ICGG 2020 - Proceedings of the 19th International Conference on Geometry and Graphics

The role of quadrics in Euclidean 3-space is similar to that of conics. Therefore, it is natural to ask for string constructions of quadrics, as spatial analogues of the gardener’s construction of ellipses. The first solution given in 1882 by O. Staude is based on an ellipse e and its focal hyperbola h. A string of a given length, fixed with one end at a focal point of h, is passed behind the nearest branch of h and in front of e and finally attached to the vertex of the second branch of h. If the string is stretched at a point P, then P traces a patch of an ellipsoid $$\mathcal {E}$$ E confocal with e and h. Later, Staude presented a second type of string constructions where e and h are replaced by an ellipsoid $$\mathcal {E}_0$$ E 0 and a confocal hyperboloid $$\mathcal {H}_0$$ H 0 . Here the ends of the string follow the two branches of the curvature line $$\mathcal {E}_0 \cap \mathcal {H}_0$$ E 0 ∩ H 0 . We provide a synthetic approach to these constructions and extend them to paraboloids.

We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of non-straight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also look at the set of osculating conics of a curve in projective space, suggest geometrically meaningful examples and briefly discuss and illustrate their corresponding motions.

The isoptic curve $$c_\alpha $$ c α of a planar curve c is defined as the locus of points where pairs of tangents of c intersect at an angle of $$\alpha \in \,(0,\pi )\,$$ α ∈ ( 0 , π ) . The definition of isoptic curves (in the plane) cannot be carried over to three-dimensional spaces. We present a generalization of isoptic curves to a special class of ruled surfaces. For that purpose, we assume that a developable (torsal) ruled surface R is given. Since R is enveloped by its one-parameter family of tangent planes, we can ask for pairs of tangent planes that enclose a fixed angle $$\alpha \in \,(0,\pi )\,$$ α ∈ ( 0 , π ) . The lines of intersection of all such pairs of tangent planes will then be defined as the isoptic ruled surface $$J_\alpha $$ J α of R. Especially, if $$\alpha ={\pi \over 2}$$ α = π 2 , we shall call $$J_{\pi \over 2}$$ J π 2 the orthoptic ruled surface. We shall give some general results on isoptic ruled surfaces together with some examples.

Points in the plane of a given triangle whose trilinear distances form a constant product gather on a planar cubic curve. All these cubics constitute a pencil of cubics in which the three-fold ideal line of the triangle plane and the three side lines of the base triangle are the only two degenerate cubics in the pencil. Among the non-degenerate cubics, there is only one rational curve with an isolated node at the centroid of the triangle. Independent of the chosen distance (product), the inflection points of the cubics are the ideal points of the triangle sides. It turns out that the harmonic polars of the inflection points are the medians of the base triangle. We shall study especially those cubics that are defined by triangle centers. Each triangle center defines its own distance product cubic and, in contrast to all other known triangle cubics, only a rather small number of centers share their cubic.

In this study, the relationship between the works of Vermeer and camera obscura light images and frame composition is considered. It has been suggested that Vermeer may have traced camera obscura images to create his paintings. Although it is unlikely that he could have traced the entire image at once, similar to scanning, it is possible that Vermeer may have taken models, motifs, or relative sizes of different objects from these light images and incorporated them directly into his compositions. One of his works during the early stages of his career is called “Milk Maid.” The results of the analysis of two works from the same period, “Woman in Blue” and “Woman with a Water Jug,” show that multiple horizon lines can be derived, indicating a geometric inaccuracy. This shows that the works of Vermeer were not geometrically accurate during the start of his career. He created stable compositions with geometric accuracy after a certain point in his career. The analysis of “Milk Maid” has created doubts regarding the possibility that Vermeer used a camera obscura to trace the surface of the floor. However, because of factors such as the necessity for focus adjustment and limitations of the light image, he may have faced problems in constructing a geometrically accurate space. This report introduces the personal opinion of the author regarding the boundary lines drawn on floors, which are essential for creating a space with a sense of depth in paintings.

This study elucidated the existence of a grid -four-dimensional Escher Grid- that creates the Droste-effect using perspective in four-dimensional space, with reference to the Escher Grid that is inherent in Escher’s work “Print Gallery”. Considering that Escher Grid is two-dimensional, it is conjectured that the grid itself that creates the Droste-effect for a four-dimensional person can be created in a three-dimensional space. Since the grid can be interpreted as being based on the figure obtained by rotating and expanding the square of the Escher Grid, it is interpreted that the four-dimensional Escher Grid is a figure obtained by rotating and expanding a cube. To generate the four-dimensional Escher Grid, it was first mentioned the relationship between the nesting, perspective, and dimension, and it was revealed that there is a “vanishing line” that has an increased dimension of vanishing point. The rotation axis of the square in the Escher Grid is the vanishing point of the painting, and that point is a point symmetrical with respect to x and y on the two-dimensional plane. Considering that, it can be seen that the vanishing line, which is the axis of rotation of the four-dimensional Escher Grid, is a line symmetrical to the cube in the x, y, z directions in the three-dimensional space, that is, the body diagonal of the cube. In conclusion, it can be said that the base of the four-dimensional Escher Grid is a figure obtained by rotating and expanding a cube around the body diagonal axis.

The contribution deals with tessellations of convex uniform honeycombs that consist of only Platonic and Archimedean solids. The Archimedean truncated cuboctahedron is the hull of a 3D model of the 9D cube. Its edges have nine different orientations and are parallel to those of solids applied in the considered space-filling mosaics. The segments connecting the centroids of same cells of a tessellation, have also nine different orientations. These belong to two sets and can be the initial edges of two different 3D models of the 9D cube. Sequences of elements along the segments can be considered a compound edge of a compound model. The elements of the models can be marked by different colours according to their shapes, orientation and roles. The above described correlations make it possible to mark all compound models of all considered convex uniform honeycombs and their tessellations in all original ones by colouring. Using the compound models of the solids, new compound solids and tessellations can be built and this process can be repeated. This way, fractals or fractal like structures can be created. Differently oriented planes cut out periodical plane-tiling patterns from the gained spatial tessellations. An intersecting plane moved parallel to itself creates a series of patterns transforming into each other. These can be queued up as frames of an animation or give the level line depictions of the tessellations.

