# Force Directed Near-Orthogonal Grid Generation on Surfaces

Abstract

Single curved surfaces can always easily be covered by meshes that result into an equilateral and orthogonal grid when the surface is developed. On double curved surfaces, however, we can never find a mesh consisting of ‘squares on the surface’. Nevertheless, there is a need for ‘orthogonal and locally almost equilateral meshes’ on such surfaces in several fields, e.g., in architecture (fair and easy to build, increased rigidity) and computer graphics (undistorted mapping of textures, good tessellation for rendering purposes and also for aesthetical reasons). We present an iterative force-directed algorithm that is capable of optimizing given grids with rectangular topology and yields the task in an optimal way. It allows to cover arbitrary parametric double-curved surfaces with grids that are almost orthogonal and, optionally, locally have almost constant grid size in both directions.

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- Article
- Jun 1979

This paper deals with the problem of placing components on a carrier, such as a printed circuit board (PCB). A new mathematical formulation of the concept of force directed placement is presented and an efficient computational procedure for solving the resulting system of equations. The placement procedure is broken down into two phases, the ″relative location phase″ , and the ″slot assignment or component overlap resolution phase″ . In Phase I, a set of simultaneous equations, based upon the interconnection topology of the system of components is solved. The results of Phase I are often unacceptable from a physical standpoint because there is a great deal of overlap among the components. Phase II eliminates component overlap by either of two methods, depending upon the physical properties of the carrier. A parametric analysis of the procedure is given based upon 12 different PCB's. These results show comparisons of this method to the work of others, and provide some insight into the method's absolute merits. - ArticleFull-text available
- Jan 2008

Cluster graphs are a valuable concept to visualize structu-red relational information. Hierarchical cluster graphs im-pose further levels of granularity which may be controlled by the user. In this paper we present a force directed layout adjustment algorithm for hierarchical cluster graphs. Clus-ters and cluster hierarchies respectively, can be dynamically closed or opened which is vital to selectively reduce the in-formation presented by large graphs. In our case such an operation only rearranges the graph locally, therefore pre-serving the mental map of the user. We also present results achieved with our algorithm in the domain of semantic net exploration. - In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform glass structures. Moreover, they possess a natural offsetting operation and provide a support structure orthogonal to the mesh. Being a discrete analogue of the network of principal curvature lines, they represent fundamental shape characteristics. We show how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical. Combining this perturbation with subdivision yields a powerful new modeling tool for all types of quad meshes with planar faces, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.
- An improved method for nearly orthogonal grid generation is presented in this study. The generating system is based on solution of a system of partial differential equations with finite difference discretization. To prevent grid lines from collapsing onto each other, the grid cell aspect ratio is controlled by functions that limit excessive ratios. Bounding all the aspect ratios is essential for high-quality numerical approximations using such grid-based methods as finite elements, finite differences, or finite volumes. The influence of the number of grid points, type of boundary, and intensity of the grid quality control function and grid properties are investigated. Specification of both boundary point distribution on all sides and moving boundaries is used. The proposed method is applied to various test problems from the literature. This method is shown to provide a good balance between controlling grid orthogonality and cell aspect ratio.
- Conference PaperFull-text available
- Jan 2005

Cluttered drawings of graphs cannot efiectively con- vey the information of graphs. Two issues might cause node overlapping when one draws a picture of a graph. The flrst issue occurs when applying a layout algorithm for an abstract graph to a practical appli- cation in which nodes are labeled. The second is the changing of a node's size in a dynamic drawing sys- tem. This paper presents two algorithms, DNLS and ODNLS, for removing the two kinds of overlapping. The algorithms are based on the well-known spring embedder model. The outputs of the algorithms provide the features of spring algorithms. Experi- ments are carried out to compare DNLS and ODNLS to the Force Scan(FS) algorithm and its variants. The results demonstrate the advantages of DNLS and ODNLS in terms of some aesthetic criteria. - Conference Paper
- Jan 2007

The design of free form surfaces is usually based on NURBS and it works well to quickly get shapes that a designer intends to create. Such surfaces then have desired properties like given border lines and C1 or C2 continuity along lines where several surfaces touch. Our approach is to create surfaces with certain physical properties that designers often need. Given a closed or not closed border line, can we then find an elastic surface (comparable with a rubber surface) with the property requiring that in each point the tension is equally distributed? This is – simplified spoken – the condition for a minimal surface. Our solution does not use any differential equations but rather the following idea: We start from a patch that may be planar or part of a cylinder or any easy to define surface. This patch is tesselated in such a way that the vertices have roughly equal distances. Each point is considered to be magnetic. Now we start a converging real-time-iteration that allows the points to move according to the rules of magnetism. Border lines or parts of them may be fixed and manipulated. The corresponding algorithm is adapted from earlier algorithms by Fruchterman et al. The result is an approximation to a minimal surface that is defined by the fixed border lines. The advantage of such a surface design is twofold: First, the problem is hard to solve exactly by means of differential equations, and second the algorithm works interactively in real time. This means that the designer can change shapes almost as quickly as with conventional free form surfaces. Finally, the surface is already suitably triangulated. - Article
- Oct 2007

In this paper, we propose a method to simulate soft bodies by using gravitational force, spring and damping forces between surface points, and internal molecular pressure forces. We consider a 3D soft body model composed of mesh points that define the body’s surface such that the points are connected by springs and influenced by internal molecular pressure forces. These pressure forces have been modeled on gaseous molecular interactions. Simulation of soft body with internal pressure forces is known to become unstable when high constants are used and is averted using an implicit integration method. We propose an approximation to this implicit integration method that considerably reduces the number of computations in the algorithm. Our results show that the proposed method realistically simulates soft bodies and improves performance of the implicit integration method. - Conference Paper
- Feb 1

This paper presents an algorithm for drawing a sequence of graphs that contain an inherent grouping of their vertex set into clusters. It differs from previous work on dynamic graph drawing in the emphasis that is put on maintaining the clustered structure of the graph during incremental layout. The algorithm works online and allows arbitrary modifications to the graph. It is generic and can be implemented using a wide range of static force-directed graph layout tools. The paper introduces several metrics for measuring layout quality of dynamic clustered graphs. The performance of our algorithm is analyzed using these metrics. The algorithm has been successfully applied to visualizing mobile object software - Article
- Jul 1979
- IEEE Trans Circ Syst

This paper deals with the problem of placing components on a carrier, such as a printed circuit board (PCB). We present a new mathematical formulation of the concept of force directed placement, and describe an efficient computational procedure for solving the resulting system of equations. The placement procedure is broken down into two phases, Phase I being the "relative location phase," and Phase II being the "slot assignment or component overlap resolution phase." In Phase I of the procedure, we solve a set of simultaneous equations, based upon the interconnection topology of the system of components, in an endeavor to determine the optimum relative location of every component with respect to every other component. The equations are set up such that there are attractive forces between components sharing a common signal, and repulsive forces between components having no signals in common. The results of Phase I are often unacceptable from a physical standpoint because there is a great deal of overlap among the components. Phase II eliminates component overlap by either of two methods, depending upon the physical properties of the carrier. If the carrier is subdivided into slots, then the components are assigned to these slots using a criteria which minimiZes the total distance that all components need be moved. We perform this assignment by using the linear assignment algorithm. If the carrier is such that components are allowed to reside anywhere, then a different technique to resolve component overlap is used. A parametric analysis of the procedure is given based upon 12 different PCB's. These results show comparisons of this method to the work of others, and provide some insight into the method's absolute merits.