hdu 1154 Cutting a Polygon (Simple Geometry)

http://acm.hdu.edu.cn/showproblem.php?pid=1154

　　简单的计算几何题。因为没有看清题意，把直线看成线段了，所以WA了几次。

　　做法相当简单，我们只需要把直线于多边形的交点都求出来，然后每相邻两个交点取其中点，判断中点是否在多边形内。如果中点在多边形内，则当前线段在多边形内。这是一个很常见的判断方法。

代码（带模板）如下：

#include <cstdio>
#include <cstring>
#include <cmath>
#include <set>
#include <vector>
#include <iostream>
#include <algorithm>

using namespace std;

// Point class
struct Point {
double x, y;
Point() {}
Point(double x, double y) : x(x), y(y) {}
} ;
template<class T> T sqr(T x) { return x * x;}
inline double ptDis(Point a, Point b) { return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));}

// basic calculations
typedef Point Vec;
Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);}
Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);}
Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);}
Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);}

const double EPS = 1e-8;
const double PI = acos(-1.0);
inline int sgn(double x) { return fabs(x) < EPS ? 0 : (x < 0 ? -1 : 1);}
bool operator < (Point a, Point b) { return a.x < b.x || (a.x == b.x && a.y < b.y);}
bool operator == (Point a, Point b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;}

inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;}
inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;}
inline double crossDet(Point o, Point a, Point b) { return crossDet(a - o, b - o);}
inline double vecLen(Vec x) { return sqrt(dotDet(x, x));}
inline double toRad(double deg) { return deg / 180.0 * PI;}
inline double angle(Vec v) { return atan2(v.y, v.x);}
inline double angle(Vec a, Vec b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));}
inline double triArea(Point a, Point b, Point c) { return fabs(crossDet(b - a, c - a));}
inline Vec vecUnit(Vec x) { return x / vecLen(x);}
inline Vec rotate(Vec x, double rad) { return Vec(x.x * cos(rad) - x.y * sin(rad), x.x * sin(rad) + x.y * cos(rad));}
Vec normal(Vec x) {
double len = vecLen(x);
return Vec(- x.y / len, x.x / len);
}

// Line class
struct Line {
Point s, t;
Line() {}
Line(Point s, Point t) : s(s), t(t) {}
Point point(double x) {
return s + (t - s) * x;
}
Line move(double x) { // while x > 0 move to (s->t)'s left
Vec nor = normal(t - s);
nor = nor * x;
return Line(s + nor, t + nor);
}
Vec vec() { return t - s;}
} ;
typedef Line Seg;

inline bool onLine(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0;}
inline bool onLine(Point x, Line l) { return onLine(x, l.s, l.t);}
inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(a - x, b - x)) == 0 && sgn(dotDet(a - x, b - x)) < 0;}
inline bool onSeg(Point x, Seg s) { return onSeg(x, s.s, s.t);}

// 0 : not intersect
// 1 : proper intersect
// 2 : improper intersect
int segIntersect(Point a, Point c, Point b, Point d) {
Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d;
int a_bc = sgn(crossDet(v1, v2));
int b_cd = sgn(crossDet(v2, v3));
int c_da = sgn(crossDet(v3, v4));
int d_ab = sgn(crossDet(v4, v1));
if (a_bc * c_da > 0 && b_cd * d_ab > 0) return 1;
if (onSeg(b, a, c) && c_da) return 2;
if (onSeg(c, b, d) && d_ab) return 2;
if (onSeg(d, c, a) && a_bc) return 2;
if (onSeg(a, d, b) && b_cd) return 2;
return 0;
}
inline int segIntersect(Seg a, Seg b) { return segIntersect(a.s, a.t, b.s, b.t);}

// point of the intersection of 2 lines
Point lineIntersect(Point P, Vec v, Point Q, Vec w) {
Vec u = P - Q;
double t = crossDet(w, u) / crossDet(v, w);
return P + v * t;
}
inline Point lineIntersect(Line a, Line b) { return lineIntersect(a.s, a.t - a.s, b.s, b.t - b.s);}

