On Shortest Paths Amidst Convex Polyhedra
On Shortest Paths Amidst Convex Polyhedra
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To quickly assess the difficulty of the text, read a short excerpt:
E. E 6 ^{A, B).
Furthermore, let e be an edge which intersects both 81 and 82. Let e' be a subsegment of e whose endpoints pi, p2 ^^ ^^ Si» ^2 respectively, and -5- whose interior is wholly contained in R. We claim that if 7r(A, B) is indeed contained in R then it must intersect e' . This is because e' splits R into two disjoint subregions, one of which contains A on its boundary and the other contains B on its boundary. Thus any continuous arc connecting A to B within R must cross e' .
This im...plies that if 'tt(A^) is contained in R, then the subsegment e' defined above must be unique, for if e contained two such subsegments, 'n{A, B) would have to intersect both of them, contradicting the fact that a shortest path cannot cross an edge of K more than once.
Finally, suppose that tt{AJB) is contained in R, and let J, e be two edges of K intersecting both 81, 82. Let d' = P1P2. ^' = ^1^2. Be their unique subsegments connecting 8^ to 82 as defined above, withpi, q-^ 6 8^, andp»2. ^2 ^ 82- Then we claim that pi is nearer to A along 81 than q-^ if and only if P2 is nearer to A along 82 than ^2- This property follows by general topological considerations from the fact that d' and e' do not intersect one another, and that each of them partitions R into two disjoint portions, so that the other subsegment is contained in just one of them.
What to read after On Shortest Paths Amidst Convex Polyhedra?
You can find similar books in the "Read Also" column, or choose other free books by Micha Sharir to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
E. E 6 ^{A, B).
Furthermore, let e be an edge which intersects both 81 and 82. Let e' be a subsegment of e whose endpoints pi, p2 ^^ ^^ Si» ^2 respectively, and -5- whose interior is wholly contained in R. We claim that if 7r(A, B) is indeed contained in R then it must intersect e' . This is because e' splits R into two disjoint subregions, one of which contains A on its boundary and the other contains B on its boundary. Thus any continuous arc connecting A to B within R must cross e' .
This im...plies that if 'tt(A^) is contained in R, then the subsegment e' defined above must be unique, for if e contained two such subsegments, 'n{A, B) would have to intersect both of them, contradicting the fact that a shortest path cannot cross an edge of K more than once.
Finally, suppose that tt{AJB) is contained in R, and let J, e be two edges of K intersecting both 81, 82. Let d' = P1P2. ^' = ^1^2. Be their unique subsegments connecting 8^ to 82 as defined above, withpi, q-^ 6 8^, andp»2. ^2 ^ 82- Then we claim that pi is nearer to A along 81 than q-^ if and only if P2 is nearer to A along 82 than ^2- This property follows by general topological considerations from the fact that d' and e' do not intersect one another, and that each of them partitions R into two disjoint portions, so that the other subsegment is contained in just one of them.
What to read after On Shortest Paths Amidst Convex Polyhedra?
You can find similar books in the "Read Also" column, or choose other free books by Micha Sharir to read onlineMoreLess
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