2,287 research outputs found
Convex Hull of Planar H-Polyhedra
Suppose are planar (convex) H-polyhedra, that is, $A_i \in
\mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i =
\{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 +
n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron
with the smallest such that
On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph
This paper studied the geometric and combinatorial aspects of the classical
Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega
be a height function which lifts the vertices of A into R3. Every flip in
triangulations of A can be associated with a direction. We first established a
relatively obvious relation between monotone sequences of directed flips
between triangulations of A and triangulations of the lifted point set of A in
R3. We then studied the structural properties of a directed flip graph (a
poset) on the set of all triangulations of A. We proved several general
properties of this poset which clearly explain when Lawson's algorithm works
and why it may fail in general. We further characterised the triangulations
which cause failure of Lawson's algorithm, and showed that they must contain
redundant interior vertices which are not removable by directed flips. A
special case if this result in 3d has been shown by B.Joe in 1989. As an
application, we described a simple algorithm to triangulate a special class of
3d non-convex polyhedra. We proved sufficient conditions for the termination of
this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure
Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)
The alternating sum of volumes (ASV) decomposition is a widely used
technique for converting a B-Rep into a CSG model. The obtained CSG
tree has convex primitives at its leaf nodes, while the contents of
its internal nodes alternate between the set union and difference
operators.
This work first shows that the obtained CSG tree T can also be
expressed as the regularized Exclusive-OR operation among all the
convex primitives at the leaf nodes of T, regardless the structure and
internal nodes of T. This is an important result in the case in which
EVM represented orthogonal polyhedra are used because in this model
the Exclusive-OR operation runs much faster than set union and
difference operations. Therefore this work applies this result to EVM
represented orthogonal polyhedra. It also presents experimental
results that corroborate the theoretical results and includes some
practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version
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