CodeForces 659D Bicycle Race (判断点是否为危险点)

D - Bicycle Race

Time Limit:1000MS     Memory Limit:262144KB     64bit IO Format:%I64d & %I64u

Submit Status Practice CodeForces 659D

Description

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 (x_i, y_i) ( - 10 000 ≤ x_i, y_i ≤ 10 000). The first of these points is the starting position. The i-th straight section of the track begins at the point (x_i, y_i) and ends at the point (x_i + 1, y_i + 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.

Sample Input

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

题意：一个多边形围成的湖，一个人绕着走。如果在某个位置不转弯直走的话会掉入湖里，

　　　则认为这个弯是危险的。问有多少个这样的弯。

题解：第一种解法：从题目中的加粗字体可知人是从左下角往上走的，即顺时针走，画顺时针走的图可知向量b在向量a的逆时针方向，

　　　即叉积大于0。注意千万不能看做逆时针走，一定要注意方向，否则结果会是n-ans。

　　　第二种解法：可以发现,在内角为270°的地方才有可能掉到水里,设这样的地方有x个,

　　　则内角和 180 * (n - 2) = 270 * x + (n - x) * 90,得到 x=(n-4)/2 。

代码一：

#include <iostream>
using namespace std;
int main()
{
    int n;
    while(cin>>n)
    {
        int i,ans=0;
        int a[1050],b[1050];
        for(i=0;i<=n;i++)
        cin>>a[i]>>b[i];
        for(i=2;i<=n;i++)
        if(((a[i-1]-a[i-2])*(b[i]-b[i-2])-(a[i]-a[i-2])*(b[i-1]-b[i-2]))>0)
        ans++;
        cout<<ans<<endl;
    }
    return 0;
}

代码二：

#include <cstdio>
using namespace std;
int main(){
        int n;
        scanf("%d",&n);
        printf("%d\n",(n - 4)/2);
        return 0;
}