Unity【Bounds Vector3 Cross】- 如何判断一个物体是否在一个凸边体三维区域内

如图所示，本文介绍如何判断一个物体是否被一个凸边体区域所囊括，本文将该功能的实现拆分成了如下步骤：

1.如何判断两条线段是否相交

2.如何判断一个点是否在一个凸边形范围内（2D、xz轴构成的平面）

3.如何判断一个点是否在一个凸边体范围内（3D）

4.如何判断一个物体是否在一个凸边体范围内

依次实现：

1.如何判断两条线段是否相交：

通过矢量叉积的符号可以判断两矢量相互之间的顺逆时针关系，如下图所示，点A和点B分别在线段CD两侧，点C和点D分别在线段AB两侧，这时可以判断它们相交。判断点A和点B是否在线段CD两侧，也就是判断向量A-D和向量B-D在向量C-D的两侧，也就是叉积的结果是异号的，即：（A-D）X（C-D）*（B-D）X（C-D）< 0。同样的，判断点C和点B是否在线段AB的两侧：（D-A）X（B-A）*（C-A）X（B-A）< 0，以上这两个条件成立时，可判断两线段相交。

当然，出现以下这种情况，即（A-D）X（C-D）*（B-D）X（C-D）= 0时，两条线段也是相交的：

在Unity中封装该判断函数：

//判断AB与CD是否相交
private bool IsIntersection(Vector3 A, Vector3 B, Vector3 C, Vector3 D)
{//A-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign1 = Mathf.Sign(Vector3.Cross(A - D, C - D).y);//B-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign2 = Mathf.Sign(Vector3.Cross(B - D, C - D).y);//C-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign3 = Mathf.Sign(Vector3.Cross(C - A, B - A).y);//D-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign4 = Mathf.Sign(Vector3.Cross(D - A, B - A).y);//AB与CD相交返回true 否则返回falsereturn !Mathf.Approximately(sign1, sign2) && !Mathf.Approximately(sign3, sign4);
}

测试脚本如下：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private Transform a;[SerializeField] private Transform b;[SerializeField] private Transform c;[SerializeField] private Transform d;//判断AB与CD是否相交private bool IsIntersection(Vector3 A, Vector3 B, Vector3 C, Vector3 D){//A-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign1 = Mathf.Sign(Vector3.Cross(A - D, C - D).y);//B-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign2 = Mathf.Sign(Vector3.Cross(B - D, C - D).y);//C-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign3 = Mathf.Sign(Vector3.Cross(C - A, B - A).y);//D-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign4 = Mathf.Sign(Vector3.Cross(D - A, B - A).y);//AB与CD相交返回true 否则返回falsereturn !Mathf.Approximately(sign1, sign2) && !Mathf.Approximately(sign3, sign4);}private void OnDrawGizmos(){if (a == null || b == null || c == null || d == null) return;Handles.Label(a.position, "A");Handles.Label(b.position, "B");Handles.Label(c.position, "C");Handles.Label(d.position, "D");bool flag = IsIntersection(a.position, b.position, c.position, d.position);Handles.color = flag ? Color.cyan : Color.red;Handles.DrawLine(a.position, b.position);Handles.DrawLine(c.position, d.position);}
}

2.如何判断一个点是否在一个凸边形范围内（2D、xz轴构成的平面）：

若从该点发出的射线与平面内凸边形的交点的个数为偶数，则点在凸边形外，若为奇数，则点在凸边形内。因此取一条从该点向凸边形发出的射线，遍历凸边形的每一条边，判断射线与边的交点个数，若个数为奇数，则可以判断该点在凸边形范围内。

//判断点A是否在凸边型范围内
private bool IsInRange(Transform[] points, Vector3 A)
{//取第一条边中点Vector3 half01 = (points[0].position + points[1].position) * .5f;//中点延伸（射线）half01 += (half01 - A).normalized * 100000;//用于记录交点的个数int count = 0;//遍历for (int i = 0; i < points.Length; i++){var a = points[i % points.Length];var b = points[(i + 1) % points.Length];//判断是否相交if (IsIntersection(a.position, b.position, A, half01)) count++;}//交点个数为奇数则表示点A在凸边型范围内return count % 2 == 1;
}

