碰撞检测之Ray-Box检测

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 framevoid 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