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This thesis presents an overview of some approaches to compute Gaussian and mean curvature on discrete surfaces and discusses discrete minimal surfaces. The variety of applications of differential geometry in visualization and shape design leads to great interest in studying discrete surfaces. With the rich smooth surface theory in hand, one would hope that this elegant theory can still be applied to the discrete counter part. Such a generalization, however, is not always successful. While discrete surfaces have the advantage of being finite dimensional, thus easier to treat, their geometric properties such as curvatures are not well defined in the classical sense. Furthermore, the powerful calculus tool can hardly be applied. The methods in this thesis, including angular defect formula, cotangent formula, parallel meshes, relative geometry etc. are approaches based on offset meshes or generalized offset meshes. As an important application, we discuss discrete minimal surfaces and discrete Koenigs meshes.

Discrete Differential Geometry (DDG) concerns discrete counterparts of notions and methods in differential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy different geometric or physical constraints. We study a combination of geometry and physics – the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences – a particular type of congruences defined by linear interpolation of vertex normals. 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. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognition.

The Grassmannian space GC(2, 4) embedded in CP5 as the Klein quadric of twistor theory has a natural interpretation in terms of the geometry of “round” 2-spheres in S4. The incidence of two lines in CP3 corresponds to the contact properties of two 2- spheres, where contact is generalized from tangency to include “half-tangency:” 2-spheres may be in contact at two isolated points. There is a connection between the contact properties of 2-spheres and soliton geometry through the classical Ribaucour and Darboux transformations. The transformation theory of surfaces in S4 is investigated using the recently developed theory of “Discrete Differential Geometry” with results leading to the conclusion that the discrete conformal maps into C of Hertrich-Jeromin, McIntosh, Norman and Pedit may be defined in terms a discrete integrable system employing halftangency in S4.

Geometric Circle Packings are of interest not only for their aesthetic appeal but also their relation to discrete analytic function theory. This thesis presents new computational methods which enable additional practical applications for circle packing geometry along with providing a new discrete analytic interpretation of the classical Schwarzian derivative and traditional univalence criterion of classical analytic function theory. To this end I present a new method of computing the maximal packing and solving the circle packing layout problem for a simplicial 2-complex along with additional geometric variants and applications. This thesis also presents a geometric discrete Schwarzian quantity whose value is associated with the classical Schwarzian derivative. Following Hille, I present a characterization of circle packings as the ratio of two linearly independent solutions of a discrete difference equation taking the discrete Schwarzian as a parameter. This characterization then gives a discrete interpretation of the classical univalence criterion of Nehari in the circle packing setting.

La problématique centrale de cette thèse est l'élaboration d'un opérateur de Laplace--Beltrami discret sur les surfaces digitales. Ces surfaces proviennent de la théorie de la géométrie discrète, c’est-à-dire la géométrie qui s'intéresse à des sous-ensembles des entiers relatifs. Nous nous plaçons ici dans un cadre théorique où les surfaces digitales sont le résultat d'une approximation, ou processus de discrétisation, d'une surface continue sous-jacente. Cette méthode permet à la fois de prouver des théorèmes de convergence des quantités discrètes vers les quantités continues, mais aussi, par des analyses numériques, de confirmer expérimentalement ces résultats. Pour la discrétisation de l’opérateur, nous faisons face à deux problèmes : d'un côté, notre surface n'est qu'une approximation de la surface continue sous-jacente, et de l'autre côté, l'estimation triviale de quantités géométriques sur la surface digitale ne nous apporte pas en général une bonne estimation de cette quantité. Nous possédons déjà des réponses au second problème : ces dernières années, de nombreux articles se sont attachés à développer des méthodes pour approximer certaines quantités géométriques sur les surfaces digitales (comme par exemple les normales ou bien la courbure), méthodes que nous décrirons dans cette thèse. Ces nouvelles techniques d'approximation nous permettent d'injecter des informations de mesure sur les éléments de notre surface. Nous utilisons donc l'estimation de normales pour répondre au premier problème, qui nous permet en fait d'approximer de façon précise le plan tangent en un point de la surface et, via une méthode d'intégration, palier à des problèmes topologiques liées à la surface discrète. Nous présentons un résultat théorique de convergence du nouvel opérateur discrétisé, puis nous illustrons ensuite ses propriétés à l’aide d’une analyse numérique de l’opérateur. Nous effectuons une comparaison détaillée du nouvel opérateur par rapport à ceux de la littérature adaptés sur les surfaces digitales, ce qui nous permet, au moins pour la convergence, de montrer que seul notre opérateur possède cette propriété. Nous illustrons également l’opérateur via quelques unes de ces applications comme sa décomposition spectrale ou bien encore le flot de courbure moyenne / The central issue of this thesis is the development of a discrete Laplace--Beltrami operator on digital surfaces. These surfaces come from the theory of discrete geometry, i.e. geometry that focuses on subsets of relative integers. We place ourselves here in a theoretical framework where digital surfaces are the result of an approximation, or discretization process, of an underlying smooth surface. This method makes it possible both to prove theorems of convergence of discrete quantities towards continuous quantities, but also, through numerical analyses, to experimentally confirm these results. For the discretization of the operator, we face two problems: on the one hand, our surface is only an approximation of the underlying continuous surface, and on the other hand, the trivial estimation of geometric quantities on the digital surface does not generally give us a good estimate of this quantity. We already have answers to the second problem: in recent years, many articles have focused on developing methods to approximate certain geometric quantities on digital surfaces (such as normals or curvature), methods that we will describe in this thesis. These new approximation techniques allow us to inject measurement information into the elements of our surface. We therefore use the estimation of normals to answer the first problem, which in fact allows us to accurately approximate the tangent plane at a point on the surface and, through an integration method, to overcome topological problems related to the discrete surface. We present a theoretical convergence result of the discretized new operator, then we illustrate its properties using a numerical analysis of it. We carry out a detailed comparison of the new operator with those in the literature adapted on digital surfaces, which allows, at least for convergence, to show that only our operator has this property. We also illustrate the operator via some of these applications such as its spectral decomposition or the mean curvature flow