hdu-1071 The area

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本文介绍了一种使用计算几何的方法来求解由抛物线和直线围成的土地面积问题。通过给定三个关键点的位置坐标，利用定积分原理，推导出面积计算公式，并给出具体的实现代码。

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计算几何（定积分求面积）

The area

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 6614 Accepted Submission(s): 4633

Problem Description

Ignatius bought a land last week, but he didn't know the area of the land because the land is enclosed by a parabola and a straight line. The picture below shows the area. Now given all the intersectant points shows in the picture, can you tell Ignatius the area of the land?

Note: The point P1 in the picture is the vertex of the parabola.

Input

The input contains several test cases. The first line of the input is a single integer T which is the number of test cases. T test cases follow.
Each test case contains three intersectant points which shows in the picture, they are given in the order of P1, P2, P3. Each point is described by two floating-point numbers X and Y(0.0<=X,Y<=1000.0).

Output

For each test case, you should output the area of the land, the result should be rounded to 2 decimal places.

Sample Input

2
5.000000 5.000000
0.000000 0.000000
10.000000 0.000000
10.000000 10.000000
1.000000 1.000000
14.000000 8.222222

Sample Output

33.33
40.69

Hint
For float may be not accurate enough, please use double instead of float.

解题思路：

计算量还是挺大的

设
直线方程：y=kx+t…………………………………………………………（1）
抛物线方程：y=ax^2+bx+c……………………………………………………（2）

已知抛物线顶点p1（x1，y1），两线交点p2（x2，y2）和p3（x3，y3）

斜率k=(y3-y2)/(x3-x2)……………………………………………………（3）

把p3点代入（1）式结合（3）式可得：t=y3-(k*x3)

又因为p1是抛物线的顶点，可得关系：x1=-b/2a即b=-2a*x1………………（4）

把p1点代入（2）式结合（4）式可得：a*x1*x1-2a*x1*x1+c=y1化简得c=y1+a*x1*x1……（5）

把p2点代入（2）式结合（4）式和（5）式可得：a=(y2-y1)/((x1-x2)*(x1-x2))

于是通过3点求出了k，t，a，b，c即两个方程式已求出

题目时求面积s

通过积分可知：s=f（x2->x3）(积分符号)（ax^2+bx+c-（kx+t））
=f（x2->x3）(积分符号)（ax^2+（b-k）x+c-t）
=[a/3*x^3+(b-k)/2*x^2+(c-t)x](x2->x3)
=a/3*x3*x3*x3+(b-k)/2*x3*x3+(c-t)*x3-(a/3*x2*x2*x2+(b-k)/2*x2*x2+(c-t)*x2)

化简得：
面积公式：s=-(y2-y1)/（(x2-x1)*(x2-x1)）*（(x3-x2)*(x3-x2)*(x3-x2)）/6;

#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<iostream>
#include<algorithm>
using namespace std;
int main()
{
    int t;
    double x1,x2,x3,y1,y2,y3;
    scanf("%d",&t);
    while(t--)
    {
        scanf("%lf%lf",&x1,&y1);
        scanf("%lf%lf",&x2,&y2);
        scanf("%lf%lf",&x3,&y3);
        double temp1=y1-y2;
        double temp2=(x2-x1)*(x2-x1);
        double temp3=(x3-x2)*(x3-x2)*(x3-x2);
        printf("%.2lf\n",temp1/temp2*temp3/6.0);
    }
    return 0;
}