CodeForces - 659D Bicycle Race（几何问题）

D. Bicycle Race

Maria participates in a bicycle race.

The speedway takes place on the shores of Lake Lucerne, just repeating its contour. As you know, the lake shore consists only of straight sections, directed to the north, south, east or west.

Let’s introduce a system of coordinates, directing the Ox axis from west to east, and the Oy axis from south to north. As a starting position of the race the southernmost point of the track is selected (and if there are several such points, the most western among them). The participants start the race, moving to the north. At all straight sections of the track, the participants travel in one of the four directions (north, south, east or west) and change the direction of movement only in bends between the straight sections. The participants, of course, never turn back, that is, they do not change the direction of movement from north to south or from east to west (or vice versa).

Maria is still young, so she does not feel confident at some turns. Namely, Maria feels insecure if at a failed or untimely turn, she gets into the water. In other words, Maria considers the turn dangerous if she immediately gets into the water if it is ignored.

Help Maria get ready for the competition — determine the number of dangerous turns on the track.

Input
The first line of the input contains an integer n (4 ≤ n ≤ 1000) — the number of straight sections of the track.

The following (n + 1)-th line contains pairs of integers (xi, yi) ( - 10 000 ≤ xi, yi ≤ 10 000). The first of these points is the starting position. The i-th straight section of the track begins at the point (xi, yi) and ends at the point (xi + 1, yi + 1).

It is guaranteed that:

the first straight section is directed to the north;
the southernmost (and if there are several, then the most western of among them) point of the track is the first point;
the last point coincides with the first one (i.e., the start position);
any pair of straight sections of the track has no shared points (except for the neighboring ones, they share exactly one point);
no pair of points (except for the first and last one) is the same;
no two adjacent straight sections are directed in the same direction or in opposite directions.

Output
Print a single integer — the number of dangerous turns on the track.

input
6
0 0
0 1
1 1
1 2
2 2
2 0
0 0
output
1
input
16
1 1
1 5
3 5
3 7
2 7
2 9
6 9
6 7
5 7
5 3
4 3
4 4
3 4
3 2
5 2
5 1
1 1
output
6
Note
The first sample corresponds to the picture:

The picture shows that you can get in the water under unfortunate circumstances only at turn at the point (1, 1). Thus, the answer is 1.

题意：按顺时针给n+1个点，这些点围成一个n边形，到拐点时要拐弯，问主人公若在拐点不拐弯掉河里的次数。

我是用的差积，那么点i和它后面的两个点分别形成的向量的差积一定小于0，此时为顺时针拐弯，当差积大于0时肯定是要逆时针拐弯了。设图中A(0,0),B(0,1),C(1,1),D(1,2)，向量AB向量AC小于0为顺时针拐弯，向量BC向量BD大于0为逆时针拐弯，会掉河里。

#include <iostream>
#include <cstdio>
#include <bits/stdc++.h>
using namespace std;
#define mod 1000000007
typedef long long LL;

int main()
{
    int n;
    while(scanf("%d", &n)!=EOF)
    {
        int a[1005][2]={0};
        for(int i=0; i<n+1; i++)
            scanf("%d%d", &a[i][0], &a[i][1]);
        int sum=0;
        a[n+1][0]=a[1][0], a[n+1][1]=a[1][1];//记得把第二个点也放到末尾，形成环路
        for(int i=0; i<n; i++)
        {
            int x0=a[i][0], y0=a[i][1];
            int x1=a[i+1][0], y1=a[i+1][1];
            int x2=a[i+2][0], y2=a[i+2][1];
            int x01=x1-x0, y01=y1-y0;
            int x02=x2-x0, y02=y2-y0;
            if(((x01*y02)-(y01*x02))>0)
                sum++;
        }
        printf("%d\n", sum);
    }
}

标签：Bicycle,point,track,straight,CodeForces,Race,north,sections,first
来源： https://blog.csdn.net/weixin_44606097/article/details/100355925