Vertex Normals and Face Curvatures

Caigui Jiang, Helmut Pottmann, Xiang Sun, Johannes Wallner


The system of lines orthogonal to a surface (called the normal congruence of that surface) has close relations to the surface’s curvatures and is a well studied object of classical differential geometry. It is quite surprising that this natural correspondence has not been extensively exploited in discrete differential geometry: most notions of discrete curvature are constructed in a way not involving normals, or involving normals only implicitly. There are however applications such as support structures and shading/lighting systems in architectural geometry where line congruences, and in particular normal congruences, come into play.

The study of discrete differential geometry of triangle meshes is continued, elaborating on discrete normal congruences in more depth and presenting a novel discrete curvature theory for triangle meshes which is based on discrete line congruences. In particular discussed was when a congruence defined by linear interpolation of vertex normals deserves to be called a ‘normal’ congruence. The main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula.


  • Xian Sun, Caigui Jiang, Johannes Wallner, and Helmut Pottmann.
    Vertex normals and face curvatures of triangle meshes.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.

Prof. Dr. Helmut Pottmann   +

Projects: C01
University: TU Wien
E-Mail: pottmann[at]
University: King Abdullah University of Science and Technology
E-Mail: helmut.pottmann[at]

Prof. Dr. Johannes Wallner   +