题解

问题分析

本题要求计算经过多次折叠后切割纸张展开后的片数。关键在于模拟折叠过程对切割线的影响，通过几何变换确定切割线在展开后纸张上的所有可能位置，进而计算这些位置与原纸张的交点及交点间的分割关系。

关键思路

折叠的几何变换：每次折叠相当于将纸张沿折叠线做镜像反射。切割线在折叠后的位置可通过对称变换反向展开，得到其在原纸张上的所有可能位置。 对称变换：点关于直线的对称点计算是核心，通过直线的一般式 $Ax+By+C=0$ 推导对称点坐标。 收集所有切割线：从最后一次折叠开始反向处理，每次折叠会引入对称的切割线，收集所有可能的切割线位置。 交点计算：计算每条切割线与原纸$[0,1]×[0,1]$的有效线段，统计这些线段之间的交点数，片数为交点数加 $1$。 #py import math

EPS = 1e-8

class Point: def init(self, x, y): self.x = x self.y = y

def __sub__(self, other):
    return Point(self.x - other.x, self.y - other.y)

def __add__(self, other):
    return Point(self.x + other.x, self.y + other.y)

def __mul__(self, k):
    return Point(self.x * k, self.y * k)

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 __eq__(self, other):
    return abs(self.x - other.x) < EPS and abs(self.y - other.y) < EPS

def __hash__(self):
    return hash((round(self.x, 9), round(self.y, 9)))

def __repr__(self):
    return f"({self.x:.6f}, {self.y:.6f})"

def line_from_points(p1, p2): a = p2.y - p1.y b = p1.x - p2.x c = p2.x * p1.y - p1.x * p2.y return (a, b, c)

def symmetric_point(p, a, b, c): d = a2 + b2 if d < EPS: return p x = p.x y = p.y tx = x - 2 * a * (ax + by + c) / d ty = y - 2 * b * (ax + by + c) / d return Point(tx, ty)

def segment_intersection(s1, s2): p, q, r, s = s1[0], s1[1], s2[0], s2[1] a = q - p b = s - r c = p - r cross_ab = a.cross(b) cross_ac = a.cross(c) cross_bc = b.cross(c)

if abs(cross_ab) < EPS:
    if abs(cross_ac) > EPS or abs(cross_bc) > EPS:
        return None
    else:
        points = [p, q, r, s]
        unique = list({p, q, r, s})
        return unique if len(unique) > 1 else None
t = cross_bc / cross_ab
u = cross_ac / cross_ab
if -EPS <= t <= 1 + EPS and -EPS <= u <= 1 + EPS:
    x = p.x + t * a.x
    y = p.y + t * a.y
    return Point(x, y)
return None

def clip_segment(p1, p2): def inside(p): return 0 - EPS <= p.x <= 1 + EPS and 0 - EPS <= p.y <= 1 + EPS def compute_intersection(p, q, a, b, c): d = a * q.x + b * q.y + c if abs(d) < EPS: return q t = (-a * p.x - b * p.y - c) / d return Point(p.x + t * (q.x - p.x), p.y + t * (q.y - p.y))

clips = [
    (lambda p: p.x >= -EPS, lambda p, q: compute_intersection(p, q, -1, 0, 0)),
    (lambda p: p.x <= 1 + EPS, lambda p, q: compute_intersection(p, q, 1, 0, -1)),
    (lambda p: p.y >= -EPS, lambda p, q: compute_intersection(p, q, 0, -1, 0)),
    (lambda p: p.y <= 1 + EPS, lambda p, q: compute_intersection(p, q, 0, 1, -1)),
]

for cond, inter in clips:
    if cond(p1) and cond(p2):
        continue
    if cond(p1):
        p1, p2 = p2, p1
    new_p = inter(p1, p2)
    if inside(new_p):
        p1 = new_p
    else:
        return None
return (p1, p2)

def main(): import sys input = sys.stdin.read().split() ptr = 0 t = int(input[ptr]) ptr += 1 for _ in range(t): n = int(input[ptr]) ptr += 1 folds = [] for __ in range(n): x1, y1, x2, y2 = map(float, input[ptr:ptr+4]) ptr += 4 p1 = Point(x1, y1) p2 = Point(x2, y2) folds.append(line_from_points(p1, p2)) cx1, cy1, cx2, cy2 = map(float, input[ptr:ptr+4]) ptr += 4 cut_points = [Point(cx1, cy1), Point(cx2, cy2)] cut_lines = [cut_points]

    for fold in reversed(folds):
        a, b, c = fold
        new_cuts = []
        for cp in cut_lines:
            p1, p2 = cp
            sp1 = symmetric_point(p1, a, b, c)
            sp2 = symmetric_point(p2, a, b, c)
            new_cuts.append(cp)
            new_cuts.append([sp1, sp2])
        cut_lines = new_cuts
    
    segments = []
    for cp in cut_lines:
        p1, p2 = cp
        seg = clip_segment(p1, p2)
        if seg:
            segments.append(seg)
    
    points = set()
    for seg in segments:
        p1, p2 = seg
        points.add(p1)
        points.add(p2)
    
    intersections = set()
    for i in range(len(segments)):
        for j in range(i+1, len(segments)):
            seg1 = segments[i]
            seg2 = segments[j]
            p = segment_intersection(seg1, seg2)
            if p:
                intersections.add(p)
    
    total = len(intersections) + 1
    print(total)

if name == "main": main()