[POJ 2074] Line of Sight【计算几何】

• 2017-12-28
• 0
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Description

An architect is very proud of his new home and wants to be sure it can be seen by people passing by his property line along the street. The property contains various trees, shrubs, hedges, and other obstructions that may block the view. For the purpose of this problem, model the house, property line, and obstructions as straight lines parallel to the x axis:
To satisfy the architect's need to know how visible the house is, you must write a program that accepts as input the locations of the house, property line, and surrounding obstructions and calculates the longest continuous portion of the property line from which the entire house can be seen, with no part blocked by any obstruction.

Input

Output

Sample Input

2 6 6
0 15 0
3
1 2 1
3 4 1
12 13 1
1 5 5
0 10 0
1
0 15 1
0 0 0

Sample Output

8.80
No View

Source

Solution:

首先注意以下几点：

1. 视线恰好穿过障碍物(obstruction)边缘时也算被遮挡，即答案不会输出 0.00 ，取而代之的应该是 No View 。
2. 对于不严格在房子(house)和地界线(property line)之间的障碍物直接忽略。

计算每个障碍物能挡住的区间 [lY, rY]（如图所示，不考虑其他障碍物时，<lY 的部分和 >rY 的部分可以看到整个房子）。

将障碍物按 lY 排序，从左到右扫描记下当前最大rY，并即时计算并更新最大连续区间。

计算时要注意 lY 和 rY 是否超出地界线边缘，可以简单地规约为以下公式：

• maxsight = max{min(curlY, pl.Y2) - max(maxrY, pl.Y1)}，其中 curlY 为当前障碍物的 lY，maxrY 为先前所有障碍物中 rY 的最大值。

实现时，为计算直线斜率和函数值方便，可以将 x 和 y 坐标交换。

Code: O(Tnlogn) [188K, 0MS]

#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<cassert>
#include<iostream>
#include<algorithm>
#define eps 1e-8
#define dmax 1e10
using namespace std;

inline int dsgn(double x){
	if(fabs(x) <= eps) return 0;
	return x < 0 ? -1 : 1;
}

int n;

struct Object{  // For slope calculation, X-coordinate is exchanged with Y-coordinate 
	double Y1, Y2, X;
	
	// Declarations and implementations below are only available for "obstructions"
	
	double lY, rY;
	// lY is the leftest Y-coordinate that can see the left of House exactly through the right of this obstruction
	// rY is the rightest Y-coordinate that can see the right of House exactly through the left of this obstruction
	
	inline bool operator < (const Object &obj2) {return dsgn(lY - obj2.lY) < 0;}
} hs, pl, obs[55];

struct Line{
	double k, b;
	
	inline bool Construct(const double &PX, const double &PY, const double &QX, const double &QY){
		if(dsgn(PX - QX) == 0) return 1;  // The slope is infinite, construction fails
		k = (QY - PY) / (QX - PX);
		b = PY - k * PX;
		assert(dsgn(k * QX + b - QY) == 0);
		return 0;
	}
	
	inline double calcY(const double &x) {return k * x + b;}
} lin;

int main(){
	while(scanf("%lf%lf%lf", &hs.Y1, &hs.Y2, &hs.X) && (dsgn(hs.Y1) > 0 || dsgn(hs.Y2) > 0 || dsgn(hs.X) > 0)){
		scanf("%lf%lf%lf%d", &pl.Y1, &pl.Y2, &pl.X, &n);
		for(register int i = 1; i <= n; i++){
			scanf("%lf%lf%lf", &obs[i].Y1, &obs[i].Y2, &obs[i].X);
			if(dsgn(obs[i].X - hs.X) >= 0 || dsgn(obs[i].X - pl.X) <= 0) i--, n--;
			// If the obstructions are not between the house and the property line, ignore it
		}
		for(register int i = 1; i <= n; i++){
			lin.Construct(obs[i].X, obs[i].Y1, hs.X, hs.Y2);
			obs[i].lY = lin.calcY(pl.X);
			lin.Construct(obs[i].X, obs[i].Y2, hs.X, hs.Y1);
			obs[i].rY = lin.calcY(pl.X);
		}
		sort(obs + 1, obs + n + 1);
		double maxsight = -dmax, rmax = pl.Y1;
		maxsight = max(maxsight, min(obs[1].lY, pl.Y2) - rmax);
		for(register int i = 1; i < n; i++){
			rmax = max(rmax, obs[i].rY);
			maxsight = max(maxsight, min(obs[i + 1].lY, pl.Y2) - rmax);
		}
		rmax = max(rmax, obs[n].rY);
		maxsight = max(maxsight, pl.Y2 - rmax);
		if(dsgn(maxsight) <= 0) puts("No View");  // Only one point is considered as "No View", not "0.00"
		else printf("%.2f\n", maxsight);
	}
	return 0;
}