On Shortest Paths Between Two Convex Polyhedra
On Shortest Paths Between Two Convex Polyhedra
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Thus Cg splits e into at most two remaining portions, each of which is visible from every point on g to which e is not (totally) obscure.
Proof: Let /t/, (resp, fiJj) denote the faces of A (resp. B) incident to g (resp. E), and let 11^, 11/ (resp. TliJlj) be the two planes containing /t/; (resp. FiJj) . By convexity of A, rijt (resp. FI/) is a supporting plane of A. Let Hi- (resp. Hf) denote the closed half space bounded by Fljt (resp. IT/) whose interior is disjoint from A, and let H^ (resp, H...f) denote the complementary open space.
Suppose for the moment that the faces of B were transparent, so that e could be seen behind them. Three cases can arise: (a) e lies entirely in H^dHf. In this case e is obscure to g.
(b) e lies entirely in H^UHf. In this case e is visible from g.
(c) e crosses one or both planes 11^, 17, . In this case e is split into at most three subsegments, one of which {eg = er\H^nHf) is obscure to g, and the other one or two (e — eDH^nHf) are visible from g.
So far we have assumed that the faces of B are transparent, but obviously they are not.
What to read after On Shortest Paths Between Two Convex Polyhedra?
You can find similar books in the "Read Also" column, or choose other free books by Avikam Baltsan to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Thus Cg splits e into at most two remaining portions, each of which is visible from every point on g to which e is not (totally) obscure.
Proof: Let /t/, (resp, fiJj) denote the faces of A (resp. B) incident to g (resp. E), and let 11^, 11/ (resp. TliJlj) be the two planes containing /t/; (resp. FiJj) . By convexity of A, rijt (resp. FI/) is a supporting plane of A. Let Hi- (resp. Hf) denote the closed half space bounded by Fljt (resp. IT/) whose interior is disjoint from A, and let H^ (resp, H...f) denote the complementary open space.
Suppose for the moment that the faces of B were transparent, so that e could be seen behind them. Three cases can arise: (a) e lies entirely in H^dHf. In this case e is obscure to g.
(b) e lies entirely in H^UHf. In this case e is visible from g.
(c) e crosses one or both planes 11^, 17, . In this case e is split into at most three subsegments, one of which {eg = er\H^nHf) is obscure to g, and the other one or two (e — eDH^nHf) are visible from g.
So far we have assumed that the faces of B are transparent, but obviously they are not.
What to read after On Shortest Paths Between Two Convex Polyhedra?
You can find similar books in the "Read Also" column, or choose other free books by Avikam Baltsan to read onlineMoreLess
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