测试代码如下：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private Transform point;[SerializeField] private Transform[] points; //凸边型顶点集合//判断点A是否在凸边型范围内private bool IsInRange(Transform[] points, Vector3 A){//取第一条边中点Vector3 half01 = (points[0].position + points[1].position) * .5f;//中点延伸（射线）half01 += (half01 - A).normalized * 100000;//用于记录交点的个数int count = 0;//遍历for (int i = 0; i < points.Length; i++){var a = points[i % points.Length];var b = points[(i + 1) % points.Length];//判断是否相交if (IsIntersection(a.position, b.position, A, half01)) count++;}//交点个数为奇数则表示点A在凸边型范围内return count % 2 == 1;}//判断AB与CD是否相交private bool IsIntersection(Vector3 A, Vector3 B, Vector3 C, Vector3 D){//A-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign1 = Mathf.Sign(Vector3.Cross(A - D, C - D).y);//B-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign2 = Mathf.Sign(Vector3.Cross(B - D, C - D).y);//C-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign3 = Mathf.Sign(Vector3.Cross(C - A, B - A).y);//D-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign4 = Mathf.Sign(Vector3.Cross(D - A, B - A).y);//AB与CD相交返回true 否则返回falsereturn !Mathf.Approximately(sign1, sign2) && !Mathf.Approximately(sign3, sign4);}private void OnDrawGizmos(){if (points.Length < 3 || point == null) return;bool flag = IsInRange(points, point.position);Handles.Label(point.position, "A");Vector3 half01 = (points[0].position + points[1].position) * .5f;half01 += (half01 - point.position).normalized * 100000;for (int i = 0; i < points.Length; i++){var a = points[i % points.Length];var b = points[(i + 1) % points.Length];Handles.color = flag ? Color.cyan : Color.red;Handles.DrawLine(a.position, b.position);Handles.Label(points[i].position, $"顶点{i + 1}");Handles.color = Color.yellow;if (IsIntersection(a.position, b.position, point.position, half01)){Handles.DrawLine(point.position, half01);}}}
}

3.如何判断一个点是否在一个凸边体范围内（3D）：

上述部分我们在xz轴所在的平面构建了一个凸边形，现在我们给其一个高度，即可构成一个凸边体空间区域：

要判断一个点是否在该凸边体范围内，只需要在满足处于xz轴所在的凸边形范围内的同时，其坐标点的y值既小等于凸边体height高度值的一半，又大等于负的高度值的一半：

封装判断函数：

//判断点A是否在凸边体范围内
private bool IsInRange(Transform[] points, float height, Vector3 A)
{bool flag = true;flag &= A.y <= height * .5f;flag &= A.y >= -height * .5f;flag &= IsInRange(points, A);return flag;
}