// Warning: This is a DIRECTED Distance!!!
double pt2Line(Point x, Point a, Point b) {
Vec v1 = b - a, v2 = x - a;
return crossDet(v1, v2) / vecLen(v1);
}
inline double pt2Line(Point x, Line L) { return pt2Line(x, L.s, L.t);}

double pt2Seg(Point x, Point a, Point b) {
if (a == b) return vecLen(x - a);
Vec v1 = b - a, v2 = x - a, v3 = x - b;
if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2);
if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3);
return fabs(crossDet(v1, v2)) / vecLen(v1);
}
inline double pt2Seg(Point x, Seg s) { return pt2Seg(x, s.s, s.t);}

Point ptOnLine(Point p, Point a, Point b) {
Vec v = b - a;
return a + v * (dotDet(v, p - a) / dotDet(v, v));
}
inline Point ptOnLine(Point p, Line x) { return ptOnLine(p, x.s, x.t);}

// Polygon class
struct Poly {
vector<Point> pt;
Poly() { pt.clear();}
~Poly() {}
Poly(vector<Point> pt) : pt(pt) {}
Point operator [] (int x) const { return pt[x];}
int size() { return pt.size();}
double area() {
double ret = 0.0;
int sz = pt.size();
for (int i = 1; i < sz; i++) {
ret += crossDet(pt[i], pt[i - 1]);
}
return fabs(ret / 2.0);
}
} ;

// Circle class
struct Circle {
Point c;
double r;
Circle() {}
Circle(Point c, double r) : c(c), r(r) {}
Point point(double a) {
return Point(c.x + cos(a) * r, c.y + sin(a) * r);
}
} ;

inline bool ptOnCircle(Point x, Circle c) { return sgn(ptDis(c.c, x) - c.r) == 0;}

// Cirlce operations
int lineCircleIntersect(Line L, Circle C, double &t1, double &t2, vector<Point> &sol) {
double a = L.s.x, b = L.t.x - C.c.x, c = L.s.y, d = L.t.y - C.c.y;
double e = sqr(a) + sqr(c), f = 2 * (a * b + c * d), g = sqr(b) + sqr(d) - sqr(C.r);
double delta = sqr(f) - 4.0 * e * g;
if (sgn(delta) < 0) return 0;
if (sgn(delta) == 0) {
t1 = t2 = -f / (2.0 * e);
sol.push_back(L.point(t1));
return 1;
}
t1 = (-f - sqrt(delta)) / (2.0 * e);
sol.push_back(L.point(t1));
t2 = (-f + sqrt(delta)) / (2.0 * e);
sol.push_back(L.point(t2));
return 2;
}

int lineCircleIntersect(Line L, Circle C, vector<Point> &sol) {
Vec dir = L.t - L.s, nor = normal(dir);
Point mid = lineIntersect(C.c, nor, L.s, dir);
double len = sqr(C.r) - sqr(ptDis(C.c, mid));
if (sgn(len) < 0) return 0;
if (sgn(len) == 0) {
sol.push_back(mid);
return 1;
}
Vec dis = vecUnit(dir);
len = sqrt(len);
sol.push_back(mid + dis * len);
sol.push_back(mid - dis * len);
return 2;
}

// -1 : coincide
int circleCircleIntersect(Circle C1, Circle C2, vector<Point> &sol) {
double d = vecLen(C1.c - C2.c);
if (sgn(d) == 0) {
if (sgn(C1.r - C2.r) == 0) {
return -1;
}
return 0;
}
if (sgn(C1.r + C2.r - d) < 0) return 0;
if (sgn(fabs(C1.r - C2.r) - d) > 0) return 0;
double a = angle(C2.c - C1.c);
double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
Point p1 = C1.point(a - da), p2 = C1.point(a + da);
sol.push_back(p1);
if (p1 == p2) return 1;
sol.push_back(p2);
return 2;
}