This paper deals with the works of the so-called “white carpenters”, people directly or indirectly designated for tasks related to the construction of buildings. Focusing on the carpentry works for creating building roofs in the Spanish territory, from the VIII century onwards we can find two types of carpentry: Muslim and Castilian. The architectural influences between these two types of woodworkers originated a new solution for interlaced framework structures, called “armaduras de lazo”. They consisted in a particular “par y nudillo” system (with jointed rafters) applied to timber roofs having four (or more) pitches, which presented decorative motifs derived from the Islamic geometries. The paper aims to analyze in detail the geometric rules that characterize the architectural and structural composition of the “armaduras de lazo” roofs. In particular, the generation process of the “par y nudillo” structures are based on the assembly of sets of eight, nine or ten pairs of rafters composing an interlaced wheel (the so-called “rueda de lazo”). Such a wheel constituted the basis module of any geometric motif.

By investigation of locus equations we sometimes encounter problems with factorization of resulting polynomials. Commands on factorization of polynomials over the field of rational numbers are implemented in most mathematical software usually by the command factor. We can also use commands on factorization of polynomials over some extension of the field of rational numbers, for instance command AFactor in Maple. Factorization over real or complex numbers is much more difficult. In two examples we will show how to make factorization using dynamic geometry systems in such cases when related commands fail.

Motivation to this problem arose from the Wallace–Simson theorem which states, that feet of perpendiculars from a point P to three lines are collinear if and only if the point P belongs to the circumcircle of the triangle given by these three lines. 3D generalization of the Wallace–Simson theorem might be as follows: Determine the locus of the point P such that feet of normals from P to four arbitrary straight lines in three dimensional Euclidean space are coplanar. In this text we investigate a special case of straight lines being parallel to a fixed plane. We will show how to transfer this case to the planar one. Finally, we state a theorem, which is a generalization of the Wallace–Simson theorem in plane.

The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space. We present an interactive animation of the stereographic projection of a hyperspherical hexahedron on a 3-sphere embedded in the 4-space. Described are synthetic constructions of stereographic images of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their double orthogonal projections. Consequently, the double-orthogonal projection of a freehand curve on a 3-sphere is created inversely from its stereographic image. Furthermore, we show an application to a synthetic construction of a spherical inversion and visualizations of double orthogonal projections and stereographic images of Hopf tori on a 3-sphere generated from Clelia curves on a 2-sphere.

The perfect circles and the amicable triangles are the structures based on any reference triangle ABC. Main part of these structures was presented in Montreal during the ICGG 2012. The perfect circles in the triangle ABC are the family of circles beginning at the Fermat-point (rx = 0), coming through the incircle (rx = r) and ending on the circumcircle (rx = R). The centers of these circles lie on the locus (called here as μ-curve), which is continuous and differentiable. The function of μ-curve is up to the present day unknown, however the mentioned family of the circles has many interesting properties, which could help to find the sought function in the future. The M-points are existent in real only for rx ≤ r. The Soddy-, Eppstein-, Griffith- and Rigby-points have been defined only for the incircle. The perfect circles allowed to generalize them for 0 ≤ rx ≤ R. The both M-points (Mi and Mo) are the centers of the circles coming through the intersections of three vertical circles. These circles are for rx = r (Soddy circles) tangent (on the sides a, b and c of the triangle ABC) and for rx ≤ r intersect mutually at 6 points (3 inner- and 3 outer-intersections). The circle coming through the inner-intersections will be called Mi-circle and the outer – Mo-circle. The both centers of these circles are so Mi-center and Mo-center. They have many very interesting properties similar to the points, which also lie on the generalized Soddy-line. There appear two new circles, pedal points, their mutual relations and proportions. We also managed to define several derived points, including the vertices of two so-called “square root angles” and a point with a maximum value of a certain proportion.

Insolation is one of the most important factors of human comfort and health in premises. Nevertheless, in a cooling period, the solar radiation brings additional heat gains decreasing the energy efficiency of buildings. The best practice is to allow the maximum solar radiation in heating and transition periods, and to limit it to the minimum permissible level during a cooling period. Usually, the problem is solved by technical measures, such as automated sun-blind-transformers, automatic jalousie or south orientation of glazing, equipped by a non-transformed canopy. In a previous work of the authors, a solution using “green structures” with deciduous plants was proposed. It avoids automation, allows using the most of orientations for glazing, does not decline the view, provides emotional comfort by the most natural appearance, and also improves the sanitary conditions by phytoncides. The goal of the work is the provision of minimum normative insolation in a cooling period by the design of a “green structure”. A geometric model is used for the sunrays, which passed through the planting. The model considers the shape of plant crowns and light transmission through them avoiding direct simulation of the shape of each leaf. An approach for the simple determination of the light transmission through different plants without special apparatus is proposed. The light transmission through different plant crowns was determined for different plants, which can be used in “green structures”. The best practice is a combination of plants with dense and openwork crowns.

An algorithm of spatial generation and analysis of intersection of offset curves forming a family and modeling cutting tool the trajectories for machining of a pocket with islands is presented in the present paper. The technique of analysis and detection of various non-working segments of the offset curves for the cases of self-intersection as well as intersection between curves of the opposing fronts is presented. The spatial geometric model of formation of a family of offset curves is based on the cyclographic method of mapping of space E3 on plane z = 0. The proposed method of detection of the non-working segments generated upon intersections of the offset curves of the opposing fronts is the method of testing ray. A geometric model of formation of a family of offset curves given a multiply connected domain with boundary contours of complex shapes is proposed. This model allows us to acquire parametric equations of a working family of offset curves as a result of calculation. This significantly simplifies solution to the task of automated cutting tool trajectory design in pocket machining.

