Separating Axis Theorem(分离轴理论)Raycast

Separating Axis Theorem(分离轴理论)

在学习Ray-Box检测之前，首先来学习一些这个分离轴理论!

先说二维情况，一句话

Two convex polygons do not intersect if and only if there exists a line such that the projections of the two polygons onto the line do not intersect.

咳咳，翻译一下

两个凸包多边形，当且仅当存在一条线，这两个多边形在这条线上的投影不相交，则这两个多边形也不相交， 如下图所示

将多边形换成多面体，线变成面，就变成了三维空间中的分离轴了。

对于矩形，假设出现碰撞的情况，则存在分离轴平行矩形的一条边。（这个后面会证明）

Ray - Rect检测

在到三维之前，还是来看二维的情况,也就是Ray-Rect检测。

假定Rect的中心为原点，所以就是下面这样

首先要面对的一个问题就是射线的原点是否在矩形的内部，这里就用到了分离轴的定理。

将矩形投影到对应的轴上，如果没有和射线原点的投影重合，那么就不在矩形里面。

接下来判断是否相交，这里提到了一个简单 slab method，简单说来，首先将矩形的四条边无限延伸，那么整个平面就被矩形分割成了几个部分，用这个”井字“去切割射线，如果得得到的线段在矩形内，那么就相交了。如下图所示，绿色的射线是相交的，红色的没有相交。

简单的代码

bool intersection(box b, ray r) {
    double tmin = -INFINITY, tmax = INFINITY;

    if (ray.n.x != 0.0) {
        double tx1 = (b.min.x - r.x0.x)/r.n.x;
        double tx2 = (b.max.x - r.x0.x)/r.n.x;

        tmin = max(tmin, min(tx1, tx2));
        tmax = min(tmax, max(tx1, tx2));
    }

    if (ray.n.y != 0.0) {
        double ty1 = (b.min.y - r.x0.y)/r.n.y;
        double ty2 = (b.max.y - r.x0.y)/r.n.y;

        tmin = max(tmin, min(ty1, ty2));
        tmax = min(tmax, max(ty1, ty2));
    }

    return tmax >= tmin;
}

三维空间

直接贴代码了。

加了坐标系的转换，代码是参考PhysX里优化过的代码，但原理基本不变。

public static bool Raycast(Ray ray, float distance, Box box, out RaycastHitInfo hitInfo)
      {
          Quaternion inverRot = Quaternion.Inverse(box.rotation);
          Vector3 origin = ray.origin - box.center;

          Vector3 localOrigin = inverRot * origin;
          Vector3 localDir = inverRot * ray.direction;
          Ray localRay = new Ray(localOrigin, localDir);

          if (!IntersectRayAABB(localRay, distance, 0.5f * box.extents, out hitInfo))
          {
              return false ;
          }
          hitInfo.normal = box.rotation * hitInfo.normal;
          hitInfo.point = box.rotation * hitInfo.point + box.center;
          return true ;
      }

      public static bool IntersectRayAABB(Ray ray, float distance, Vector3 dimension, out RaycastHitInfo hitInfo)
      {
          float RAYAABB_EPSILON = 0.00001f;
          hitInfo = new RaycastHitInfo();
          Vector3 minPos = -dimension;
          Vector3 maxPos = dimension;
          Vector3 maxT = -Vector3.one;
          bool isInside = true ;

          for ( int i = 0; i < 3; i++)
          {
              if (ray.origin[i] < minPos[i])
              {
                  hitInfo.point[i] = minPos[i];
                  isInside = false ;
                  if (ray.direction[i] != 0)
                      maxT[i] = (minPos[i] - ray.origin[i]) / ray.direction[i];
              }
              else if (ray.origin[i] > maxPos[i])
              {
                  hitInfo.point[i] = maxPos[i];
                  isInside = false ;
                  if (ray.direction[i] != 0)
                      maxT[i] = (maxPos[i] - ray.origin[i]) / ray.direction[i];
              }
          }

          // Ray origin inside bounding box
          if (isInside)
          {
              hitInfo.point = ray.origin;
              hitInfo.distance = 0;
              hitInfo.normal = -ray.direction;
              return true ;
          }

          // Get largest of the maxT's for final choice of intersection
          int whichPlane = 0;
          if (maxT[1] > maxT[whichPlane]) whichPlane = 1;
          if (maxT[2] > maxT[whichPlane]) whichPlane = 2;

          //Ray distance large than ray cast ditance
          if (maxT[whichPlane] > distance)
          { return false ; }

          // Check final candidate actually inside box
          for ( int i = 0; i < 3; i++)
          {
              if (i != whichPlane)
              {
                  hitInfo.point[i] = ray.origin[i] + maxT[whichPlane] * ray.direction[i];
                  if (hitInfo.point[i] < minPos[i] - RAYAABB_EPSILON || hitInfo.point[i] > maxPos[i] + RAYAABB_EPSILON)
                      return false ;
                  //  if (hitInfo.point[i] < minPos[i] || hitInfo.point[i] > maxPos[i])
                  //  return false;
              }
          }

          hitInfo.distance = maxT[whichPlane];
          Vector3 normal = Vector3.zero;
          normal[whichPlane] = (hitInfo.point[whichPlane] > 0) ? 1 : -1;
          hitInfo.normal = normal;
          return true ;
      }

测试代码

public class RayBoxTester : MonoBehaviour {

    public GameObject box;
    Box _box;
    // Use this for initialization
    void Start () {
        _box = new Box(Vector3.zero, Vector3.one, Quaternion.identity);
    }

    // Update is called once per frame
    void Update () {
        //Ray OBB test.
        Ray ray2 = new Ray(Vector3.zero, new Vector3(1, 1, 1));
        RaycastHitInfo hitinfo2;
        //ray2.origin = rayOrigin.transform.position;
        float castDistance = 10f;

        _box.center = box.transform.position;
        _box.extents = box.transform.localScale;
        _box.rotation = box.transform.rotation;
        if (NRaycastTests.Raycast(ray2, castDistance, _box, out hitinfo2))
        {
            Debug.DrawLine(ray2.origin, ray2.origin + ray2.direction * hitinfo2.distance, Color.red, 0, false );
            Debug.DrawLine(hitinfo2.point, hitinfo2.point + hitinfo2.normal, Color.green, 0, false );
        }
        else
        {
            Debug.DrawLine(ray2.origin, ray2.origin + ray2.direction * castDistance, Color.blue, 0, false );
        }
    }
}

结果

       

收工。

参考

FAST, BRANCHLESS RAY/BOUNDING BOX INTERSECTIONS  - https://tavianator.com/fast-branchless-raybounding-box-intersections/

Hyperplane_separation_theorem - https://en.wikipedia.org/wiki/Hyperplane_separation_theorem

Ray - Box Intersection  - http://www.siggraph.org/education/materials/HyperGraph/raytrace/rtinter3.htm

PhysX 3.3  source code

碰撞检测之Ray-Box检测