碰撞检测之Ray-Box检测
Posted 拳四郎
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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
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