The currently established tendency in computer-aided design consists in creating three-dimensional geometric models of the designed objects. Road design is no exception. The development of information technology has allowed us to perform the transition from 2D systems to the spatial design. This has raised the demand for modern mathematical models applicable for spatial design.The basic element of road surface design is road axis defined as a spatial curve. With that in mind, the mathematical models of road surface formation proposed in the present paper are based on the cyclographic method of modeling of space and its objects. In the present paper the algorithms of acquiring the basic lines and surfaces applied in road surface design are proposed with particular consideration for smoothness of their connection. Special attention has been given to local particularities of acquiring lines and road surfaces of transition segments.Application of the cyclographic method allows us to acquire a computationally simple geometric model featuring an analytical, i.e. exact solution. The proposed model can be applied in development of modern automobile road design CAD.

The use of ICT is becoming common in architectural design. It is expected that the increasing use of simulations and various AI technologies will lead to further improvements in the evaluation of spatial performance during the design stage. This study develops a data-driven agent model to simulate consumer shopping behavior in a commercial facility, with the ultimate aim of identifying the optimal store arrangement and passage shape. The subject commercial facility is a shopping center with 232 stores and 68,640 square meters of floor space. Four stories of the facility are above ground and one story is below. The data obtained from the facility and used in the model included (1) the number of entering/exiting visitors, (2) the purchase histories of membership cardholders, and (3) the number of visitors passing various points in the facility’s passageways. The membership cards of shoppers gave the authors access to 300,000 to 400,000 purchase histories of approximately 100,000 cardholders per month. To model consumer shopping behavior, the transition probabilities of moving between stores were generated from the available purchase histories. The study’s reproducibility was verified by comparing the simulated results to actual data on the number of remaining visitors, exiting visitors and visitors passing through specific points in the passages.

In the advanced equipment manufacturing industry, the flattening computing of complex surfaces is often involved, and its application range is very wide. These methods generally focus on the theoretical calculation of unfold, but lack of consideration of mechanical processing technology, unfold error and other factors. Generally, when the thickness of the plate is less than 1.2 mm, the influence of the thickness of the plate on the accuracy of flattening and forming can be ignored. If the thickness of the plate is greater than 1.2 mm, it will have a certain impact on the forming of the curved structure. Welding is a manufacturing process and technology for joining metals or other thermoplastic materials by heating, high temperature or high pressure. Therefore, fillers will be added during the welding, which will definitely affect the accuracy of surface forming. At that time, the surface processing technology must be combined to make the flattening computing more accurate. A curved surface flattening computing method combined with machining process is proposed based on the geometric mapping method in this paper. Factors such as sheet thickness and welding process are introduced, and they are designed as corresponding parameters. By adjusting the parameter values, the surface flattening errors caused by different sheet thicknesses and welding methods are modified to improve the accuracy of surface flattening molding and meet the actual application requirements.

Reverse engineering is a process in which a man-made object or system is deconstructed to reveal its design, architecture, or to extract useful design information and knowledge from it. In mechanical product design, it has also been used for digitalization of the geometry of an existing product for archiving or further development. This paper presents a research on the application of reverse engineering to produce the 3D CAD models of free-form surfaces, going beyond meshes normally generated in the same process. The model generated allows a design to be assessed and modified to meet new design requirements and manufacturing plans to be generated. Five methods for measurement, modeling and assessment for the CAD modeling of free-form surfaces through reverse engineering are covered in this paper.

This paper describes the strokes, rotation, and modeling of an icosahedron in the orthogonal projection to produce a 3D print model, two cardboard models, and an augmented reality application with the same STL file. Each model and procedure - 3D print, stacked sheets, interlaced sheets, and the creation of a marker - required the utilization of different computer programs. Our study shows that the programs used to create the two cardboard 3D models and the marker modified the position of the 3D model in space from how it was originally saved in the STL file. On the other hand, for the 3D print, the position of the model was preserved according to the original STL file. This investigation determines that each program encodes the model’s geometry differently. Therefore, we conclude that the knowledge and application of descriptive geometry to troubleshoot 3D modeling issues is more important than any technological advances.

The problem of origami design is how to express the intended shape by folding a single sheet of paper. A two-layer origami tessellation (TLOT) is a category of flat origami which makes a polygonal figure that is slightly lifted up from the surrounding paper layers. To construct the crease pattern of a TLOT origami piece, a Voronoi diagram that corresponds to the figure is needed. Although a method to obtain such a Voronoi diagram is known, it is not clear how to evaluate the Voronoi diagram in order to obtain better crease patterns in which there are few too-short creases that are difficult to fold physically. In this paper, we propose a method to evaluate the lengths of the creases from the Voronoi diagram. We also propose a method to interactively construct Voronoi diagrams that create a crease pattern with relatively long creases. We show the effectiveness of the methods by implementing a prototype system on a PC. We confirmed that it is possible to construct appropriate Voronoi diagrams for TLOT.

The teeth segmentation plays a key role in the orthodontics analysis, but, due to their irregularity, it is a complex task. Irregular surfaces segmentation is still an open issue. Whereas some state-of-art algorithms are either very time consuming or not very precise, the region growth algorithm performs well in both aspects. In this project, the region growth algorithm was adapted to reduce noise and improve the performance with a pre-processing step. In this step, the curvature was analysed in order to obtain the boundaries and apply the region growth. Two method were developed: a manual algorithm and an automatic application. The manual algorithm was capable of segmenting all teeth correctly in the experiments. However, the automatic variant did not achieve the same results. Moreover, some holes must be removed in order to perform simulation on the results.

Biodesign means life, living forms, it’s often associated with organic shapes and patterns and it has its wide use in landscape architecture. In the past few decades, there has been a great deal of progress in the study of natural systems as a means of developing contemporary design solutions. The biomimetic approach involves a close connection between biology and technology, which requires special attention because these are two areas that are fundamentally different. Biology is descriptive, broad, and variable, while technology is more numerical, with many guidelines and limitations. By this way, biomimetic as a design tool has so much potential that in the future it has yet to find comprehensive application in projects developed in the landscape architecture profession. A design solution that uses biomimicry can be as highly stylized on different ways, but the aesthetics will always be dictated by the perfect shapes of the nature. This research is inspired by the specific shape and unique beauty of the orchid flower. The study involves exploring the biomimetic pattern of a flower in its parametric modeling and its application in landscape design. This paper deals with the research of a species of the genus Phalaenopsis sp., it contains analysis of its natural form using a bionic pattern and parametric modeling of its flower. The result is the development of a conceptual solution for urban furniture designed for representative city areas.

