49 research outputs found
Transversals to Line Segments in Three-Dimensional Space
We completely describe the structure of the connected components of transversals to a collection of n line segments in R3. We show that n \u3e 3 arbitrary line segments in R3 admit 0, 1, . . . , n or infinitely many line transversals. In the latter case, the transversals form up to n connected components
Transversals to line segments in three-dimensional space
http://www.springerlink.com/We completely describe the structure of the connected components of transversals to a collection of line segments in . Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that arbitrary line segments in admit at most connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of on the number of geometric permutations of line segments in
On the Number of Maximal Free Line Segments Tangent to Arbitrary Three-dimensional Convex Polyhedra
We prove that the lines tangent to four possibly intersecting convex polyhedra in with edges in total form connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of possibly intersecting convex polyhedra with a total of edges admits, in the worst case, connected components of maximal free line segments tangent to any four of the polytopes. This bound also holds for the number of connected components of possibly occluded lines tangent to any four of the polytopes
Semiregular Polytopes and Amalgamated C-groups
In the classical setting, a convex polytope is said to be semiregular if its
facets are regular and its symmetry group is transitive on vertices. This paper
studies semiregular abstract polytopes, which have abstract regular facets,
still with combinatorial automorphism group transitive on vertices. We analyze
the structure of the automorphism group, focusing in particular on polytopes
with two kinds of regular facets occurring in an "alternating" fashion. In
particular we use group amalgamations to prove that given two compatible
n-polytopes P and Q, there exists a universal abstract semiregular
(n+1)-polytope which is obtained by "freely" assembling alternate copies of P
and Q. We also employ modular reduction techniques to construct finite
semiregular polytopes from reflection groups over finite fields.Comment: Advances in Mathematics (to appear, 28 pages