Bounding Volumes - Real-Time Collision Detection - page 103

Table 4.1 The 15 separating axis tests needed to determine OBB-OBB intersection. Super-
scripts indicate which OBB the value comes from.

L | T · L | r A r B

u 0 e 0 e 0 | e 1 | e 2 |

| t 0 | r 00 | + r 01 | + r 02 |

u 1 e 1 e 0 | e 1 | e 2 | | t 1 | r 10 | + r 11 | + r 12 |

u 2 e 2 e 0 | e 1 | e 2 |

| t 2 | r 20 | + r 21 | + r 22 |

u 0 e 0 | e 1 | e 2 | e 0

| t 0 r 00 + t 1 r 10 + t 2 r 20 | r 00 | + r 10 | + r 20 |

u 1

|

t 2 r 21 | e 0 | r 01 | + e 1 | r 11 | + e 2 | r 21 | e 1 t 0 r 01 + t 1 r 11 +

u 2 e 0 | e 1 | e 2 | e 2

| t 0 r 02 + t 1 r 12 + t 2 r 22 | r 02 | + r 12 | + r 22 |

u 0 u 0 e 1 | e 2 | e 1 | e 2 |

× | − | | + | | + |

t 2 r 10 t 1 r 20 r 20 r 10 r 02 r 01

u 0 u 1 e 1 | e 2 | e 0 | e 2 | × | t 2 r 11 − t 1 r 21 | r 21 | + r 11 | r 02 | + r 00 |

u 0 u 2 e 1 | e 2 | e 0 | e 1 |

× | t 2 r 12 − t 1 r 22 | r 22 | + r 12 | r 01 | + r 00 |

u 1 u 0 e 0 | e 2 | e 1 | e 2 | × | t 0 r 20 − t 2 r 00 | r 20 | + r 00 | r 12 | + r 11 |

u 1 u 1 e 0 | e 2 | e 0 | e 2 | × | t 0 r 21 − t 2 r 01 | r 21 | + r 01 | r 12 | + r 10 |

u 1 u 2

|

t 2 r 02 | e 0 | r 22 | + e 2 | r 02 | e 0 | r 11 | + e 1 | r 10 | × t 0 r 22 −

u 2 u 0 e 0 | e 1 | e 1 | e 2 | × | t 1 r 00 − t 0 r 10 | r 10 | + r 00 | r 22 | + r 21 |

u 2 u 1 e 0 | e 1 | e 0 | e 2 |

× | − | | + | | + |

t 1 r 01 t 0 r 11 r 11 r 01 r 22 r 20

u 2 u 2 e 0 | e 1 | e 0 | e 1 | × | t 1 r 02 − t 0 r 12 | r 12 | + r 02 | r 21 | + r 20 |

matrix bringing B into A 's coordinate frame), the tests that must be performed for the
different axes L are summarized in Table 4.1.
This test can be implemented as follows:

int TestOBBOBB(OBB &a, OBB &b)
{

float ra, rb;
Matrix33 R, AbsR;

// Compute rotation matrix expressing b in a's coordinate frame
for(inti=0;i<3;i++)
for(intj=0;j<3;j++)
R[i][j] = Dot(a.u[i], b.u[j]);