Discretization of Planar Geometric Cover Problems

We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union of the translates in the collection contains all the points in $P$. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.Comment: 16 pages, 5 figure

Minimal surfaces from circle patterns: Geometry from combinatorics

We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal surfaces. The data used for the construction are purely combinatorial--the combinatorics of the curvature line pattern. A Weierstrass-type representation and an associated family are derived. We show the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended introduction. Minor changes in presentation. v3: revision according to the referee's suggestions, improved & expanded exposition, references added, minor mistakes correcte

Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators

In this paper, a discrete model is adopted, as proposed by Hencky for elastica based on rigid bars and lumped rotational springs, to design the control of a lightweight planar manipulator with multiple highly flexible links. This model is particularly suited to deal with nonlinear equations of motion as those associated with multilink robot arms, because it does not include any simplification due to linearization, as in the assumed modes method. The aim of the control is to track a trajectory of the end effector of the robot arm, without the onset of vibrations. To this end, an energy-based method is proposed. Numerical simulations show the effectiveness of the presented approach

On Weingarten transformations of hyperbolic nets

Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated in great detail in the area of (integrable) discrete differential geometry and can be traced back at least to the early 1950s. Here, we propose a canonical analogue of (discrete) Weingarten transformations for hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove the existence of Weingarten pairs and analyse their geometric and algebraic properties.Comment: 41 pages, 30 figure

Graphical Modelling with Computer Extended Descriptive Geometry (CeDG): Description and Comparison with CAD

We present a Computer Extended Descriptive Geometry (CeDG) approach for modelling spatial geometric systems that surpasses several CAD limitations. A first concept proof has shown that the CeDG can be implemented on the dynamic geometry software (DGS) paradigm to generate parametric models based on descriptive geometric techniques. Thereliabilityandperformanceofthe CeDG approachwascomparedto CAD throughtwo study cases from the sheet metal and mechanisms design fields. The outcomes demonstrate that CeDG is able to compute the design geometrical parameters concurrently with the modelling process and to obtain planar cutouts of 3D surfaces, in situations where CAD systems can not do it. The implementation was performed in Geogebra© for CeDG and Solid Edge© 2009 for CAD, which were selected because of their cutting edge technology. As main conclusion, the CeDG approach is a Descriptive Geometry (DG) - based computer parametric graphical modelling that may complement the CAD technology with accuracy and reliability