The quadrangular sandstone pyramid over the vault of the city’s founder Margrave Charles III William is an important symbolic object for Karlsruhe. An idea to transform this geometric body and make it dynamic was initiated by the artist Sabine Classen who is working with large-scale clay sculptures. The process of eversion, established with the Evertible Cube investigated by Paul Schatz in 1929, was taken as reference for moving the pyramid. The inherent transformation process can provide new insight to apparently well-known objects. To make the pyramid movable it is sliced up into triangle-shaped lamellas that can be turned around separately but together form a morphing body. Defining the exact movements and the resulting ruled surfaces over triangle edges is subject of this paper. A parametric virtual model of the lamella pyramid was set up in Rhinoceros with Grasshopper and used for exploration. Considerations about mechanical structures to realize a full-scale dynamic model that could be installed for public are concluding the current approach.

Presented reflections are attempts to confront mathematical and artistic approach to measuring various aspects of creativity, imagination and our understanding of aesthetic values hidden in the produced artifacts in fine art and in science, geometry in particular. Phenomena that are carriers of these subtle values enable comparison of the concepts of measures and measurement strategies in mathematics and art. Common intersection of mathematics and art is not just the utilisation of geometric forms in fine arts. Into a much greater extent this can be found on abstract level, in searching for description and visualisation of natural relations, principles and rules of life, beauty and truth. Mathematical preposition or geometric relation and fine art work are different representations of the observed reality, two abstractions appearing seemingly on the opposite poles of human mind, two different illustrations of truth and efforts to comprise it and present in a compound way. These two representations of “world truths” should perfectly complement each other. Short revision of the surface measure definitions and its application within the two environments is presented, together with continuous efforts of finding their common features and differences.

The current scenario of the development of design processes may consider methods that have the potential to cover complex concepts and forms, which previously had limited reproduction, such as minimal surfaces, but can now be contextualized for professional and teaching practice. Considering these concepts, this article sought to propose and describe a parametric and simulation model that responds to surface attributes, such as those defined by Freii Otto. Initially, a theoretical study was made on the type of surface addressed and the techniques and digital tools to obtain such a model were recognized. Then, a parametric modeling process was structured by visual programming in two stages: the first sought to reproduce Otto’s model from the parametric modeling of an initial flat surface and obtain the conformation of a curved geometry in non-parametric modeling; the second, sought to verify the curvature of the model in a parametric environment. The results of the study indicated that the construction process of the surface from the simulation of the acting physical forces provides a more reliable model of minimum surface, with optimization properties as a function of the null curvature in most points of the surface. This process in a parametric environment advances towards the systematization of new methods of modeling in architecture, important to professional practice and training processes. By integrating the simulation and evaluation of the surface in terms of its physical characteristics in the design process, there is more aligned with contemporary architectural design processes.

A graphic methodology for the structural analysis of domes and other surfaces of revolution, based in the combined use of funicular and projective geometry, is presented. It is considered a dome as a network of lines of latitude and longitude and it is analyzed the equilibrium of this network in both horizontal and vertical projection. The resulting dual configuration is also a spatial system that can be considered by its projection in a horizontal and a vertical plane. The dome is divided by latitude and longitude into an arbitrary number of sectors. The equilibrium can be enforced at each node, that is, at each intersection of latitude and longitude lines. The tangential forces can be considered for their net effect at each node; the net effect of two tangential forces, equal in magnitude, at a node is a radially directed force in the plane of the line of latitude: acting outwards (compression) or inwards (tension). By considering their horizontal projection, and its dual form, it is possible to choose the shape of the radial force diagram (the vertical projection of the force diagram), and identify the radial forces associated with it, and thus the tangential forces. The new methodology is presented through its application to a hemispherical brick dome of small thickness. The model of hemispherical dome analyzed has been referenced to a real dome: the hemispherical dome of brick over the transept, in Basilica of San Juan de Dios in Granada (Spain). The hemispherical brick dome has been also analyzed by applying the slicing technique in the frame of limit analysis. The structural analysis of the brick dome using both methodologies allow us the contrasting of the results obtained.

Traditionally, euclidean lines served as the basis for intuitive understanding of coordinates. Everyone must have noticed that the approximate estimate of the distance in the mountains or in highly rugged terrain does not match the actual distance traveled. If we estimate that we will overcome the site in half an hour, then it really turns out that we spent almost an hour. This is due to the fact that we usually think of lines as smooth, but in fact, in nature, almost all lines are strongly cut and curved. B. B. Mandelbrot called such lines fractal.Recent research has shown that an effective way to model undifferentiated surfaces for solving problems is to use fractal geometry methods. Undifferentiated surfaces include the earth’s surface, which in the theory of fractal geometry is represented by a stochastic fractal.The article considers the problem of constructing a digital geometric elevation model based on fractal theory. The author presents a method for calculating the fractal dimension of self-similar structures of natural origin, describes the differences in approaches to fractal analysis of abstract mathematical fractals and real fractals, for calculating the fractal dimension of which it is necessary to use a numerical algorithm. The method of finding the dimension that allows to avoid uncertainties in the selection phase approximation based on the number of cells covering the size of the cell. This method can be used for both deterministic and natural (stochastic) fractals. This method is verified by the fractal coordinates of the earth’s surface.

This paper presents a new method for inverse designing an airfoil shape in compressible flow by target pressure distribution profile. This work is the first step to reach to the aerodynamic shape optimization of an airfoil in viscous and non-viscous flows. The computation of the flow coefficients was performed in the presented algorithm by the AUSM+ method. From the aerodynamic aspect, the most important factor for designing an airfoil is the determination of the shape geometry. FlexFoil is a model of a modern airfoil deformed with a complicated internal mechanism. Choosing the optimum shape for these airfoils is crucial and this research is a solution to shape the deformation of the airfoil in different flight conditions. In our process, by presenting the target pressure distribution profile on the surface of the airfoil and determining the initial airfoil, the shape of the airfoil is acquired by the specified pressure distribution. The first section of this article uses this method to verify the accuracy of the numerical generated code during the laminar flow, such that it has implemented for a test sample of laminar flows around the airfoil and its results have reviewed. Then by the inverse design of an airfoil, as an innovation, the optimization block is evaluated by continuous extension method.

