Ray Box 碰撞检测

Box采用OBB定义：

enum AxisDefine{    X ,     Y ,     Z};// data member Vector3f Center;Vector3f Axis[3];float extent[3];    // 边长的一半

Ray采用参数式Point=Origin+t∗Direction,0≤t≤∞

// dataVector3f Origin;Vector3f Direction;

Ray.Origin⇒O
Ray.Direction⇒Dir−→−
Box.Center⇒C
Box.Extent[i]⇒ei
Box.Axis[i]⇒Axis[i]−→−−−

采用分离轴方法来进行碰撞检测：
一：首先我们使用Box的面法线（即坐标轴）Aixs[i]−→−−−作为分离轴

在这种情况下Box在分离轴上的投影即为该轴向的边

计算光线起点在分离轴上的投影：

delta−→−−=O−C=CO−→−
ProRay=delta−→−−⋅Axis[i]−→−−−

Box在分离轴上的投影为：
−ei≤ProBox≤ei

当满足如下两种情况：

1. ProRay>ei且Dir−→−⋅Axis[i]−→−−−≥0(即光线方向与分离轴的方向范围为(−90。,90。)
2. ProRay<−ei且Dir−→−⋅Axis[i]−→−−−≤0

将二者结合：|ProRay|>ei and (Dir−→−⋅Axis[i]−→−−−)∗ProRay≥0

二：射线方向向量与面法线的叉乘作为分离轴
为了使得计算简单，我们可以将OBB转换为AABB

Dir−→−⇒Dir−→− ′=(Dir−→−⋅Axis[X]−→−−−−,Dir−→−⋅Axis[Y]−→−−−−,Dir−→−⋅Axis[Z]−→−−−−)

O⇒O ′=(delta−→−−⋅Axis[X]−→−−−−,delta−→−−⋅Axis[Y]−→−−−−,delta−→−−⋅Axis[Z]−→−−−−)
则Box的三个轴为：
Axis[X]−→−−−−⇒Axis[X]−→−−−− ′=(1,0,0)
Axis[Y]−→−−−−⇒Axis[Y]−→−−−− ′=(0,1,0)
Axis[Z]−→−−−−⇒Axis[Z]−→−−−− ′=(0,0,1)

则分离轴为：
v→=Dir−→− ′×Axis[i]−→−−− ′

Axis[i]−→−−− ′ v→ case Axis[X]−→−−−− ′=(1,0,0) (0,Dir−→−⋅Axis[Z]−→−−−−,−Dir−→−⋅Axis[Y]−→−−−−) (1) Axis[Y]−→−−−− ′=(0,1,0) (−Dir−→−⋅Axis[Z]−→−−−−,0,Dir−→−⋅Axis[X]−→−−−−) (2) Axis[Z]−→−−−− ′=(0,0,1) (Dir−→−⋅Axis[Y]−→−−−−,−Dir−→−⋅Axis[X]−→−−−−,0) (3)

Box在v→上的投影：
ProBox=e0∗|Axis[X]−→−−−− ′⋅v→|+e1∗|Axis[Y]−→−−−− ′⋅v→|+e2∗|Axis[Z]−→−−−− ′⋅v→|

v→ ProBox case (0,Dir−→−⋅Axis[Z]−→−−−−,−Dir−→−⋅Axis[Y]−→−−−−) e1∗|Dir−→−⋅Axis[Z]−→−−−−+e2∗|Dir−→−⋅Axis[Y]−→−−−−| (1) (−Dir−→−⋅Axis[Z]−→−−−−,0,Dir−→−⋅Axis[X]−→−−−−) e0∗|Dir−→−⋅Axis[Z]−→−−−−|+e2∗|Dir−→−⋅Axis[X]−→−−−−| (2) (Dir−→−⋅Axis[Y]−→−−−−,−Dir−→−⋅Axis[X]−→−−−−,0) e0∗|Dir−→−⋅Axis[Y]−→−−−−|+e1∗|Dir−→−⋅Axis[X]−→−−−−| (3)

Ray起点在其上的投影：
ProRay=O ′⋅v→=(delta−→−−⋅Axis[X]−→−−−−,delta−→−−⋅Axis[Y]−→−−−−,delta−→−−⋅Axis[Z]−→−−−−)⋅v→

v→与Ray.Direction垂直，因此Ray本身投影在分离轴v→为一个点

若|ProRay|>ProBox，则未碰撞

以下看看(1)情况下的v→来计算ProRay：
v→=(0,Dir−→−⋅Axis[Z]−→−−−−,−Dir−→−⋅Axis[Y]−→−−−−)
ProRay=(delta−→−−⋅Axis[X]−→−−−−,delta−→−−⋅Axis[Y]−→−−−−,delta−→−−⋅Axis[Z]−→−−−−)⋅(0,Dir−→−⋅Axis[Z]−→−−−−,−Dir−→−⋅Axis[Y]−→−−−−)
=(delta−→−−⋅Axis[Y]−→−−−−)∗(Dir−→−⋅Axis[Z]−→−−−−)−(delta−→−−⋅Axis[Z]−→−−−−)∗(Dir−→−⋅Axis[Y]−→−−−−)
=(delta−→−−.x∗Axis[Y]−→−−−−.x+delta−→−−.y∗Axis[Y]−→−−−−.y+delta−→−−.z∗Axis[Y]−→−−−−.z)∗(Dir−→−.x∗Axis[Z]−→−−−−.x+Dir−→−.y∗Axis[Z]−→−−−−.y+Dir−→−.z∗Axis[Z]−→−−−−.z)−(delta−→−−.x∗Axis[Z]−→−−−−.x+delta−→−−.y∗Axis[Z]−→−−−−.y+delta−→−−.z∗Axis[Z]−→−−−−.z)∗(Dir−→−.x∗Axis[Y]−→−−−−.x+Dir−→−.y∗Axis[Y]−→−−−−.y+Dir−→−.z∗Axis[Y]−→−−−−.z)
=(delta−→−−.z∗Dir−→−.y,delta−→−−.x∗Dir−→−.z,delta−→−−.y∗Dir−→−.x)⋅(−Axis[X]−→−−−−)+(delta−→−−.y∗Dir−→−.z,delta−→−−.z∗Dir−→−.x,delta−→−−.x∗Dir−→−.y)⋅(Axis[X]−→−−−−)
=(delta−→−−×Dir−→−)⋅Axis[X]−→−−−−

其他情况类似，因此：

v→ ProRay case (0,Dir−→−⋅Axis[Z]−→−−−−,−Dir−→−⋅Axis[Y]−→−−−−) (delta−→−−×Dir−→−)⋅Axis[X]−→−−−− (1) (−Dir−→−⋅Axis[Z]−→−−−−,0,Dir−→−⋅Axis[X]−→−−−−) (delta−→−−×Dir−→−)⋅Axis[Y]−→−−−− (2) (Dir−→−⋅Axis[Y]−→−−−−,−Dir−→−⋅Axis[X]−→−−−−,0) (delta−→−−×Dir−→−)⋅Axis[Z]−→−−−− (3)