凸包(Convex Hull)问题求解--Gift-Wrapping 算法

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凸包问题（Convex Hull）求解--卷包裹(Gift-Wrapping) 算法

1.前言

最近在做MIT 6.031的问题集0时遇到了要计算凸包的问题，题中提示要用Gift Wrapping算法。作为一个在实际工程中需要应用的求解算法来讲它并不是最好的，因为它有着 $O(nh)$ 的时间复杂度，但是我们依然可以通过它更好地理解问题的实质。更好地学习和应用这个基本算法。

2.Convex Hull 问题概述

百度百科中给出的定义为：

凸包（Convex Hull）是一个计算几何（图形学）中的概念。

在一个实数向量空间V中，对于给定集合X，所有包含X的凸集的交集S被称为X的凸包。X的凸包可以用X内所有点(X1，...Xn)的凸组合来构造.

在二维欧几里得空间中，凸包可想象为一条刚好包著所有点的橡皮圈。

用不严谨的话来讲，给定二维平面上的点集，凸包就是将最外层的点连接起来构成的凸多边形，它能包含点集中所有的点。

通俗来说，二维的凸包就是在平面上给定的若干个点组成的点集中选取最外围的点，使得他们的连线组成的多边形能够覆盖全部的点。并且这些点应当满足以下两个条件：

（1）组成的凸多边形能够覆盖所有的点

（2）所选取的点数越少越好

MIT 6.031 problem Set 0 中给出了具体的设计规约：

    /*** Given a set of points, compute the convex hull, the smallest convex set that contains all the points * in a set of input points. The gift-wrapping algorithm is one simple approach to this problem, and * there are other algorithms too.* * @param points a set of points with xCoords and yCoords. It might be empty, contain only 1 point, two points or more.* @return minimal subset of the input points that form the vertices of the perimeter of the convex hull*/

3.卷包裹（Gift-Wrapping 算法）

3.1算法思想

该算法的思想为

1、首先选取一个最靠边界的点（例如最左上或最右下，我选的最左上）作为起始点，以这个点为基准开始选择下一个点。

2、遍历点集，考察它们相对于基准点所偏转的角度：（即目标点与基准点连线与当前基准点朝向的方向形成的射线所形成的角度），第一个点所朝向的角度设置为0。其中将以north（正上）方向为基准的顺时针偏转角定义为朝向的角度。在点集中选出偏转角最小的点作为下一个点并将其加入结果点集，同时将基准点设置为该点。

3、重复过程2，直至选取的点为起始点。

需要注意的是：在两个点偏转角度相同时，为保证所选取的点数最少，应该选取与基准点距离更大的点。

3.2代码实现（使用Java实现）

首先将点（Point）定义如下：

public class Point {private final double x;private final double y;/*** Construct a point at the given coordinates.* @param x x-coordinate* @param y y-coordinate*/public Point(double x, double y) {this.x = x;this.y = y;}/*** @return x-coordinate of the point*/public double x() {return x;}/*** @return y-coordinate of the point*/public double y() {return y;}
}

然后是具体的方法：

import java.util.Set;
import java.util.HashSet;
public class TurtleSoup {/*** Given the current direction, current location, and a target location, calculate the Bearing* towards the target point.* * The return value is the angle input to turn() that would point the turtle in the direction of* the target point (targetX,targetY), given that the turtle is already at the point* (currentX,currentY) and is facing at angle currentBearing. The angle must be expressed in* degrees, where 0 <= angle < 360. ** * @param currentBearing current direction as clockwise from north* @param currentX current location x-coordinate* @param currentY current location y-coordinate* @param targetX target point x-coordinate* @param targetY target point y-coordinate* @return adjustment to Bearing (right turn amount) to get to target point,*         must be 0 <= angle < 360*/public static double newCalculateBearingToPoint(double currentBearing, double currentX, double currentY,double targetX, double targetY) {//计算偏转角度double hei=Math.abs(currentY-targetY);double wid=Math.abs(currentX-targetX);double slop = Math.sqrt(hei*hei+wid*wid);double CAngle =Math.toDegrees(Math.asin(wid/slop));double TAngle;if(currentX>=targetX&&currentY>targetY) {TAngle=180+CAngle;}else if(currentX>targetX&&currentY<=targetY) {TAngle=360-CAngle;}else if(currentX<targetX&&currentY>=targetY) {TAngle=180-CAngle;}else if(currentX<=targetX&&currentY<targetY) {TAngle=CAngle;}else {return 359;}return (TAngle>=currentBearing)?(TAngle-currentBearing):(360-(currentBearing-TAngle));}public static double calculateDistance(double currentX,double currentY,double targetX,double targetY) {//计算两点距离double wid = Math.abs(currentX-targetX);double hei = Math.abs(currentY-targetY);return Math.sqrt(wid*wid+hei*hei);}/*** Given a set of points, compute the convex hull, the smallest convex set that contains all the points * in a set of input points. The gift-wrapping algorithm is one simple approach to this problem, and * there are other algorithms too.* * @param points a set of points with xCoords and yCoords. It might be empty, contain only 1 point, two points or more.* @return minimal subset of the input points that form the vertices of the perimeter of the convex hull*/public static Set<Point> convexHull(Set<Point> points) {if(points.size()<=2) {return points;}HashSet<Point> result =new HashSet<Point>();Point tmp=points.iterator().next();Point start = tmp;Point targ = tmp;double angle = 0,a1=0,at=0;for(Point p:points) {if(p.x()<start.x()||p.x()==start.x()&&p.y()>start.y())start = p;}result.add(start);Point ptr = start;while(true) {at=TurtleSoup.newCalculateBearingToPoint(angle, ptr.x(), ptr.y(), targ.x(), targ.y());for(Point q:points) {if(targ==q)continue;a1=TurtleSoup.newCalculateBearingToPoint(angle, ptr.x(), ptr.y(), q.x(), q.y());if(a1<at) {//选择偏转角度最小的targ =q;at=a1;}else if(a1==at) {//选择距离更大的double dist=TurtleSoup.calculateDistance(ptr.x(), ptr.y(), targ.x(), targ.y());double dis1=TurtleSoup.calculateDistance(ptr.x(), ptr.y(), q.x(), q.y());if(dis1>dist) {targ=q;at=a1;}}}if(targ == start)//终止条件break;else {angle=at;result.add(targ);ptr =targ;}}return result;}
}