LA 3263 欧拉定理

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#include<cstdio>#include<cmath>#include<algorithm>using namespace std;const double eps = 1e-10;int dcmp(double x) {  if(fabs(x) < eps) return 0; else return x < 0 ? -1 : 1;}struct Point {  double x, y;  Point(double x=0, double y=0):x(x),y(y) { }};typedef Point Vector;Vector operator + (const Vector& A, const Vector& B) { return Vector(A.x+B.x, A.y+B.y); }Vector operator - (const Point& A, const Point& B) { return Vector(A.x-B.x, A.y-B.y); }Vector operator * (const Vector& A, double p) { return Vector(A.x*p, A.y*p); }bool operator < (const Point& a, const Point& b) {  return a.x < b.x || (a.x == b.x && a.y < b.y);}bool operator == (const Point& a, const Point &b) {  return dcmp(a.x-b.x) == 0 && dcmp(a.y-b.y) == 0;}double Dot(const Vector& A, const Vector& B) { return A.x*B.x + A.y*B.y; }double Cross(const Vector& A, const Vector& B) { return A.x*B.y - A.y*B.x; }Point GetLineIntersection(const Point& P, const Vector& v, const Point& Q, const Vector& w) {   Vector u = P-Q;  double t = Cross(w, u) / Cross(v, w);  return P+v*t;}bool SegmentProperIntersection(const Point& a1, const Point& a2, const Point& b1, const Point& b2) {  double c1 = Cross(a2-a1,b1-a1), c2 = Cross(a2-a1,b2-a1),  c3 = Cross(b2-b1,a1-b1), c4=Cross(b2-b1,a2-b1);  return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;}bool OnSegment(const Point& p, const Point& a1, const Point& a2) {  return dcmp(Cross(a1-p, a2-p)) == 0 && dcmp(Dot(a1-p, a2-p)) < 0;}const int maxn = 300 + 10;Point P[maxn], V[maxn*maxn];int main() {  int n, kase = 0;  while(scanf("%d", &n) == 1 && n) {    for(int i = 0; i < n; i++) { scanf("%lf%lf", &P[i].x, &P[i].y); V[i] = P[i]; }    n--;    int c = n, e = n;    for(int i = 0; i < n; i++)      for(int j = i+1; j < n; j++)        if(SegmentProperIntersection(P[i], P[i+1], P[j], P[j+1]))          V[c++] = GetLineIntersection (P[i], P[i+1]-P[i], P[j], P[j+1]-P[j]);    sort(V, V+c);    c = unique(V, V+c) - V;    for(int i = 0; i < c; i++)      for(int j = 0; j < n; j++)        if(OnSegment(V[i], P[j], P[j+1])) e++;    printf("Case %d: There are %d pieces.\n", ++kase, e+2-c);  }  return 0;}

书上的代码：

欧拉定理：

设平面图的顶点数 V ，边数 E ，面数 F ，则 F = E+2-V

分析：

1：算出顶点数，包括交点，有可能三条线段交于一点，需要去重。

2：算出线段数，如果原来的线段新增加一个点，那么一条线段一为二，所以一条线段每新增一个点就多出一条线段。

即可

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