Orthogonal drawings and crossing numbers of the Kronecker product of two cycles

An orthogonal drawing of a graph is an embedding of the graph in the plane such that each edge is representable as a chain of alternately horizontal and vertical line segments. This style of drawing finds applications in areas such as optoelectronic systems, information visualization and VLSI circuits. We present orthogonal drawings of the Kronecker product of two cycles around vertex partitions of the graph into grids. In the process, we derive upper bounds on the crossing number of the graph. The resulting upper bounds are within a constant multiple of the lower bounds. Unlike the Cartesian product that is amenable to an inductive treatment, the Kronecker product entails a case-to-case analysis since the results depend heavily on the parameters corresponding to the lengths of the two cycles.     

基于科技文档的AutoCAD 插图精准出图方法

插图中的文字高度、图线线宽和图纸比例的精准打印输出,对制图类教材或教学 课件等科技文档十分重要。为了能精确出图,在分析文字信息版面占优势的科技文档中 AutoCAD 插图制作方法的基础上,从几个绘图概念出发,研究了插图的显示特性、图形对象的 图线宽度和图中文字高度的属性特征。研究表明,只需将Word 软件所测得的插图原始尺寸看 成制作AutoCAD 图形的绘图画布和物理纸张的图幅尺寸,精准设置图形的打印出图比例、线 宽和文字高度的属性值,就能通过科技文档打印输出既精准又符合制图规范的插图。     

一种包括隐含图形信息提取的工程图识别算法

工程图识别应该模仿人阅读工程图，把隐含图形信息提取出来，文中提出一种由稀疏的行程段直接提取整条图线，包括隐含图形信息提取的工程图识别算法，该算法首先提取整条图线的信息，再确定图线交点，通过分析线素，间隔等的规律确定线型，最后，基于工程图知识对图线拓扑关系进行校正。     

Drawing graphs with right angle crossings

Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing. This paper initiates the study of combinatorial and algorithmic questions related to the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed.     

Evolutionary search for faces from line drawings

Single 2D line drawing is a straightforward method to illustrate 3D objects. The faces of an object depicted by a line drawing give very useful information for the reconstruction of its 3D geometry. Two recently proposed methods for face identification from line drawings are based on two steps: finding a set of circuits that may be faces and searching for real faces from the set according to some criteria. The two steps, however, involve two combinatorial problems. The number of the circuits generated in the first step grows exponentially with the number of edges of a line drawing. These circuits are then used as the input to the second combinatorial search step. When dealing with objects having more faces, the combinatorial explosion prevents these methods from finding solutions within feasible time. This paper proposes a new method to tackle the face identification problem by a variable-length genetic algorithm with novel heuristic and geometric constraints incorporated for local search. The hybrid GA solves the two combinatorial problems simultaneously. Experimental results show that our algorithm can find the faces of a line drawing having more than 30 faces much more efficiently. In addition, simulated annealing for solving the face identification problem is also implemented for comparison.     

Arc crossing minimization in hierarchical digraphs with tabu search

Graphs are used commonly as a basic modeling tool in areas such as project management, production scheduling, line balancing, business process reengineering, and software visualization. An important problem in the area of graph drawing is to minimize arc crossings in a multi-layer hierarchical digraph. Existing solution methods for this problem are based on simple ordering rules for single layers that may lead to inferior drawings. This article first introduces an extensive review of relevant work previously published in this area. Then a tabu search implementation is presented that seeks high-quality drawings by means of an intensification phase that finds a local optimum according to an insertion mechanism and two levels of diversification. Computational experiments with 200 graphs with up to 30 nodes per layer and up to 30 layers are presented to assess the merit of the method.     

A Graph‐based Approach to Continuous Line Illustrations with Variable Levels of Detail

This paper introduces a method for automatically generating continuous line illustrations, drawings consisting of a single line, from a given input image. Our approach begins by inferring a graph from a set of edges extracted from the image in question and obtaining a path that traverses through all edges of the said graph. The resulting path is then subjected to a series of post‐processing operations to transform it into a continuous line drawing. Moreover, our approach allows us to manipulate the amount of detail portrayed in our line illustrations, which is particularly useful for simplifying the overall illustration while still retaining its most significant features. We also present several experimental results to demonstrate that our approach can automatically synthesize continuous line illustrations comparable to those of some contemporary artists.     

Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

Recovering solid geometric object from single line drawing image

Many educational materials contain a lot of solid geometric figures. The solid geometric objects in these figures are usually drawn as 2D line drawings thus have lost their 3D information. This paper presents a method to recover the 3D information of the solid geometric object from single line drawing image taken from the geometric books, which would be used to help the users better present and understand the solid geometric object on their mobile devices. The main advantage of our method is the abilitYTo handle inaccurately processed sketches as opposed to the previous methods which require perfect line drawings as inputs. Our method consists of three main steps as follows. First, the sketch of the input line drawing image is automatically extracted and further represented as an undirected graph. Second, candidate 3D models from the pre-built 3D model database are found by graph similarity-based searching and sub-graph isomorphism matching. Third, for each candidate 3D model, the model parameters, the rotation and the translation aligning the model with the sketch are found by minimizing an objective function which is composed of the residuals between the vertices of the sketch and the 2D projections of the candidate model’s vertices, and an optimal reconstruction solution is further selected as the final result. Extensive experimental results demonstrate the effectiveness and robustness of our method for recovering the solid geometric object from single line drawing image.