Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps

  • Víctor-Manuel Soto-FrancésEmail author
  • Emilio-José Sarabia-Escrivá
  • José-Manuel Pinazo-Ojer


The modelling of real world objects is not a straightforward subject. There are many different schemes; constructive solid geometry (CSG), cell decomposition, boundary representation, etcetera. Obviously, somehow, any scheme will be related to any other since they have a common goal. The paper shows how to model general polyhedra as an unordered discrete and finite set of geometric numbers of a projective Clifford Algebra or Geometric Algebra (GA). Clearly, not any randomly generated finite set of geometric numbers will have the structure of an object, this set must have some well defined properties. The topological properties extracted from this set are mapped to a boundary representation scheme based on a type of combinatorial map called generalised map or n-gmap. The n-gmaps have different types of orbits (in the mathematical sense) to which an attribute can be attached. When the attribute has a geometrical meaning, it is said that it is the geometrical embedding of the n-gmap. In this way the n-gmap holds explicitly the topology or structure already defined by the discrete geometry. In our proposal, each single element of a n-gmap is consistently embedded into a geometrical number also known as multi-vector. The scheme has been implemented by modifying an open source code [46] of n-gmaps. This representation has interesting properties. GA and n-gmaps complement and reinforce each other. For instance; it improves the robustness when computing the structure from the geometrical information. It is capable of computing lengths, areas and volumes of any polyhedral complex (with or without holes) using the orbits of the n-gmap (some examples are given). Finally the paper gives hints about other potentialities.


Geometric algebra Clifford algebra Multi-vectors n-gmaps Building energy simulation Solid modelling Combinatorial maps Flags Darts 




Vectors in \({\mathbb {R}}^3\)

Greek Symbols

\(\alpha \)

Involution function used to sew darts

\(\beta \)

Scalar number \(\beta \in {\mathbb {R}}\)

Other Symbols

\(\bullet \)

Dot product (inner product)

\(\cdot \)

Geometric product (also, juxtaposition of symbols)

\(\circ \)

Composition of functions

\(\dagger \)

Conjugate operator in Geometric Algebra


Reversion operator in Geometric Algebra

\(\wedge \)

Wedge product (outer product)





We would like to thank: The help of Guillaume Damiand for his help in solving doubts about handling the [46] software and for sharing it freely. Randolph Franklin by his kind advice about the computation of lengths, areas and volumes. The reviewers specialized in both disciplines; n-gmaps and GA, for their constructive and helpful comments.


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Authors and Affiliations

  1. 1.Departamento de Termodinámica AplicadaETSII, Universitat Politècnica de ValènciaValenciaSpain

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