Approximate Fitting of a Circular Arc When Two Points Are Known

The task of approximating points with circular arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. However, the development of algorithms that perform a significant amount of circular arcs fitting requires an efficient way of fitting circular arcs with complexity O(1). The elegant solution to this task based on an eigenvector problem for a square nonsymmetrical matrix is described in [1]. For the compression algorithm described in [2], it is necessary to solve this task when two points on the arc are known. This paper describes a different approach to efficiently fitting the arcs and solves the task when one or two points are known.Comment: 15 pages, 4 figures, extended abstract published at the conferenc

Tolerance Zone-Based Grouping Method for Online Multiple Overtracing Freehand Sketches

Multiple overtracing strokes are common drawing behaviors in freehand sketching; that is, additional strokes are often drawn repeatedly over the existing ones to add more details. This paper proposes a method based on stroke-tolerance zones to group multiple overtraced strokes which are drawn to express a 2D primitive, aiming to convert online freehand sketches into 2D line drawings, which is a base for further 3D reconstruction. Firstly, after the user inputs a new stroke, a tolerance zone around the stroke is constructed by reference to its polygonal approximation points obtained from the stroke preprocessing. Then, the input strokes are divided into stroke groups, each representing a primitive through the stroke grouping process based on the overtraced ratio of two strokes. At last, each stroke group is fitted into one or more 2D geometric primitives including line segments, polylines, ellipses, and arcs. The proposed method groups two strokes together based on their screen-space proximity directly instead of classifying and fitting them firstly, so that it can group strokes of arbitrary shapes. A sketch-recognition prototype system has been implemented to test the effectiveness of the proposed method. The results showed that the proposed method could support online multiple overtracing freehand sketching with no limitation on drawing sequence, but it only deals with strokes with relatively high overtraced ratio

Dynamic Reconstruction of Complex Planar Objects on Irregular Isothetic Grids

International audienceThe vectorization of discrete regular images has been widely developed in many image processing and synthesis applications, where images are considered as a regular static data. Regardless of final application, we have proposed in [14] a reconstruction algorithm of planar graphical elements on irregular isothetic grids. In this paper, we present a dynamic version of this algorithm to control the reconstruction. Indeed, we handle local refinements to update efficiently our complete shape representation. We also illustrate an application of our contribution for interactive approximation of implicit curves by lines, controlling the topology of the reconstruction

Special Curve Patterns for Freeform Architecture

In recent years, freeform shapes are gaining more and more popularity in architecture. Such shapes are often challenging to manufacture, and have motivated an active research field called architectural geometry. In this thesis, we investigate patterns of special curves on surfaces, which find applications in design and realization of freeform architectural shapes. We first consider families of geodesic curves or piecewise geodesic curves on a surface, which are important for panelization of the surface and for interior design. We propose a method to propagate a series of such curves across a surface, starting from a given source curve, so that the distance functions between neighboring curves are close to given target distance functions. We use Jacobi fields as first order approximation of the distance functions from a curve to its neighboring curves, and select a Jacobi field which is closest to the target distance function. A neighboring curve is then computed according to the selected Jacobi field by solving an optimization problem. Using different target distance functions, we can generate different patterns of geodesic/piecewise geodesic curves. Our method provides an intuitive and controllable way to design geodesic patterns on freeform surfaces. We then present a method to compute functional webs, which are three families of curves with regular connectivity, where the curves have given special properties. We consider planar, circular and geodesic properties of the curves, which facilitate the fabrication of curve elements. We discretize a web as a regular triangle mesh, where the curves are represented by edge polylines of the mesh. The shape of the web is determined by optimizing a target functional which penalizes the deviation of the curves from their target properties. Furthermore, for webs where all curves are planar, we also show they can be computed in an exact way using three families of planes. By enabling the design of webs composed of curve elements which are easily manufacturable, our method addresses the challenge in realization of webs which have emerged in recent architectural designs

Geometric Planar Networks on Bichromatic Points

We study four classical graph problems – Hamiltonian path, Traveling salesman, Minimum spanning tree, and Minimum perfect matching on geometric graphs induced by bichromatic ( Open image in new window and Open image in new window ) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP -hard. In this work, we consider these problems in two restricted settings: (i) collinear points and (ii) equidistant points on a circle. We show that almost all of these problems can be solved in linear time in these constrained, yet non-trivial settings.acceptedVersio