Constructive Geometric Models with Imaginary Objects

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

Photogrammetric reconstruction software as a cost-efficient support tool in conservation research

A crucial activity in architectural and archaeological conservation research is the process of synthesising information in which the researcher records collected field data in the form of a planar drawing. This labour-intensive stage is significantly improved by automated systems which support the measurement work. Some of these are programs that convert sets of photographs into virtual and spatial models. The author compares the reasonably priced software options, shares the experience which was gathered during their use and presents the results of the research. The paper also presents the economic aspect and practical examples and highlights the development potential of these tools.

A Heuristic Approach Towards Drawings of Graphs With High Crossing Resolution

The crossing resolution of a non-planar drawing of a graph is the value of the minimum angle formed by any pair of crossing edges. Recent experiments suggest that the larger the crossing resolution is, the easier it is to read and interpret a drawing of a graph. However, maximizing the crossing resolution turns out to be an NP-hard problem in general, and only heuristic algorithms are known that are mainly based on appropriately adjusting force-directed algorithms. In this paper, we propose a new heuristic algorithm for the crossing resolution maximization problem and we experimentally compare it against the known approaches from the literature. Our experimental evaluation indicates that the new heuristic produces drawings with better crossing resolution, but this comes at the cost of slightly higher edge-length ratio, especially when the input graph is large.

Every Plane Graph is Facially-Non-Repetitively $C$-choosable

A sequence $\left(x_1,x_2,\ldots,x_{2n}\right)$ of even length is a repetition if $\left(x_1,\ldots,x_n\right) =\left(x_{n+1},\ldots,x_{2n}\right)$. We prove existence of a constant $C < 10^{4 \cdot 10^7}$ such that given any planar drawing of a graph $G$, and a list $L(v)$ of $C$ permissible colors for each vertex $v$ in $G$, there is a choice of a permissible color for each vertex such that the sequence of colors of the vertices on any facial simple path in $G$ is not a repetition.

How pigeons couple three-dimensional elbow and wrist motion to morph their wings

Birds change the shape and area of their wings to an exceptional degree, surpassing insects, bats and aircraft in their ability to morph their wings for a variety of tasks. This morphing is governed by a musculoskeletal system, which couples elbow and wrist motion. Since the discovery of this effect in 1839, the planar ‘drawing parallels’ mechanism has been used to explain the coupling. Remarkably, this mechanism has never been corroborated from quantitative motion data. Therefore, we measured how the wing skeleton of a pigeon ( Columba livia ) moves during morphing. Despite earlier planar assumptions, we found that the skeletal motion paths are highly three-dimensional and do not lie in the anatomical plane, ruling out the ‘drawing parallels’ mechanism. Furthermore, micro-computed tomography scans in seven consecutive poses show how the two wrist bones contribute to morphing, particularly the sliding ulnare. From these data, we infer the joint types for all six bones that form the wing morphing mechanism and corroborate the most parsimonious mechanism based on least-squares error minimization. Remarkably, the algorithm shows that all optimal four-bar mechanisms either lock, are unable to track the highly three-dimensional bone motion paths, or require the radius and ulna to cross for accuracy, which is anatomically unrealistic. In contrast, the algorithm finds that a six-bar mechanism recreates the measured motion accurately with a parallel radius and ulna and a sliding ulnare. This revises our mechanistic understanding of how birds morph their wings, and offers quantitative inspiration for engineering morphing wings.

Unified Hanani–Tutte Theorem

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.

An efficient algorithm for planar drawing of RNA structures with pseudoknots of any type

An RNA pseudoknot is a tertiary structural element in which bases of a loop pair with complementary bases are outside the loop. A drawing of RNA secondary structures is a tree, but a drawing of RNA pseudoknots is a graph that has an inner cycle within a pseudoknot and possibly outer cycles formed between the pseudoknot and other structural elements. Visualizing a large-scale RNA structure with pseudoknots as a planar drawing is challenging because a planar drawing of an RNA structure requires both pseudoknots and an entire structure enclosing the pseudoknots to be embedded into a plane without overlapping or crossing. This paper presents an efficient heuristic algorithm for visualizing a pseudoknotted RNA structure as a planar drawing. The algorithm consists of several parts for finding crossing stems and page mapping the stems, for the layout of stem-loops and pseudoknots, and for overlap detection between structural elements and resolving it. Unlike previous algorithms, our algorithm generates a planar drawing for a large RNA structure with pseudoknots of any type and provides a bracket view of the structure. It generates a compact and aesthetic structure graph for a large pseudoknotted RNA structure in O([Formula: see text]) time, where n is the number of stems of the RNA structure.

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph.

Constrained Point Set Embedding of a Balanced Binary Tree

Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P, a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P. A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm for embedding an n-vertex balanced binary tree BBT on a set S of n points inside a simple m-gon P in O(m4/3+ϵ + nlog2 n + m log n) time with at most O(m) bends per edge.

Study on the Correlation between Drawing Education and Special Cognitive Ability Evaluation Based on the MCT

Use the international universal theory of evaluation of special cognitive ability with Mental Cutting Test (MCT) to test the students of Science and Technology University. Compare the MCT average scores of fore-and-aft drawing education, it shows that the effect of descriptive geometry teaching is positive correlative with the scores of MCT, the students improved their special cognitive ability from planar drawing to three-dimensional drawing significantly. The students who got lower MCT scores before taking drawing courses improved their MCT scores obviously. The effect of mechanical drawing teaching is feeble correlative with the scores of MCT; it shows feeble improvement in students’ special cognitive ability.