Electrical Impedance Tomography (EIT) is a noninvasive, fast-growing technique for image reconstruction. In a typical EIT modeling configuration, electrodes are assumed to be positioned at regularly spaced intervals along the contour of an object. Thus, the resulting reconstructed image – which could be obtained by different methods, such as Artificial Neural Networks (ANNs) - has deviations due to the common existence of errors in the measurement, which is related to wrong electrode positioning in real applications. When a discrepancy exists between model and experiment, the distribution obtained by the inverse problem solver will differ from the real case. We propose herein a framework for an EIT problem with non-uniform electrodes distribution. This proposal takes into account prior knowledge about the object interior, in order to make it permit the investigation of electrode positions. A framework to generate training data for the EIT problem was developed and tested with a simple experiment with three cucumbers inside a circular tank. The proposed framework is very flexible as it allows for different experimental setups, with different geometries for the contour and its interior physical structures, by converting the shapes to simple polygons. Moreover, it allows for non-uniform electrode distribution, which can be even more important when medical applications are considered.

The task of determining a set of points defining the injured zone in an injured skull is a hard task, once these points do not have a specific geometric property. This work proposes a semi-automatic iterative algorithm to determine these points. The user manually defines two point clouds, one representing the injured part and another the healthy one. An iterative automatic algorithm is used to fill the gaps of the injured zone, in which it is divided in k clusters and ordered in each iteration. A piecewise Bézier curve is determined with the adaptive neighborhood simulated annealing. This curve is a reference curve used to add and remove points in the injured zone. The proposed methodology is successfully tested in two examples. The first one is a simple example with just convex areas, and the second one has some concave areas.

The topology of mechanical parts has a crucial effect on their performance, and many methods have been already presented to optimize the topology of different structures. In this paper, Coannealing algorithm was used for topology optimization problems with more than one objective function. This algorithm took many concepts from archived multi-objective simulated annealing (AMOSA); however, the acceptance criterion of the new solution is less complex. Also, a filtering technique has been used to limit the search area, thereby decreasing computation costs and improving the resolution of the final material distribution. The results were validated using the same input values and the results in the literature with a maximum error of 2.7%. The viable solutions in the Pareto front curve from this method can be used to select the most suitable one.

This study examines the experience of architectural spaces and their evaluation using AR (augmented reality) and VR (virtual reality).We developed AR models of the Haus-hyazinth, the Row House in Sumiyoshi and the Farnsworth House. Using smartphones, users can perceive their structure and experience their internal spaces on real scale. By using these AR models, we carried out a space evaluation experiment among students who have studied these houses using drawings and photos. We presented two types of AR models: one with textures on their floors, walls, etc., and the other without textures. Then, we asked the students to examine and describe their perception of the space dimensions, largeness, ceiling heights, etc. Next, we developed a VR model of the Jiyu Gakuen Myonichikan. By using VR glasses, users can experience being inside the building. In a space evaluation experiment using this model, at first, we set up the examinee to visit the actual architecture; then, we presented a VR model, which allowed them to change the height, depth, and width of the space freely, to adjust the dimensions to the experienced space. Each of the three buildings have a small space, a single large space, and a space with varying ceiling height. We examined how the dimensions and the textures of the spaces are perceived through AR and VR models. We consider that AR and VR are not only tools for evaluating space, but also methods that complement the understanding of spatial characteristics. Needless to say, studying architectural spaces with the help of drawings and photographs is important. However, complementing this learning with spatial experience through AR and VR makes it easier to comprehend the characteristics of the space.

This article reports a geometric investigation. The object under study is the main piece of the construction toy set called ReptanglesTM, created by Jonathan Walker Stapleton and manufactured by Fat brain Toy Co. In the form of a stylized turtle, this plastic block holds a series of relationships with well-known polyhedra, starting from the cube. Inspired by a video entitled Contortionist Cubes, performed by the mathematician Burkard Polster, and the animation entitled Ars Qubica, created by the graphic designer Cristóbal Vila, both available on the Internet, we also compare other toys and puzzles related to the first one by their polyhedral nature. Similarities between the turtle-shaped piece from ReptanglesTM and the parts from Tri MagsTM (former MagBlocksTM), GeoBender®, the Star Burr Puzzle, and the Yoshimoto Cubes are noticed.

The efficiency in the use of energy is a subject that transcends our current affairs, it is one of the needs and demands of contemporary times. As designers it is our responsibility to devise resources that pose our position in the face of the challenges that arise in the era of new technologies. The objective of this study was to design parametric processes that allow us to determine the best locations, from the energy efficiency of the weather factors, for different devices and devices through digital simulations concerning the real environment. Experiences were carried out from passive and active energy optimization such as shadow study, solar radiation and renewable energy implementation. Being an investigation for project purposes, three types of actions were carried out: analyze, project and evaluate. Finally, the research allowed us to appreciate the potential of emerging techniques and how these dynamic systems can generate a family of solutions to a certain problem such as energy efficiency. The visual interface proposed by the research allows simple visualization and orders a lot of complex information. As they are parameterized studies from the geographical location, they can be repeated quickly at any latitude, saving time and expanding the project possibilities based on weather conditions.

During the twentieth century, the ramp was an architectural element valued by modern architects. Among the architects who most explored this element was Oscar Niemeyer. The plasticity of his buildings is notorious. Since 2017, data and information were collected, resulting in the identification of 497 projects. A database was created for each of them. From these data, it was decided to study outdoor ramps, with curvilinear geometry, in cultural projects, resulting in 32 projects. The methodological procedures for analysis and synthesis were as follows: i) ramp redrawing; ii) geometry analysis; iii) identification of geometric characteristics in tangent curves and lines; iv) criteria for the identification of ramp types; v) comparative analysis of the types of ramps identified; vi) elaboration of a synthetic table with the types of ramps. The parameters were extracted from the preliminary analysis of original drawings of the ramp element. The ramps were parametrically modeled with the plug-in Grasshopper. The classification of ramp types occurred based on the identification of the sequential order of straight lines and curves tangents. It was possible to identify and classify types of ramps according to their geometrical characteristics. This paper contributes to fill the gap in the study of Oscar Niemeyer’s intrinsic architecture geometry.