测试代码如下：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private Transform point;[SerializeField] private Transform[] points; //凸边型顶点集合[SerializeField] private float height = 1f;//判断点A是否在凸边体范围内private bool IsInRange(Transform[] points, float height, Vector3 A){bool flag = true;flag &= A.y <= height * .5f;flag &= A.y >= -height * .5f;flag &= IsInRange(points, A);return flag;}//判断点A是否在凸边型范围内private bool IsInRange(Transform[] points, Vector3 A){//取第一条边中点Vector3 half01 = (points[0].position + points[1].position) * .5f;//中点延伸（射线）half01 += (half01 - A).normalized * 100000;//用于记录交点的个数int count = 0;//遍历for (int i = 0; i < points.Length; i++){var a = points[i % points.Length];var b = points[(i + 1) % points.Length];//判断是否相交if (IsIntersection(a.position, b.position, A, half01)) count++;}//交点个数为奇数则表示点A在凸边型范围内return count % 2 == 1;}//判断AB与CD是否相交private bool IsIntersection(Vector3 A, Vector3 B, Vector3 C, Vector3 D){//A-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign1 = Mathf.Sign(Vector3.Cross(A - D, C - D).y);//B-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign2 = Mathf.Sign(Vector3.Cross(B - D, C - D).y);//C-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign3 = Mathf.Sign(Vector3.Cross(C - A, B - A).y);//D-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign4 = Mathf.Sign(Vector3.Cross(D - A, B - A).y);//AB与CD相交返回true 否则返回falsereturn !Mathf.Approximately(sign1, sign2) && !Mathf.Approximately(sign3, sign4);}private void OnDrawGizmos(){if (points.Length < 3 || point == null) return;bool flag = IsInRange(points, height, point.position);Handles.color = flag ? Color.cyan : Color.red;Handles.Label(point.position, "A");for (int i = 0; i < points.Length; i++){Handles.Label(points[i].position - new Vector3(0, height * .5f, 0), $"顶点{i + 1}");Handles.Label(points[i].position + new Vector3(0, height * .5f, 0), $"顶点{i + 1 + points.Length}");var a = points[i % points.Length];var b = points[(i + 1) % points.Length];var minA = a.position - new Vector3(0, height * .5f, 0);var maxA = a.position + new Vector3(0, height * .5f, 0);var minB = b.position - new Vector3(0, height * .5f, 0);var maxB = b.position + new Vector3(0, height * .5f, 0);Handles.DrawAAPolyLine(minA, minB);Handles.DrawAAPolyLine(maxA, maxB);Handles.DrawAAPolyLine(minA, maxA);Handles.DrawAAPolyLine(minB, maxB);}}
}

4.如何判断一个物体是否在一个凸边体范围内：

上述部分判断的是一个坐标点是否在一个凸边体范围内，要判断一个物体是否被该凸边体区域所囊括，需要获取该物体及其子物体构成的Bounds边界盒，如果Bounds边界盒的每一个顶点都在该凸边体范围内，则可以大致推断该物体被这个凸边体所囊括，当然要想更精确还是要获取该物体的更为精确的边界点，在这里不做延申。

首先来看Unity圣典中关于Bounds边界盒及其核心变量的介绍：

其中max、min分别是最大、最小点，可以通过这两点获取到其它各顶点的坐标，测试代码如下：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private GameObject obj;private void OnDrawGizmos(){if (obj == null) return;Bounds bounds = obj.GetComponent<MeshRenderer>().bounds;Vector3 point1 = bounds.min;Vector3 point2 = bounds.max;Vector3 point3 = new Vector3(point1.x, point1.y, point2.z);Vector3 point4 = new Vector3(point1.x, point2.y, point1.z);Vector3 point5 = new Vector3(point2.x, point1.y, point1.z);Vector3 point6 = new Vector3(point1.x, point2.y, point2.z);Vector3 point7 = new Vector3(point2.x, point1.y, point2.z);Vector3 point8 = new Vector3(point2.x, point2.y, point1.z);Handles.DrawLine(point6, point2);Handles.DrawLine(point2, point8);Handles.DrawLine(point8, point4);Handles.DrawLine(point4, point6);Handles.DrawLine(point3, point7);Handles.DrawLine(point7, point5);Handles.DrawLine(point5, point1);Handles.DrawLine(point1, point3);Handles.DrawLine(point6, point3);Handles.DrawLine(point2, point7);Handles.DrawLine(point8, point5);Handles.DrawLine(point4, point1);Handles.Label(point1, "顶点1");Handles.Label(point2, "顶点2");Handles.Label(point3, "顶点3");Handles.Label(point4, "顶点4");Handles.Label(point5, "顶点5");Handles.Label(point6, "顶点6");Handles.Label(point7, "顶点7");Handles.Label(point8, "顶点8");}
}

