:
c1, c2 = self, other
elif self.radius > other.radius:
c2, c1 = self, other
else:
d = other.centre-self.centre
n = d.normal_unit_vector()*self.radius
a1, b1 = self.centre+n, self.centre-n
a2, b2 = other.centre+n, other.centre-n
return [la, lb, Line(a1, a2), Line(b1, b2)]
q = c2.centre+(c1.centre-c2.centre)/(c2.radius-c1.radius)*c2.radius
a1, b1 = self.tangent_Point_from_Point(q)
a2, b2 = other.tangent_Point_from_Point(q)
return [la, lb, Line(a1, a2), Line(b1, b2)]
# 内部なら2, 境界上なら1, 外部なら0
def is_contain_Point(self, other):
if type(other) is not Point:
raise ValueError
s = sgn(other.distance_from_Point_sq(self.centre)-self.radius**2)
if s < 0:
return 2
elif s == 0:
return 1
return 0
def is_intersect_with_Point(self, other) -> bool:
if type(other) is not Point:
raise ValueError
return self.is_contain_Point(other) == 1
def is_intersect_with_Line(self, other) -> bool:
if type(other) is not Line:
raise ValueError
p = self.centre.projection(other)
d = p.distance_from_Point(self.centre)
return sgn(self.radius-d) >= 0
def is_intersect_with_Circle(self, other) -> bool:
if type(other) is not Circle:
raise ValueError
return 1 <= self.tangent_num_with_Circle(other) <= 3
def intersection_with_Point(self, other) -> bool:
if type(other) is not Point:
raise ValueError
if sgn(other.distance_from_Point_sq(self.centre)-self.radius**2) == 0:
return other
return None
def intersection_with_Line(self, other):
if type(other) is not Line:
raise ValueError
p = self.centre.projection(other)
d = p.distance_from_Point(self.centre)
if sgn(self.radius-d) == 0:
return p, None
elif sgn(self.radius-d) > 0:
e = sqrt(self.radius**2-d**2)
return other.vec().normalized()*e+p, other.vec().normalized()*-e+p
return None, None
def intersection_with_Circle(self, other):
if type(other) is not Circle:
raise ValueError
tn = self.tangent_num_with_Circle(other)
if tn == 1:
if self.radius < other.radius:
c2, c1 = self, other
else:
c1, c2 = self, other
a = c1.centre+(c2.centre-c1.centre).normalized()*c1.radius
return a, None
elif tn == 3:
a = self.centre+(other.centre-self.centre).normalized()*self.radius
return a, None
elif tn == 2:
d = self.centre.distance_from_Point_sq(other.centre)
r1, r2 = self.radius, other.radius
arg = acos((r1**2+d-r2**2)/(2*r1*sqrt(d)))
a = self.centre+((other.centre-self.centre).normalized()*r1).rotate(arg)
b = self.centre+((other.centre-self.centre).normalized()*r1).rotate(-arg)
return a, b
return None, None
def get_Ray(x, y, d):
if d == "R":
return Ray(Point(x, y), Point(x+1, y))
elif d == "L":
return Ray(Point(x, y), Point(x-1, y))
elif d == "U":
return Ray(Point(x, y), Point(x, y+1))
else:
return Ray(Point(x, y), Point(x, y-1))
def solve():
x1, y1, d1 = input().split()
x2, y2, d2 = input().split()
ray1, ray2 = get_Ray(int(x1), int(y1), d1), get_Ray(int(x2), int(y2), d2)
intersection = ray1.intersection_with_Ray(ray2)
if intersection is None:
return "No"
elif isinstance(intersection, Line):
return "Yes" if d1 != d2 else "No"
else:
return "Yes" if intersection.distance_from_Point_sq(ray1.begin) == intersection.distance_from_Point_sq(ray2.begin) else "No"
T = int(input())
for _ in range(T):
print(solve())