Ruled Laguerre minimal surfaces

A Laguerre minimal surface is an immersed surface in the Euclidean space being an extremal of the functional \int (H^2/K - 1) dA. In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C, D are fixed real numbers. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.Comment: 28 pages, 9 figures. Minor correction: missed assumption (*) added to Propositions 1-2 and Theorem 2, missed case (nested circles having nonempty envelope) added in the proof of Pencil Theorem 4, missed proof that the arcs cut off by the envelope are disjoint added in the proof of Lemma

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Geometric computing for freeform architecture

Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and construction-aware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes which allow multilayer constructions (which is related to discrete surfaces and their curvatures), triangles meshes with circle-packing properties (which is related to conformal uniformization), and with the paneling problem. We emphasize the combination of numerical optimization and geometric knowledge.

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $\mu$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $\mu$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature

Discrete spherical means of directional derivatives and Veronese maps

We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

Modeling of 2D and 3D Assemblies Taking Into Account Form Errors of Plane Surfaces

The tolerancing process links the virtual and the real worlds. From the former, tolerances define a variational geometrical language (geometric parameters). From the latter, there are values limiting those parameters. The beginning of a tolerancing process is in this duality. As high precision assemblies cannot be analyzed with the assumption that form errors are negligible, we propose to apply this process to assemblies with form errors through a new way of allowing to parameterize forms and solve their assemblies. The assembly process is calculated through a method of allowing to solve the 3D assemblies of pairs of surfaces having form errors using a static equilibrium. We have built a geometrical model based on the modal shapes of the ideal surface. We compute for the completely deterministic contact points between this pair of shapes according to a given assembly process. The solution gives an accurate evaluation of the assembly performance. Then we compare the results with or without taking into account the form errors. When we analyze a batch of assemblies, the problem is to compute for the nonconformity rate of a pilot production according to the functional requirements. We input probable errors of surfaces (position, orientation, and form) in our calculus and we evaluate the quality of the results compared with the functional requirements. The pilot production then can or cannot be validated

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Geometric Mechanics of Curved Crease Origami

Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is corroborated by numerical simulations which allow us to generalize our analysis to study multiply folded structures