void circleCircleIntersect(Circle C1, Circle C2, vector<double> &sol) {
double d = vecLen(C1.c - C2.c);
if (sgn(d) == 0) return ;
if (sgn(C1.r + C2.r - d) < 0) return ;
if (sgn(fabs(C1.r - C2.r) - d) > 0) return ;
double a = angle(C2.c - C1.c);
double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2.0 * C1.r * d));
sol.push_back(a - da);
sol.push_back(a + da);
}

int tangent(Point p, Circle C, vector<Vec> &sol) {
Vec u = C.c - p;
double dist = vecLen(u);
if (dist < C.r) return 0;
if (sgn(dist - C.r) == 0) {
sol.push_back(rotate(u, PI / 2.0));
return 1;
}
double ang = asin(C.r / dist);
sol.push_back(rotate(u, -ang));
sol.push_back(rotate(u, ang));
return 2;
}

// ptA : points of tangency on circle A
// ptB : points of tangency on circle B
int tangent(Circle A, Circle B, vector<Point> &ptA, vector<Point> &ptB) {
if (A.r < B.r) {
swap(A, B);
swap(ptA, ptB);
}
double d2 = sqr(A.c.x - B.c.x) + sqr(A.c.y - B.c.y);
double rdiff = A.r - B.r, rsum = A.r + B.r;
if (d2 < sqr(rdiff)) return 0;
double base = atan2(B.c.y - A.c.y, B.c.x - A.c.x);
if (d2 == 0 && A.r == B.r) return -1;
if (d2 == sqr(rdiff)) {
ptA.push_back(A.point(base));
ptB.push_back(B.point(base));
return 1;
}
double ang = acos((A.r - B.r) / sqrt(d2));
ptA.push_back(A.point(base + ang));
ptB.push_back(B.point(base + ang));
ptA.push_back(A.point(base - ang));
ptB.push_back(B.point(base - ang));
if (d2 == sqr(rsum)) {
ptA.push_back(A.point(base));
ptB.push_back(B.point(PI + base));
} else if (d2 > sqr(rsum)) {
ang = acos((A.r + B.r) / sqrt(d2));
ptA.push_back(A.point(base + ang));
ptB.push_back(B.point(PI + base + ang));
ptA.push_back(A.point(base - ang));
ptB.push_back(B.point(PI + base - ang));
}
return (int) ptA.size();
}

// -1 : onside
// 0 : outside
// 1 : inside
int ptInPoly(Point p, Poly &poly) {
int wn = 0, sz = poly.size();
for (int i = 0; i < sz; i++) {
if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1;
int k = sgn(crossDet(poly[(i + 1) % sz] - poly[i], p - poly[i]));
int d1 = sgn(poly[i].y - p.y);
int d2 = sgn(poly[(i + 1) % sz].y - p.y);
if (k > 0 && d1 <= 0 && d2 > 0) wn++;
if (k < 0 && d2 <= 0 && d1 > 0) wn--;
}
if (wn != 0) return 1;
return 0;
}

// if DO NOT need a high precision
/*
int ptInPoly(Point p, Poly &poly) {
int sz = poly.size();
double ang = 0.0, tmp;
for (int i = 0; i < sz; i++) {
if (onSeg(p, poly[i], poly[(i + 1) % sz])) return -1;
tmp = angle(poly[i] - p) - angle(poly[(i + 1) % sz] - p) + PI;
ang += tmp - floor(tmp / (2.0 * PI)) * 2.0 * PI - PI;
}
if (sgn(ang - PI) == 0) return -1;
if (sgn(ang) == 0) return 0;
return 1;
}
*/

// Convex Hull algorithms
// return the number of points in convex hull

// andwer's algorithm
// if DO NOT want the points on the side of convex hull, change all "<" into "<="
int andrew(Point *pt, int n, Point *ch) {
sort(pt, pt + n);
int m = 0;
for (int i = 0; i < n; i++) {
while (m > 1 && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--;
ch[m++] = pt[i];
}
int k = m;
for (int i = n - 2; i >= 0; i--) {
while (m > k && crossDet(ch[m - 1] - ch[m - 2], pt[i] - ch[m - 2]) <= 0) m--;
ch[m++] = pt[i];
}
if (n > 1) m--;
return m;
}

