497 research outputs found
Robust solid modeling by avoiding redundancy for manifold objects in boundary representation
Journal ArticleThis paper describes a new approach to the robustness problem in solid modeling. We identify as t h e main cause of t h e lack of robustness that interdependent topological relations are derived from approximate data. Disregarding the interdependencies very likely violates basic properties, such as reflexivity, and transitivity, resulting in invalid data representations, such as dangling edges, missing faces, etc. We show that the boundary of manifold objects can be represented without redundant relations which avoids inconsistencies. An algorithm for regularized set operations for manifold solids which is based on the principle of avoiding and eliminating redundancy is described. This algorithm has been implemented for objects bounded by planar and natural quadric surfaces; it handles coincidence and incidence cases between surfaces and curves robustly
QuickCSG: Fast Arbitrary Boolean Combinations of N Solids
QuickCSG computes the result for general N-polyhedron boolean expressions
without an intermediate tree of solids. We propose a vertex-centric view of the
problem, which simplifies the identification of final geometric contributions,
and facilitates its spatial decomposition. The problem is then cast in a single
KD-tree exploration, geared toward the result by early pruning of any region of
space not contributing to the final surface. We assume strong regularity
properties on the input meshes and that they are in general position. This
simplifying assumption, in combination with our vertex-centric approach,
improves the speed of the approach. Complemented with a task-stealing
parallelization, the algorithm achieves breakthrough performance, one to two
orders of magnitude speedups with respect to state-of-the-art CPU algorithms,
on boolean operations over two to dozens of polyhedra. The algorithm also
outperforms GPU implementations with approximate discretizations, while
producing an output without redundant facets. Despite the restrictive
assumptions on the input, we show the usefulness of QuickCSG for applications
with large CSG problems and strong temporal constraints, e.g. modeling for 3D
printers, reconstruction from visual hulls and collision detection
Combinatorial models for topology-based geometric modeling
Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc
QuickCSG: Fast Arbitrary Boolean Combinations of N Solids
QuickCSG computes the result for general N-polyhedron boolean expressions
without an intermediate tree of solids. We propose a vertex-centric view of the
problem, which simplifies the identification of final geometric contributions,
and facilitates its spatial decomposition. The problem is then cast in a single
KD-tree exploration, geared toward the result by early pruning of any region of
space not contributing to the final surface. We assume strong regularity
properties on the input meshes and that they are in general position. This
simplifying assumption, in combination with our vertex-centric approach,
improves the speed of the approach. Complemented with a task-stealing
parallelization, the algorithm achieves breakthrough performance, one to two
orders of magnitude speedups with respect to state-of-the-art CPU algorithms,
on boolean operations over two to dozens of polyhedra. The algorithm also
outperforms GPU implementations with approximate discretizations, while
producing an output without redundant facets. Despite the restrictive
assumptions on the input, we show the usefulness of QuickCSG for applications
with large CSG problems and strong temporal constraints, e.g. modeling for 3D
printers, reconstruction from visual hulls and collision detection
Euler flag enumeration of Whitney stratified spaces
The flag vector contains all the face incidence data of a polytope, and in
the poset setting, the chain enumerative data. It is a classical result due to
Bayer and Klapper that for face lattices of polytopes, and more generally,
Eulerian graded posets, the flag vector can be written as a cd-index, a
non-commutative polynomial which removes all the linear redundancies among the
flag vector entries. This result holds for regular CW complexes.
We relax the regularity condition to show the cd-index exists for Whitney
stratified manifolds by extending the notion of a graded poset to that of a
quasi-graded poset. This is a poset endowed with an order-preserving rank
function and a weighted zeta function. This allows us to generalize the
classical notion of Eulerianness, and obtain a cd-index in the quasi-graded
poset arena. We also extend the semi-suspension operation to that of embedding
a complex in the boundary of a higher dimensional ball and study the simplicial
shelling components.Comment: 41 pages, 3 figures. Final versio
O-minimal Hauptvermutung for polyhedra II
Hilbert initiated the standpoint in foundations of mathematics. From this
standpoint, we allow only a finite number of repetitions of elementary
operations when we construct objects and morphisms. When we start from a subset
of a Euclidean space. Then we assume that any element of the line has only a
finite number of connected components. We call the set tame if the assumption
is satisfied, and define a tame morphism in the same way.
In this paper we will show that a tame topological manifold is carried by a
tame homeomorphism to the interior of a compact piecewise linear manifolds
possibly with boundary and such a piecewise linear manifold possibly with
boundary is unique up to piecewise linear homeomorphisms in the sense that if
two manifolds are such PL manifolds possibly with boundary then they are the
same as piecewise linear manifolds. We modify this to Theorem 2 so that
argument of model theory works, and we prove it. We also consider the
differentiable case.Comment: 45 page
Robust regularized set operations on polyhedra
Journal ArticleThis paper describes a provably correct and robust implementation of regularized set operations on polyhedral objects. Although the algorithm described here does not assume manifold polyhedra and handles all possible degenerate cases, it turns out to be quite simple. The geometric operations and relations are computed with floating point arithmetic which is efficient but not necessarily precise. To ensure that the results are still consistent we implemented a test that detects when dependent decisions contradict each other. The consistency test is relatively simple and can be carried out locally without having to reason about the logical dependencies of the geometric relations. The logical structure and the efficiency of the algorithm are barely influenced by the consistency test which makes this approach well suited for interactive modeling systems on modern workstations
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