1,318 research outputs found

    On organizing principles of Discrete Differential Geometry. Geometry of spheres

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    Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

    Möbius Geometry and Cyclidic Nets: A Framework for Complex Shape Generation

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    International audienceFree-form architecture challenges architects, engineers and builders. The geometrical rationalization of complex structures requires sophisticated tools. To this day, two frameworks are commonly used: NURBS modeling and mesh-based approaches. The authors propose an alternative modeling framework called generalized cyclidic nets that automatically yields optimal geometrical properties for the façade and the structure. This framework uses a base circular mesh and Dupin cyclides, which are natural objects of the geometry of circles in space, also known as Möbius geometry. This paper illustrates how new shapes can be generated from generalized cyclidic nets. Finally, it is demonstrated that this framework gives a simple method to generate curved-creases on free-forms. These findings open new perspectives for structural design of complex shells

    Discretisation design strategies: strategies to integrate design and fabrication through discretization.

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    In the present paper, we introduce a classification system, for discretisation strategies, based on the procedural differences. This paper has a particular focus on strategies explicitly positioned towards an integration between digital design, robotic fabrication and robotic assembly. In the first step, the paper introduces and analyses previous methods from the literature and built case studies and proposes a classification for discretisation approaches. This classification is based on three basic designing strategies: Top-Down, Bottom-Up and Hybrid, in a parametric design manner. The second step defines a general parametric framework for each approach based on the classification analysis. Due to the specifications and functions, these approaches can be synced and combined with other parametric design tactics, such as panelising, subdivision, or generative design. We describe and analyse the possibilities of connecting other parametric features with our discretisation definitions in each category. In the end, this paper introduces several alternative implementation avenues for each category, including a logical design strategy, without considering any specific software or tool

    Contributions to Four-Position Theory with Relative Rotations

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    We consider the geometry of four spatial displacements, arranged in cyclic order, such that the relative motion between neighbouring displacements is a pure rotation. We compute the locus of points whose homologous images lie on a circle, the locus of oriented planes whose homologous images are tangent to a cone of revolution, and the locus of oriented lines whose homologous images form a skew quadrilateral on a hyperboloid of revolution

    On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs

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    Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines which are diagonally related to lines of curvature is proven theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory
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