1 Consider Obbs A And B Let Pa And Pb Be The Centers Of A And B Respectively Let A A1 And A2 Be The Box Axes Ortho 1 (175.99 KiB) Viewed 29 times
1. Consider OBBs A and B, Let Pa and Pb be the centers of A and B respectively Let A”, A1, and A2 be the box axes (orthonormal basis) of A Let Bº, B1, and B2 be the box axes (orthonormal basis) of B Let ao, ai, and az be the radii of A Let bo, bi, and b2 be the radii of B Let L be a possible unit separating axis of A and B Let T be a vector = Pb – Pa Let V be a vector = (T-A Let W be a vector = (T.Bº = T.A1 T:A2) T.B1 T:B2) Roo Roi Roz R10 R11 R12 R20 R21 R22 Let R be a 3 x 3 matrix ( -( AO-BO A0.B1A0-B2 A1.BO A1-B1 A1-B2 A2.BO A2-B1 A2-B2 L separates the OBBs of A and B if |T.LI>laiAL[ + 16;B'. L| Σ i i a) Prove that when L = B2 [6 marks] |W2 > ao | Ro2 | + a1 R12 + a2|R22 | + b2 b) Prove that when L = A2 x B1 [6 marks] |ViRoi - VoR11 | > ao|R1| +a1 | Roi | + bo | R22|+ b2 | R20 |
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-( AO-BO A0.B1A0-B2 A1.BO A1-B1 A1-B2 A2.BO A2-B1 A2-B2 L separates the OBBs of A and B if |T.LI>laiAL[ + 16;B'. L| Σ i i a) Prove that when L = B2 [6 marks] |W2 > ao | Ro2 | + a1 R12 + a2|R22 | + b2 b) Prove that when L = A2 x B1 [6 marks] |ViRoi - VoR11 | > ao|R1| +a1 | Roi | + bo | R22|+ b2 | R20 |