Contents -- 1 Discrete Modeling for Natural Objects -- 1.1 Introduction -- 1.2 Discrete modeling -- 1.3 Interpolation -- 1.4 Examples of applications -- 1.5 Conclusions -- 2 Cellular Partitions -- 2.1 Introduction -- 2.2 Elements of topology -- 2.3 Cellular partition of an n-manifold object -- 2.4 Generalized Maps -- 2.5 Implementing GMap-based models -- 2.6 Conclusions -- 3 Tessellations -- 3.1 Introduction -- 3.2 Delaunay's tessellation -- 3.3 Non-Delaunay triangulated surfaces -- 3.4 Notion of a regular n-grid -- 3.5 Notion of an irregular n-grid -- 3.6 Implicit surfaces -- 3.7 Conclusions -- 4 Discrete Smooth Interpolation -- 4.1 Introduction -- 4.2 The DSI problem -- 4.3 Uniqueness of the DSI solution -- 4.4 The local DSI equation -- 4.5 Accounting for hard constraints -- 4.6 Accelerating the convergence -- 4.7 The fuzzy Control-Point paradigm -- 4.8 The fuzzy Control-Node paradigm -- 4.9 From a discrete to a continuous model -- 4.10 Conclusions -- 5 Elements of Differential Geometry -- 5.1 Parametric curves -- 5.2 Parametric surfaces -- 5.3 Curvature of curves drawn on a surface -- 5.4 Miscellaneous -- 5.5 Discrete modeling -- 5.6 Examples of applications to geology -- 5.7 Conclusions -- 6 Piecewise Linear Triangulated Surfaces -- 6.1 Introduction -- 6.2 Basic DSI constraints -- 6.3 Modeling a faulted surface -- 6.4 Continuity through faults -- 6.5 Global parameterization -- 6.6 Modifying the topology -- 6.7 Conclusions -- 7 Curvilinear Triangulated Surfaces -- 7.1 Introduction -- 7.2 Building a smooth curvilinear triangle -- 7.3 Gregory G[sup(1)] patchwork -- 7.4 Recursive subdivisions -- 7.5 Conclusions -- 8 Elements of Structural Geology -- 8.1 Geometry of faults and horizons -- 8.2 Modeling stratified media -- 8.3 Merging seismic data with well data -- 8.4 Deformation analysis -- 8.5 Unfolding a horizon.

8.6 Unfolding a stack of layers -- 8.7 Conclusions -- 9 Stochastic Modeling -- 9.1 Simulation versus interpolation -- 9.2 Probabilities in a nutshell -- 9.3 Random Functions -- 9.4 Random Fourier Series -- 9.5 Uniform Random Functions and P-fields -- 9.6 Stochastic simulators -- 9.7 Kriging-based methods -- 9.8 Blending-based method -- 9.9 Assessing geometric uncertainties -- 9.10 Conclusions -- 10 Discrete Smooth Partition -- 10.1 Introduction -- 10.2 The probabilistic approach -- 10.3 Structural constraints -- 10.4 Moving-Centers-based methods -- 10.5 A tutorial example -- 10.6 Conclusions -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W.