O'Rourke Chapter 7: Search & Intersection in PDF - Computer Science

Extremal Polytope Queries
Dtil Details (continued)
source: O’Rourke
• Lemma7103:Leta 7.10.3: Let a i and a i+1 be uniquely highest vertices of P i and
P i+1 . Then either (to provide constant-time search):
• a i = a i+1 or
• a i is the highest among the parents of the extreme edges L i+1 and
R i+1
• Proof Sketch: π
• Project P i+1 onto plane orthogonal to π
(e.g. xz plane)
• Let L i+1 , R i+1 = 2 edges of P i+1 projecting to 2
edges of P’ i+1 incident to a’ i+1
• Parents of edge are vertices of P i from
L’
π’ i+1 = R’ i+1
which it “derives” (during P i+1 creation)
tetrahedron
• New extreme edges are “close” to the old.
projected
onto xz plane
• See p. 281-283 for remaining details. P’ i+1
tetrahedron