A purpose of this study is to investigate the case of portable Building with spatial extension and to clarify its geometrical features and applications. A portable building in this paper follows the definition of Robert Kronengburg. Portable building indicates the space which can be carried and towed without disassemble into parts. As a result of the analysis, it could be classified into four groups. Group1 and Group 2 are both under 50 m2. And only Group1 is mass produced for travel trailer used by individual consumer. Group 2 has variety of expansion system but has not become general. Group 3 show the limits of area expansion using single trailer. And the limitation is around 70 m2. Group 4 is over 150 m2 and using two trailers to expand a space. It is considered that the mobile space is effective as a means to enhance the ability to cope with future infectious diseases and severe natural disasters. The knowledge in this paper will be basic information about usage of portable building during normal and emergency. Flexibility of expandable portable building can rise a resilience of society against pandemic and natural disaster.

The Hough transform (HT) is a popular tool for line detection due to its robustness to noise and missing data. Its main use is for transformation of raster images like PNGs, GIFs, and JPEGs to scalable vector graphics. However, the apparent complexities of equations, the difficulties to understand the voting scheme and importance of discretization parameter have prevented HT algorithm developments for generic polynomials. The main concern of this work is to describe the importance of the process of turning discrete the parameters of the equations that describe different curves to be identified in the images. Although it applies to any type of equation, the influence of discretization of open and closed conics, such as parabolas and ellipses, is specifically discussed. The proposed approach identifies the parameters and the possible dependence among them to define the search order of them. The discrete limits and the incremental value of each parameter are commented as well. They are fundamental for the voting scheme, playing a key role on the accuracy of this and promoting a considerable impact on the HT result. Both the number of parameters (that is, the size of the solution space) and the discretization values used are directly proportional to the number of accumulator cells created. Thus, both the computational cost and the processing time may compromise the implementation process. A correct order to calculate the values of the parameters in the algorithm and the appropriate limits for each one of them guarantees a reduction and simplification of operations, as well as an improvement in the efficiency of the results obtained.

One approach to optimise the heat transfer rate in heat exchangers is to increase the surface area between the two streaming fluids. During the German-Austrian joint project instaf, we designed 3-D models using space-filling curves in various iteration steps, which can be produced with additive manufacturing technology. In this paper we first report how space-filling curves can be implemented using Lindenmayer code. After that we describe how we use LEGO® MINDSTORMS® software and robots to provide a lively visualization of the design process.

This article aims to contribute to the development of articulated and expandable structures, addressing the needs of the steadfast paradigm shift, currently perceived in the Architecture and Construction field. Observed in nature and in contemporary society, dynamic polyhedral geometries are imbued with a constructive and aesthetic potential that reveal processes of change, transience and impermanence. Studying the modular units, systems, subsystems and connections in the pantographic polyhedrons can contribute for the development of expandable and contractible structures, which is vital for the technological advances of the Kinetic Architecture. The starting point of the study was the polyhedrons of Plato and Archimedes. These structures are projected in a way that its volumes may vary - i.e. it may expand or contract without the need to modify or change any of its components. The present article brings new technological tools for the development of this type of metamorphic geometries, inspired in forms found in Nature and based on the Buckminster Fuller geodesic structures and on the retractile structures of Chuck Hoberman. This method introduces “The Great – Circle Planes” as a solution to the subdivision of polyhedrons. It also presents a new theoretical/practical system called “System for the development of metamorphic geometries” based on the “Transformable Design” of Chuck Hoberman. In addition, the research presents the method developed for the construction of retractable geodesic domes with its geometry applicable for cover systems. This works also deepens the study on the dynamic geometric structures presenting physical models and virtual simulations applicable to architectural elements, based on their constructions. For this purpose, generative and parametric modeling software and digital manufacturing software were used. The Artifacts were built using a 3D printer and are intended be used in form of activities kits (building pieces) and didactic material.

The paper presents the first results of a research pertaining the communication of painted spaces to visually impaired people, particularly referring to Renaissance paintings in which perspective dictates the arrangement of space, time, narration and composition. In order to make such paintings understandable and knowable by blind visitors, it is first necessary to translate them into a tactile, three-dimensional form. The research hypotheses have been applied to a case-study, i.e. The Feast of Herod and the Beheading of Saint John the Baptist by Florentine painter Benozzo Gozzoli, which is characterized by sequential temporal moments coexisting in the same scene. The pictorial subject also provides a number of very interesting inputs which make it well suitable for the research, not only for the role of perspective and its unifying power, but also for a multisensory communication of its content. The inverse method of linear perspective has been exploited to reveal the plans and sections corresponding to the perspectival space painted by Gozzoli, from which it was possible to build a three-dimensional model of the scene. However, a tactile transposition of the painting alone is not enough; it must be supported by an educational model and a multisensory experience which takes into account the needs of a visually impaired audience and benefits everyone by adding ‘feeling’ to the visual experience.

Lampshades were developed as wind shield for light source when fire was used for light source. Though contemporary light sources, like LED lamp, fluorescent lamp and incandescent light, are working with electricity and wind shield is not required for light source, almost all light sources are still covered by lampshade to arrange or control luminous flux from light source. And combination of lampshades and light sources may give distinctive impression to surrounding people. Until now, paper folding method ‘SQEF’ (Skew Quadrilateral Elastic Folding) was proposed to construct shapes with non-flat texture in the field of lampshade design. With SQEF methods, final shape is constructed by skew quadrilateral unit with a curved surface. As each skew quadrilateral unit is composed by continuously curved surface, luminance distribution within each unit is continuous as well. The shapes of section appeared in proposed lampshades are convex and curved surface in each unit is concave. The purpose of the research described in this paper is expansion of the horizontal sections of lampshades to nonconvex shape to generate convex unit. The buffer part was inserted between concave parts to combine them. An example of the lampshade including concave parts was manufactured. And finally, luminance distribution on the lampshade was examined.

