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