Advances in Computational Mathematics

, Volume 29, Issue 3, pp 249–268 | Cite as

The focal geometry of circular and conical meshes

  • Helmut PottmannEmail author
  • Johannes Wallner


Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the three-dimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.


Discrete differential geometry Architectural design Geometric modeling Integrable systems Quadrilateral meshes Conical meshes Circular meshes Offset meshes Focal meshes 

Mathematics Subject Classifications (2000)

68U05 53A40 52C99 51B15 


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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

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