Building area extraction plays a significant role in the aspect of image scene classification, building targets recognition and 3D reconstruction of buildings. When there are multiple building targets in one image, the extraction becomes difficult. Since Gestalt principle reflects the features of human visual perception, it has advantages in detecting the man-made targets which are constructed in accordance with certain rules, such as buildings. Therefore, a building area extraction method based on the Gestalt principle is proposed. Firstly, it proposes an algorithm to calculate the salient edge based on texture features and gray feature information, which solves the excessive false extraction problem of the edge extraction method. And using the relevant point to determine the bottom boundary of a building. Secondly, aimed at the contour integrity of the building target, a new graph model method of extracting the closed contour is proposed. Thirdly, mathematical modeling is conducted regarding to the parallelism, symmetry and similarity feature of the Gestalt principle, and the energy function of the closed contour is designed in accordance with Gestalt principle. False closed contours will be deleted by energy function, and this method can precisely extract the closed contour of the building target in the image. Finally, analyzing the relationship between the location of the building area, and combining the regional texture and gray information to extract the building area. Experimental results demonstrate that the method we proposed can extract multiple building areas in the image, and the building area is more accurate and complete in contour than existing building extraction methods.

This paper presents the implementation in GeoGebra of algorithms for computing the intersection curve of a quadric surface with a torus surface. We present three approaches to get and visualise the intersection curve in GeoGebra. One of the approaches makes use of the geometric capabilities of GeoGebra. The second described approach makes use of CAS to obtain a parametrization and the corresponding visualisation of the intersection curve. Finally, the third one is based on computing the projection of the intersection curve, determining its singularities and structure, and its lifting to the 3D embedding space. The research carried out reveals some of the difficulties arising from the implementation in GeoGebra of a geometric algorithm based on the algebraic equations characterising the objects in consideration.

In view of the characteristics of polyhedron spatial structure, such as simple structure, high repeatability, less node types and joined members, good bearing, seismic resistance and ductility, the parametric modeling of polyhedron spatial structure based on BIM technology is studied. Based on the programming interface (API) provided by Revit, and combined with the mathematical theory of Weaire Phelan (WP) like polyhedron, a parametric modeling plug-in of the polyhedron is developed in Revit through the WP like controlled by the window program to realize the parametric selection and visual modeling of the polyhedron basic cell and basic combination. The correctness of the plug-in in WP like polyhedron selection process is verified by an example, which provides a convenient and fast visual design method for designers.

This study focuses on three-phase topology optimization of distributing a finite amount of material within a reference domain spatially to achieve the goal of minimizing or maximizing objective function like structural compliance. The basic objective of this study is given an abundant of materials with distinguished properties, realizing that how to distribute the whole material to optimize the objective function under volume constraint on the amount of the individual phases is viewed as the goal. And the boundary between all materials should be clear for manufacturing. In this research, a method was proposed to consider the density gradient information and control the number of filtered elements in the multi-material topology optimization design. The mathematical model of multi-material topology optimization is established, and further, the sensitivity analysis process is deduced in detail. The OC (Optimality Criteria) based optimizer is employed to solve the stated optimization problems. Finally, the validity of the model has been done in three-phase case, and it has observed that smooth structure and clear boundaries have been obtained in three phases topology optimization. Therefore, it has been concluded that the proposed method can effectively deal with the above problem about multi-material topology optimization to get clear material boundaries between different materials and make the optimized result more manufacturable.

In the palletizing task, the three-dimensional size of the stack directly affects the working distance (i.e., depth) of the manipulator in the Z-axis direction. In order to complete the palletizing task accurately and reliably and realize human-computer interaction, it is necessary to obtain the three-dimensional structural characteristics of the pallet in real time. Therefore, this paper proposes a three-dimensional reconstruction of the single image of the grab based on the palletizing task. Firstly this paper converts the RGB image of the stack to a grayscale image, then uses the Canny method to detect the edge of the corresponding grayscale image, and uses the Hough line detection algorithm to extract the straight line in the edge image. By analyzing the characteristics of the straight line cluster. A stack is divided into three groups of straight lines in the horizontal, vertical, and vertical directions according to the angle characteristics of the straight line. The RANSAC algorithm is used to analyze and obtain the best linear model of the straight line in each direction. And the minimum distance method is used to solve the vanishing points in the three directions of horizontal, vertical and vertical, and then the internal and external parameters of the camera are calibrated according to the attributes of the vanishing points. A fast and high-precision rectangle detection algorithm is used to detect the coordinates of the vertices of the cuboid. In particular, the size of each cuboid part is consistent and known, so this paper combines the camera’s internal and external parameters with the geometric characteristics of the stack can get the length, width, and height and visual model of a real-time changing stack during the palletizing task. The experimental results show that this algorithm can well build a three-dimensional wireframe model of the stack, and realize the three-dimensional visualization of the model. The maximum error of the reconstructed model is 0.950, the minimum error is 0, and the average error is 0.217; the maximum length The error is 0.560, the minimum error is 0, and the average error is 0.187; the maximum error of the width is 0.950, the minimum error is 0.064, and the average error is 0.461; the maximum error of the height is 0.420, the minimum error is 0, and the average error is 0.14; the robot Z axis The maximum error of the working distance (i.e. depth) in the direction is 0.24, the minimum error is 0, and the average error is 0.08, which meets our actual modeling requirements.

