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Register a new userUnfolding Polyhedra
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Added by
Joseph O'Rourke
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Joseph ORourke
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Starting with the unsolved Durers problem of edge-unfolding a convex polyhedron to a net, we specialize and generalize (a) the types of cuts permitted, and (b) the polyhedra shapes, to highlight both advances established and which problems remain open.
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A convex polyhedron $P$ is $k$-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of $P$ are $k$-gon for some fixed value of $k$. Since 1968, it is an open problem to construct all equiprojective polyhedra. Recently, Hasan and Lubiw [CGTA 40(2):148-155, 2008] have given a characterization of equiprojective polyhedra. Based on their characterization, in this paper we discover some new equiprojective polyhedra by cutting and gluing existing polyhedra.
We introduce polyhedra circuits. Each polyhedra circuit characterizes a geometric region in $mathbb{R}^d$. They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedra. They can be used to approximate a large class of $d$-dimensional manifolds in $mathbb{R}^d$. Barvinok developed polynomial time algorithms to compute the volume of a rational polyhedra, and to count the number of lattice points in a rational polyhedra in a fixed dimensional space $mathbb{R}^d$ with a fix $d$. Define $T_V(d,, n)$ be the polynomial time in $n$ to compute the volume of one rational polyhedra, $T_L(d,, n)$ be the polynomial time in $n$ to count the number of lattice points in one rational polyhedra with $d$ be a fixed dimensional number, $T_I(d,, n)$ be the polynomial time in $n$ to solve integer linear programming time with $d$ be the fixed dimensional number, where $n$ is the total number of linear inequalities from input polyhedra. We develop algorithms to count the number of lattice points in the geometric region determined by a polyhedra circuit in $Oleft(ndcdot r_d(n)cdot T_V(d,, n)right)$ time and to compute the volume of the geometric region determined by a polyhedra circuit in $Oleft(ncdot r_d(n)cdot T_I(d,, n)+r_d(n)T_L(d,, n)right)$ time, where $n$ is the number of input linear inequalities, $d$ is number of variables and $r_d(n)$ be the maximal number of regions that $n$ linear inequalities with $d$ variables partition $mathbb{R}^d$.
We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuum of polyhedra P each realizing T for some x on P. Three main tools in the proof are properties of the star unfolding of P, Alexandrovs gluing theorem, and a cut-locus partition lemma. The construction of P from T is surprisingly simple.
We present new examples of topologically convex edge-ununfoldable polyhedra, i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet cannot be cut along their edges and unfolded into one planar piece without overlap. One family of examples is acutely triangulated, i.e., every face is an acute triangle. Another family of examples is stacked, i.e., the result of face-to-face gluings of tetrahedra. Both families achieve another natural property, which we call very ununfoldable: for every $k$, there is an example such that every nonoverlapping multipiece edge unfolding has at least $k$ pieces.
The construction of an unbounded polyhedron from a jagged convex cap is described, and several of its properties discussed, including its relation to Alexandrovs limit angle.
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