// graham's algorithm
// if DO NOT want the points on the side of convex hull, change all "<=" into "<"
Point origin;
inline bool cmpAng(Point p1, Point p2) { return crossDet(origin, p1, p2) > 0;}
inline bool cmpDis(Point p1, Point p2) { return ptDis(p1, origin) > ptDis(p2, origin);}

void removePt(Point *pt, int &n) {
int idx = 1;
for (int i = 2; i < n; i++) {
if (sgn(crossDet(origin, pt[i], pt[idx]))) pt[++idx] = pt[i];
else if (cmpDis(pt[i], pt[idx])) pt[idx] = pt[i];
}
n = idx + 1;
}

int graham(Point *pt, int n, Point *ch) {
int top = -1;
for (int i = 1; i < n; i++) {
if (pt[i].y < pt[0].y || (pt[i].y == pt[0].y && pt[i].x < pt[0].x)) swap(pt[i], pt[0]);
}
origin = pt[0];
sort(pt + 1, pt + n, cmpAng);
removePt(pt, n);
for (int i = 0; i < n; i++) {
if (i >= 2) {
while (!(crossDet(ch[top - 1], pt[i], ch[top]) <= 0)) top--;
}
ch[++top] = pt[i];
}
return top + 1;
}

// Half Plane
// The intersected area is always a convex polygon.
// Directed Line class
struct DLine {
Point p;
Vec v;
double ang;
DLine() {}
DLine(Point p, Vec v) : p(p), v(v) { ang = atan2(v.y, v.x);}
bool operator < (const DLine &L) const { return ang < L.ang;}
DLine move(double x) { // while x > 0 move to v's left
Vec nor = normal(v);
nor = nor * x;
return DLine(p + nor, v);
}

} ;

Poly cutPoly(Poly &poly, Point a, Point b) {
Poly ret = Poly();
int n = poly.size();
for (int i = 0; i < n; i++) {
Point c = poly[i], d = poly[(i + 1) % n];
if (sgn(crossDet(b - a, c - a)) >= 0) ret.pt.push_back(c);
if (sgn(crossDet(b - a, c - d)) != 0) {
Point ip = lineIntersect(a, b - a, c, d - c);
if (onSeg(ip, c, d)) ret.pt.push_back(ip);
}
}
return ret;
}
inline Poly cutPoly(Poly &poly, DLine L) { return cutPoly(poly, L.p, L.p + L.v);}

inline bool onLeft(DLine L, Point p) { return crossDet(L.v, p - L.p) > 0;}
Point dLineIntersect(DLine a, DLine b) {
Vec u = a.p - b.p;
double t = crossDet(b.v, u) / crossDet(a.v, b.v);
return a.p + a.v * t;
}

Poly halfPlane(DLine *L, int n) {
Poly ret = Poly();
sort(L, L + n);
int fi, la;
Point *p = new Point
;
DLine *q = new DLine
;
q[fi = la = 0] = L[0];
for (int i = 1; i < n; i++) {
while (fi < la && !onLeft(L[i], p[la - 1])) la--;
while (fi < la && !onLeft(L[i], p[fi])) fi++;
q[++la] = L[i];
if (fabs(crossDet(q[la].v, q[la - 1].v)) < EPS) {
la--;
if (onLeft(q[la], L[i].p)) q[la] = L[i];
}
if (fi < la) p[la - 1] = dLineIntersect(q[la - 1], q[la]);
}
while (fi < la && !onLeft(q[fi], p[la - 1])) la--;
if (la - fi <= 1) return ret;
p[la] = dLineIntersect(q[la], q[fi]);
for (int i = fi; i <= la; i++) ret.pt.push_back(p[i]);
return ret;
}

