### 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 |

### Fingerprint

### 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

*CAD Computer Aided Design*,

*66*, 45-54. [2320]. https://doi.org/10.1016/j.cad.2015.04.003

**Extending CSG with projections : Towards formally certified geometric modeling.** / Tzoumas, George; Michelucci, Dominique; Foufou, Sebti.

Research output: Contribution to journal › Article

*CAD Computer Aided Design*, vol. 66, 2320, pp. 45-54. https://doi.org/10.1016/j.cad.2015.04.003

}

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

UR - http://www.scopus.com/inward/record.url?scp=84928793978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928793978&partnerID=8YFLogxK

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 -