一个物体可能包含若干个带有MeshRenderer组件的子物体，因此我们要获取一个囊括所有的Bounds边界盒，要使用到Bounds类中的Encapsulate函数：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private GameObject obj;private void OnDrawGizmos(){if (obj == null) return;Bounds bounds = new Bounds(Vector3.zero, Vector3.zero);//获取所有MeshRenderer 包括子物体var mrs = obj.GetComponentsInChildren<MeshRenderer>(true);Vector3 center = Vector3.zero;for (int i = 0; i < mrs.Length; i++){center += mrs[i].bounds.center;//Encapsulate函数重新计算boundsbounds.Encapsulate(mrs[i].bounds);}Vector3 point1 = bounds.min;Vector3 point2 = bounds.max;Vector3 point3 = new Vector3(point1.x, point1.y, point2.z);Vector3 point4 = new Vector3(point1.x, point2.y, point1.z);Vector3 point5 = new Vector3(point2.x, point1.y, point1.z);Vector3 point6 = new Vector3(point1.x, point2.y, point2.z);Vector3 point7 = new Vector3(point2.x, point1.y, point2.z);Vector3 point8 = new Vector3(point2.x, point2.y, point1.z);Handles.DrawLine(point6, point2);Handles.DrawLine(point2, point8);Handles.DrawLine(point8, point4);Handles.DrawLine(point4, point6);Handles.DrawLine(point3, point7);Handles.DrawLine(point7, point5);Handles.DrawLine(point5, point1);Handles.DrawLine(point1, point3);Handles.DrawLine(point6, point3);Handles.DrawLine(point2, point7);Handles.DrawLine(point8, point5);Handles.DrawLine(point4, point1);Handles.Label(point1, "顶点1");Handles.Label(point2, "顶点2");Handles.Label(point3, "顶点3");Handles.Label(point4, "顶点4");Handles.Label(point5, "顶点5");Handles.Label(point6, "顶点6");Handles.Label(point7, "顶点7");Handles.Label(point8, "顶点8");}
}

封装判断函数：

//判断一个物体是否在凸边体范围内
private bool IsInRange(Transform[] points, float height, GameObject obj)
{Bounds bounds = new Bounds(Vector3.zero, Vector3.zero);//获取所有MeshRenderer 包括子物体var mrs = obj.GetComponentsInChildren<MeshRenderer>(true);Vector3 center = Vector3.zero;for (int i = 0; i < mrs.Length; i++){center += mrs[i].bounds.center;//Encapsulate函数重新计算boundsbounds.Encapsulate(mrs[i].bounds);}Vector3 min = bounds.min;Vector3 max = bounds.max;return IsInRange(points, height, min)&& IsInRange(points, height, max)&& IsInRange(points, height, new Vector3(min.x, min.y, max.z))&& IsInRange(points, height, new Vector3(min.x, max.y, min.z))&& IsInRange(points, height, new Vector3(max.x, min.y, min.z))&& IsInRange(points, height, new Vector3(min.x, max.y, max.z))&& IsInRange(points, height, new Vector3(max.x, min.y, max.z))&& IsInRange(points, height, new Vector3(max.x, max.y, min.z));
}