// 3D Geometry
void getCoor(double R, double lat, double lng, double &x, double &y, double &z) {
lat = toRad(lat);
lng = toRad(lng);
x = R * cos(lat) * cos(lng);
y = R * cos(lat) * sin(lng);
z = R * sin(lat);
}

struct Point3 {
double x, y, z;
Point3() {}
Point3(double x, double y, double z) : x(x), y(y), z(z) {}
} ;
typedef Point3 Vec3;

Vec3 operator + (Vec3 a, Vec3 b) { return Vec3(a.x + b.x, a.y + b.y, a.z + b.z);}
Vec3 operator - (Vec3 a, Vec3 b) { return Vec3(a.x - b.x, a.y - b.y, a.z - b.z);}
Vec3 operator * (Vec3 a, double p) { return Vec3(a.x * p, a.y * p, a.z * p);}
Vec3 operator / (Vec3 a, double p) { return Vec3(a.x / p, a.y / p, a.z / p);}

bool operator < (Point3 a, Point3 b) {
if (a.x != b.x) return a.x < b.x;
if (a.y != b.y) return a.y < b.y;
return a.z < b.z;
}
bool operator == (Point3 a, Point3 b) { return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0 && sgn(a.z - b.z) == 0;}

inline double dotDet(Vec3 a, Vec3 b) { return a.x * b.x + a.y * b.y + a.z * b.z;}
inline Vec3 crossDet(Vec3 a, Vec3 b) { return Vec3(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x);}
inline double vecLen(Vec3 x) { return sqrt(dotDet(x, x));}
inline double angle(Vec3 a, Vec3 b) { return acos(dotDet(a, b) / vecLen(a) / vecLen(b));}
inline double triArea(Point3 a, Point3 b, Point3 c) { return vecLen(crossDet(b - a, c - a));}

struct Plane {
Point3 p;
Vec3 n;
Plane() {}
Plane(Point3 p, Vec3 n) : p(p), n(n) {}
} ;

// Warning: This is a DIRECTED Distance!!!
inline double pt2Plane(Point3 p, Point3 p0, Vec3 n) { return dotDet(p - p0, n) / vecLen(n);}
inline double pt2Plane(Point3 p, Plane P) { return pt2Plane(p, P.p, P.n);}
// get projection on plane
inline Point3 ptOnPlane(Point3 p, Point3 p0, Vec3 n) { return p + n * pt2Plane(p, p0, n);}
inline Point3 ptOnPlane(Point3 p, Plane P) { return ptOnPlane(p, P.p, P.n);}
inline bool ptInPlane(Point3 p, Point3 p0, Vec3 n) { return sgn(dotDet(p - p0, n)) == 0;}
inline bool ptInPlane(Point3 p, Plane P) { return ptInPlane(p, P.p, P.n);}

struct Line3 {
Point3 s, t;
Line3() {}
Line3(Point3 s, Point3 t) : s(s), t(t) {}
Vec3 vec() { return t - s;}
} ;
typedef Line3 Seg3;

double pt2Line(Point3 p, Point3 a, Point3 b) {
Vec3 v1 = b - a, v2 = p - a;
return vecLen(crossDet(v1, v2)) / vecLen(v1);
}
inline double pt2Line(Point3 p, Line3 l) { return pt2Line(p, l.s, l.t);}

double pt2Seg(Point3 p, Point3 a, Point3 b) {
if (a == b) return vecLen(p - a);
Vec3 v1 = b - a, v2 = p - a, v3 = p - b;
if (sgn(dotDet(v1, v2)) < 0) return vecLen(v2);
else if (sgn(dotDet(v1, v3)) > 0) return vecLen(v3);
else return vecLen(crossDet(v1, v2)) / vecLen(v1);
}
inline double pt2Seg(Point3 p, Seg3 s) { return pt2Seg(p, s.s, s.t);}

struct Tri {
Point3 a, b, c;
Tri() {}
Tri(Point3 a, Point3 b, Point3 c) : a(a), b(b), c(c) {}
} ;

bool ptInTri(Point3 p, Point3 a, Point3 b, Point3 c) {
double area1 = triArea(p, a, b);
double area2 = triArea(p, b, c);
double area3 = triArea(p, c, a);
double area = triArea(a, b, c);
return sgn(area1 = area2 + area3 - area) == 0;
}
inline bool ptInTri(Point3 p, Tri t) { return ptInTri(p, t.a, t.b, t.c);}

