Abstract
Abstract We extend traditional Constructive Solid Geometry (CSG) trees to support the projection operator. Existing algorithms in the literature prove various topological properties of CSG sets. Our extension readily allows these algorithms to work on a greater variety of sets, in particular parametric sets, which are extensively used in CAD/CAM systems. Constructive Solid Geometry allows for algebraic representation which makes it easy for certification tools to apply. A geometric primitive may be defined in terms of a characteristic function, which can be seen as the zero-set of a corresponding system along with inequality constraints. To handle projections, we exploit the Disjunctive Normal Form, since projection distributes over union. To handle intersections, we transform them into disjoint unions. Each point in the projected space is mapped to a contributing primitive in the original space. This way we are able to perform gradient computations on the boundary of the projected set through equivalent gradient computations in the original space. By traversing the final expression tree, we are able to automatically generate a set of equations and inequalities that express either the geometric solid or the conditions to be tested for computing various topological properties, such as homotopy equivalence. We conclude by presenting our prototype implementation and several examples.
| Original language | English (US) |
|---|---|
| Article number | 2320 |
| Pages (from-to) | 45-54 |
| Number of pages | 10 |
| Journal | CAD Computer Aided Design |
| Volume | 66 |
| DOIs | |
| State | Published - Sep 1 2015 |
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Keywords
- Constraint solving
- Constructive solid geometry
- Disjunctive normal form
- Formal methods
- Homotopy equivalence
- Projection
ASJC Scopus subject areas
- Computer Science Applications
- Computer Graphics and Computer-Aided Design
- Industrial and Manufacturing Engineering
Cite this
Extending CSG with projections : Towards formally certified geometric modeling. / Tzoumas, George; Michelucci, Dominique; Foufou, Sebti.
In: CAD Computer Aided Design, Vol. 66, 2320, 01.09.2015, p. 45-54.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Extending CSG with projections
T2 - Towards formally certified geometric modeling
AU - Tzoumas, George
AU - Michelucci, Dominique
AU - Foufou, Sebti
PY - 2015/9/1
Y1 - 2015/9/1
N2 - Abstract We extend traditional Constructive Solid Geometry (CSG) trees to support the projection operator. Existing algorithms in the literature prove various topological properties of CSG sets. Our extension readily allows these algorithms to work on a greater variety of sets, in particular parametric sets, which are extensively used in CAD/CAM systems. Constructive Solid Geometry allows for algebraic representation which makes it easy for certification tools to apply. A geometric primitive may be defined in terms of a characteristic function, which can be seen as the zero-set of a corresponding system along with inequality constraints. To handle projections, we exploit the Disjunctive Normal Form, since projection distributes over union. To handle intersections, we transform them into disjoint unions. Each point in the projected space is mapped to a contributing primitive in the original space. This way we are able to perform gradient computations on the boundary of the projected set through equivalent gradient computations in the original space. By traversing the final expression tree, we are able to automatically generate a set of equations and inequalities that express either the geometric solid or the conditions to be tested for computing various topological properties, such as homotopy equivalence. We conclude by presenting our prototype implementation and several examples.
AB - Abstract We extend traditional Constructive Solid Geometry (CSG) trees to support the projection operator. Existing algorithms in the literature prove various topological properties of CSG sets. Our extension readily allows these algorithms to work on a greater variety of sets, in particular parametric sets, which are extensively used in CAD/CAM systems. Constructive Solid Geometry allows for algebraic representation which makes it easy for certification tools to apply. A geometric primitive may be defined in terms of a characteristic function, which can be seen as the zero-set of a corresponding system along with inequality constraints. To handle projections, we exploit the Disjunctive Normal Form, since projection distributes over union. To handle intersections, we transform them into disjoint unions. Each point in the projected space is mapped to a contributing primitive in the original space. This way we are able to perform gradient computations on the boundary of the projected set through equivalent gradient computations in the original space. By traversing the final expression tree, we are able to automatically generate a set of equations and inequalities that express either the geometric solid or the conditions to be tested for computing various topological properties, such as homotopy equivalence. We conclude by presenting our prototype implementation and several examples.
KW - Constraint solving
KW - Constructive solid geometry
KW - Disjunctive normal form
KW - Formal methods
KW - Homotopy equivalence
KW - Projection
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U2 - 10.1016/j.cad.2015.04.003
DO - 10.1016/j.cad.2015.04.003
M3 - Article
VL - 66
SP - 45
EP - 54
JO - CAD Computer Aided Design
JF - CAD Computer Aided Design
SN - 0010-4485
M1 - 2320
ER -