测试代码如下：

using UnityEngine;
using UnityEditor;public class Example : MonoBehaviour
{[SerializeField] private GameObject obj;[SerializeField] private Transform[] points; //凸边型顶点集合[SerializeField] private float height = 1f;//判断一个物体是否在凸边体范围内private bool IsInRange(Transform[] points, float height, GameObject obj){Bounds bounds = new Bounds(Vector3.zero, Vector3.zero);//获取所有MeshRenderer 包括子物体var mrs = obj.GetComponentsInChildren<MeshRenderer>(true);Vector3 center = Vector3.zero;for (int i = 0; i < mrs.Length; i++){center += mrs[i].bounds.center;//Encapsulate函数重新计算boundsbounds.Encapsulate(mrs[i].bounds);}Vector3 min = bounds.min;Vector3 max = bounds.max;return IsInRange(points, height, min)&& IsInRange(points, height, max)&& IsInRange(points, height, new Vector3(min.x, min.y, max.z))&& IsInRange(points, height, new Vector3(min.x, max.y, min.z))&& IsInRange(points, height, new Vector3(max.x, min.y, min.z))&& IsInRange(points, height, new Vector3(min.x, max.y, max.z))&& IsInRange(points, height, new Vector3(max.x, min.y, max.z))&& IsInRange(points, height, new Vector3(max.x, max.y, min.z));}//判断点A是否在凸边体范围内private bool IsInRange(Transform[] points, float height, Vector3 A){bool flag = true;flag &= A.y <= height * .5f;flag &= A.y >= -height * .5f;flag &= IsInRange(points, A);return flag;}//判断点A是否在凸边型范围内private bool IsInRange(Transform[] points, Vector3 A){//取第一条边中点Vector3 half01 = (points[0].position + points[1].position) * .5f;//中点延伸（射线）half01 += (half01 - A).normalized * 100000;//用于记录交点的个数int count = 0;//遍历for (int i = 0; i < points.Length; i++){var a = points[i % points.Length];var b = points[(i + 1) % points.Length];//判断是否相交if (IsIntersection(a.position, b.position, A, half01)) count++;}//交点个数为奇数则表示点A在凸边型范围内return count % 2 == 1;}//判断AB与CD是否相交private bool IsIntersection(Vector3 A, Vector3 B, Vector3 C, Vector3 D){//A-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign1 = Mathf.Sign(Vector3.Cross(A - D, C - D).y);//B-D与C-D叉积结果大等于0时返回1 小于0时返回-1float sign2 = Mathf.Sign(Vector3.Cross(B - D, C - D).y);//C-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign3 = Mathf.Sign(Vector3.Cross(C - A, B - A).y);//D-A与B-A叉积结果大等于0时返回1 小于0时返回-1float sign4 = Mathf.Sign(Vector3.Cross(D - A, B - A).y);//AB与CD相交返回true 否则返回falsereturn !Mathf.Approximately(sign1, sign2) && !Mathf.Approximately(sign3, sign4);}private void OnDrawGizmos(){if (points.Length < 3 || obj == null) return;bool flag = IsInRange(points, height, obj);Handles.color = flag ? Color.cyan : Color.red;for (int i = 0; i < points.Length; i++){Handles.Label(points[i].position - new Vector3(0, height * .5f, 0), $"顶点{i + 1}");Handles.Label(points[i].position + new Vector3(0, height * .5f, 0), $"顶点{i + 1 + points.Length}");var a = points[i % points.Length];var b = points[(i + 1) % points.Length];var minA = a.position - new Vector3(0, height * .5f, 0);var maxA = a.position + new Vector3(0, height * .5f, 0);var minB = b.position - new Vector3(0, height * .5f, 0);var maxB = b.position + new Vector3(0, height * .5f, 0);Handles.DrawAAPolyLine(minA, minB);Handles.DrawAAPolyLine(maxA, maxB);Handles.DrawAAPolyLine(minA, maxA);Handles.DrawAAPolyLine(minB, maxB);}Bounds bounds = new Bounds(Vector3.zero, Vector3.zero);//获取所有MeshRenderer 包括子物体var mrs = obj.GetComponentsInChildren<MeshRenderer>(true);Vector3 center = Vector3.zero;for (int i = 0; i < mrs.Length; i++){center += mrs[i].bounds.center;//Encapsulate函数重新计算boundsbounds.Encapsulate(mrs[i].bounds);}Vector3 point1 = bounds.min;Vector3 point2 = bounds.max;Vector3 point3 = new Vector3(point1.x, point1.y, point2.z);Vector3 point4 = new Vector3(point1.x, point2.y, point1.z);Vector3 point5 = new Vector3(point2.x, point1.y, point1.z);Vector3 point6 = new Vector3(point1.x, point2.y, point2.z);Vector3 point7 = new Vector3(point2.x, point1.y, point2.z);Vector3 point8 = new Vector3(point2.x, point2.y, point1.z);Handles.color = Color.yellow;Handles.DrawLine(point6, point2);Handles.DrawLine(point2, point8);Handles.DrawLine(point8, point4);Handles.DrawLine(point4, point6);Handles.DrawLine(point3, point7);Handles.DrawLine(point7, point5);Handles.DrawLine(point5, point1);Handles.DrawLine(point1, point3);Handles.DrawLine(point6, point3);Handles.DrawLine(point2, point7);Handles.DrawLine(point8, point5);Handles.DrawLine(point4, point1);}
}