Curvature analysis of triangulated surfaces in structural geology

This paper addresses the problem of characterizing the shape of a geological surface on the basis of its principal curvatures. The surface is assumed to be modeled as a set of adjacent triangles defined by the location of their vertices and a method is proposed for estimating numerically the principal curvatures at the vertices of the triangles using a local C2 interpolant. Also shown is how principal curvatures can be useful for studying the deformation of a geological surface (with application to 3D balanced unfolding), and analyzing the folding or faulting of the interface between two adjacent layers.

[1]  Jean-Laurent Mallet,et al.  Discrete smooth interpolation , 1989, TOGS.

[2]  John C. Tipper.  A method and fortran program for the computerized reconstruction of three-dimensional objects from serial sections , 1977 .

[3]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[4]  H. Klein,et al.  3-D visualization of geologic structures and processes , 1992 .

[5]  Bernd Hamann.  Curvature approximation of 3D manifolds in 4D space , 1994, Comput. Aided Geom. Des..

[6]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[7]  John C. Tipper.  The Study of Geological Objects in Three Dimensions by the Computerized Reconstruction of Serial Sections , 1976, The Journal of Geology.

[8]  Charles T. Loop.  A G1 triangular spline surface of arbitrary topological type , 1994, Comput. Aided Geom. Des..

[9]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[10]  E. T. Y. Lee.  Yet another proof of knot insertion , 1989, Comput. Aided Geom. Des..

[11]  R. Millman,et al.  Elements of Differential Geometry , 2018, Applications of Tensor Analysis in Continuum Mechanics.

[12]  R. Barnhill.  SMOOTH INTERPOLATION OVER TRIANGLES , 1974 .

[13]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[14]  J. Mallet.  Three-dimensional graphic display of disconnected bodies , 1988 .

[15]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[16]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[17]  J. Mallet,et al.  Présentation d'un ensemble de méthodes et techniques de la cartographie automatique numérique , 1974 .

[18]  Jörg Peters,et al.  Bézier nets, convexity and subdivision on higher-dimensional simplices , 1995, Comput. Aided Geom. Des..

[19]  Gershon Elber,et al.  Second-order surface analysis using hybrid symbolic and numeric operators , 1993, TOGS.

[20]  Jean-Laurent Mallet,et al.  Discrete smooth interpolation in geometric modelling , 1992, Comput. Aided Des..

[21]  Paul Sablonnière,et al.  Composite finite elements of class Ck , 1985 .

[22]  Youdong Liang,et al.  GCn continuity conditions for adjacent rational parametric surfaces , 1995, Comput. Aided Geom. Des..