// 0 : not intersect
// 1 : intersect at only 1 point
// 2 : line in plane
int linePlaneIntersect(Point3 s, Point3 t, Point3 p0, Vec3 n, Point3 &x) {
double res1 = dotDet(n, p0 - s);
double res2 = dotDet(n, t - s);
if (sgn(res2)) {
if (ptInPlane(s, p0, n)) return 2;
return 0;
}
x = s + (t - s) * (res1 / res2);
// use general form : a * x + b * y + c * z + d = 0
// Vec3 v = s - t;
// double k = (a * s.x + b * s.y + c * s.z + d) / (a * v.x + b * v.y + c * v.z);
// x = s + (t - s) * k;
return 1;
}
inline int linePlaneIntersect(Line3 l, Plane p, Point3 &x) { return linePlaneIntersect(l.s, l.t, p.p, p.n, x);}

// ¡÷abc intersect with segment st
bool triSegIntersect(Point3 a, Point3 b, Point3 c, Point3 s, Point3 t, Point3 &x) {
Vec3 n = crossDet(b - a, c - a);
if (sgn(dotDet(n, t - s)) == 0) return false;
else {
double k = dotDet(n, a - s) / dotDet(n, t - s);
if (sgn(k) < 0 || sgn(k - 1) > 0) return false;
x = s + (t - s) * k;
return ptInTri(x, a, b, c);
}
}
inline bool triSegIntersect(Tri t, Line3 l, Point3 &x) { return triSegIntersect(t.a, t.b, t.c, l.s, l.t, x);}

// Warning: This is a DIRECTED Volume!!!
inline double tetraVol(Point3 a, Point3 b, Point3 c, Point3 p) { return dotDet(p - a, crossDet(b - a, c - a)) / 6.0;}
inline double tetraVol(Tri t, Point3 p) { return tetraVol(t.a, t.b, t.c, p);}

/****************** template above *******************/

Poly poly;

bool cmp(Point a, Point b) {
if (sgn(a.x - b.x)) return a.x < b.x;
return a.y < b.y;
}

double work(Seg seg) {
int sz = poly.size();
vector<Point> ip;
ip.clear();
seg = Seg(seg.s + (seg.s - seg.t) * 1e5, seg.t + (seg.t - seg.s) * 1e5);
ip.push_back(seg.s);
ip.push_back(seg.t);
for (int i = 0; i < sz; i++) {
if (segIntersect(Seg(poly[i], poly[(i + 1) % sz]), seg)) {
ip.push_back(lineIntersect(Line(poly[i], poly[(i + 1) % sz]), seg));
}
}
sort(ip.begin(), ip.end(), cmp);
sz = ip.size();
double sum = 0.0;
for (int i = 1; i < sz; i++) {
if (ptInPoly((ip[i - 1] + ip[i]) / 2.0, poly)) {
sum += ptDis(ip[i - 1], ip[i]);
}
}
return sum;
}

int main() {
//    freopen("in", "r", stdin);
int n, m;
while (cin >> n >> m && (n || m)) {
poly.pt.clear();
double x, y;
for (int i = 0; i < n; i++) {
cin >> x >> y;
poly.pt.push_back(Point(x, y));
}
Seg tmp;
for (int i = 0; i < m; i++) {
cin >> tmp.s.x >> tmp.s.y >> tmp.t.x >> tmp.t.y;
printf("%.3f\n", work(tmp));
}
}
return 0;
}

View Code

——written by Lyon