求两圆相交的面积

一个无穷大的花园,花园中修建了两个浇水喷头,每个喷头可以给以它中心的一定距离内的花浇水。请问两个喷头可以给多大面积的花园浇水?

package step1;

public class Task {

	 public double solve(double x1, double y1, double r1,
              double x2, double y2, double r2) {
		  /********* Begin *********/
		double area = Math.PI * (r1 * r1 + r2 * r2);
		double d = Math.sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
		if (d >= r1 + r2)
			return area;
		else if (Math.abs(r1 - r2) >= d) {
			if (r1 > r2)
				return Math.PI * r1 * r1;
			else
				return Math.PI * r2 * r2;
		} else {
			// 正弦定理求扇形圆心角
			double a1 = 2 * Math.acos((r1 * r1 + d * d - r2 * r2) / 2 / r1 / d);
			double a2 = 2 * Math.acos((r2 * r2 + d * d - r1 * r1) / 2 / r2 / d);
			// 两个扇形面积和减去四边形的面积即为相交区域面积
			// 四边形面积再转化为两个三角形的面积之和来计算
			double ans = r1 * r1 * a1 / 2 + r2 * r2 * a2 / 2 - r1 * r1 * Math.sin(a1) / 2 - r2 * r2 * Math.sin(a2) / 2;
			return area - ans;
		}
		  
		  
		  /********* End *********/
	  }
}

 

本程序是两圆相交交点,输入信息为两圆心坐标和半径值。如有更好的算法,欢迎交流!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
在MATLAB中计算两个相交部分的面积需要一些数学知识和编程技巧。首先,你需要确定这两个的中心点坐标 (x1, y1) 和 (x2, y2),以及它们各自的半径 r1 和 r2。假设两个不完全重叠并且有交集。 以下是基本步骤: 1. 确定心到另一个心的距离 d = sqrt((x2 - x1)^2 + (y2 - y1)^2)。 2. 计算两个半径之和 R = r1 + r2 和之差 r = |r1 - r2|。 3. 检查 d 的值:如果 d > R,则两个不会相交;如果 d < r,则两个完全包含;只有当 r <= d <= R 时,才会有交集。 4. 计算两个的公共弦长 c = 2 * sqrt(R^2 - d^2) 或者 c = 0(如果d>R)。 5. 如果c > 0,那么心距的一半和公共弦长构成三角形,可以用 Heron 公式计算出三角形面积 S = sqrt[p * (p - a) * (p - b) * (p - c)],其中 p = (a + b + c) / 2,a、b 和 c 分别是三角形的边长。 6. 最后的交集面积 A = S(如果S>0),否则A=0。 如果你想要编写一个MATLAB函数来进行这个计算,可以尝试以下示例代码: ```matlab function area_intersection = circle_intersection(x1, y1, r1, x2, y2, r2) % Step 1: Calculate distance between centers d = norm([x2-x1; y2-y1]); % Step 2: Determine if circles intersect if d > r1 + r2 || d < abs(r1 - r2) area_intersection = 0; return; end % Step 3: Calculate common chord length c = 2 * sqrt((r1 + r2)^2 - d^2); % Step 4: Form triangle and calculate intersection area a = b = c; p = a + b + c; S = sqrt(p * (p - a) * (p - b) * (p - c)); area_intersection = S; end ```
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