Extending CSG with projections: Towards formally certified geometric modeling

George Tzoumas, Dominique Michelucci, Sebti Foufou

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article number2320
Pages (from-to)45-54
Number of pages10
JournalCAD Computer Aided Design
StatePublished - Sep 1 2015


  • 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


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