Solid models of the sphere-cone tangential combination in equal circle views and the intersecting combination in cubic cell space region symmetrically and three dimensionally were implemented on CATIA Part Design and Generative Shape Design modules, and their artistic sticker were accomplished accordingly on CATIA Sticker interface. Firstly, the coordinates of cone vertex, the angle of cone-apex, the length of cone generatrix and the radius of sphere tangentially connected to the cone surface were parameterized. Secondly, the circular array patterns of cone surface model tangent to the spherical surface were formed by controlling the axes and planes of the three-dimensional orthogonal coordinate system, and the solid models of filleted sphere-cone tangential combination were generated by Close Surface operation. Thirdly, the cubic cell models were made through changing the coordinates of cone vertex and the radius of intersecting circle between the cone surface and the sphere surface respectively. Finally, selected pictures from scientific literatures were stuck to the solid models, exhibiting the aesthetic effect and actual application conceptionally. The objective of this work is that these solid models in the sticker style maybe conceptionally referenced as a way in logo, fashion and visual design.

Digital image processing is an important part of information perception of mobile robots. Unlike simple geometry, the shape of real objects is always irregular. An accurate and real-time dynamic image processing strategy for mobile robots based on deep learning is developed in this paper. To ensure the dynamic target in water always in the center of the mobile robot’s vision field, a target object detection strategy is designed according to the YOLO-V3 algorithm. 2279 pictures of the target object at different draught depths and different motion directions are collected to retrain the YOLO-V3 model. After testing, the accuracy of the model reaches 94.82%. Besides, considering the high accuracy and high efficiency of the Siamese network, SiamFC (a highly representative algorithm) is selected to support dynamic target tracking. An improved target tracking algorithm based on detection and supervision feedback is designed based on IOU (Intersection over Union) concept. Also, to guarantee the smooth motion of the mobile robot, a strategy of terrain information perception and obstacle terrain passing based on lidar scanning is designed. By analyzing the results of lidar scanning, the mobile robot can judge and avoid the obstacle terrains. The real-time and accuracy of each algorithm is verified by a comprehensive experiment of dynamic target search, detection, and tracking.

The aim of this paper is to use the contour method from the Theory of Mechanisms to approach a problem of geometry. We approach the problem mechanically and mathematically and we discuss several important cases.

Simultaneous localization and mapping (SLAM) is one of the key technologies in the field of robotics, is the key to autonomous navigation of mobile robots, and is also the core and foundation of autonomous and intelligent mobile robots. The SLAM method which relied information solely on monocular cameras is too dependent on the characteristic information of the surrounding environment. For the lack of the scene texture, the dramatic changes of the illumination and the poor performance of the dynamic scene, and the low frame rate of the visual sensor, it cannot deal with the situation of fast motion. The inertial measurement unit (IMU) can output the acceleration and angular velocity of the sensor itself at a high frame rate, and is not affected by the environment, but with serious drift. To solve this problem, this paper aims to design a SLAM system which combines binocular vision information and IMU information. It can realize robust and precise positioning in unknown environment, and provide corresponding navigation map for navigation.

Analysis of soccer videos is an area of recent interest, particularly the 3D reconstruction of soccer matches from broadcasted videos. Three algorithms are proposed that, when combined, will support this 3D reconstruction automatically. Initially, the region of the game is determined, which corresponds to the region of the image where the ball, the judge and the players are located. Subsequently, the field lines are determined using the Hough transform, while geometric and color filters are used to remove false positives. The intersections between the lines determine invariant points that are mapped to points in the field model. The correct mapping allows the correction of the perspective distortion. To determine the correct mapping, correspondence combinations are made between the invariant points and points in the field model. The mapping of the goal plane is also determined.

With the growing market of online sales, there is a new demand for tools that facilitate and augment the client online shopping experience. One example is clothing stores, which face the client preference to buy in physical stores to make sure the clothing will fit nicely. In this sense, computer vision could be used to augment the client experience by virtually reconstructing the body of the client, enabling a virtual model to show how the clothing will fit, suggest models based on the client physical characteristics and provide measurements for customized clothing fabrication. This paper describes the most important part of this technology, which is the body 3D reconstruction requiring only a camera. A video input is used to build a point cloud model and the body is extracted from the point cloud. Then a parametric model of the torso is fit using an optimization algorithm.

Nonuniform Rational Basis Splines (NURBS) are the standard mathematical formulation for the representation of geometry in Computer Aided Design and Manufacturing (CAD/CAM) systems. They are easily adapted to n-dimensional vector functions, which is good for computing additional, non-spatial variables using the same parameter basis. In this paper, n-dimensional NURBS are used to create sweep surfaces and other extruded geometries with twisting and cross-section variations. A kinematics framing methodology is adopted herein so as to extent n-dimensional NURBS to the design of kinematics systems such as roller coasters and railway track. Examples of different pseudo-extruded geometries are presented, including roller coaster track, based both on curve-dependent features and imposed extrusion parameters.

As we know, the robot is a mechanical device that integrates mechanical engineering, electronic technology, automatic control, and computer science. With the wide application of robot technology, obstacle avoidance motion planning of manipulator has become one of the hot issues in the field of robot research. This paper presents the research on the obstacle avoidance motion planning of a 6-DOF serial manipulator in a static space environment. This Obstacle avoidance motion planning is carried out by using the method of configuration space motion planning. Firstly, the obstacle avoidance problem is described mathematically, and a new collision detection method based on a capsule model envelope is proposed. Then RRT algorithm is improved from sampling space and sampling strategy to generate a collision-free motion path from the initial position to target position. The simulation experiment of the multi-obstacle working environment in MATLAB® verifies the feasibility of the improved algorithm. This study shows that the motion planning method of the improved RRT algorithm is not only feasible in theory, but also can meet the needs of practical application and ensure that there is no collision between the manipulator and the obstacles.

Performance simulation of building vegetal envelope can be very resource intensive and time consuming when made with a high number of polygons. The aim of this study is to assess the robustness of a low poly modeling strategy based on raster image sampling, with the scope of reducing the simulation burden of natural illumination performance, in several scenarios of operation. The image-based approach is implemented for the geometric reconstruction of vegetation, starting from an ivy (Hedera Helix L.) leaf, to model a double skin green facade. Due to the high influence of foliage density on natural lighting performance of green walls and its variability in real cases, the strategy behavior is evaluated for the variation of this parameter, addressed as Wall Coverage Ratio (WCR). Five scenarios of WCR are taken into account to perform the analysis of several levels of detail of the low poly mesh geometries, obtained with different sample densities of the raster image. The strategy appears to be reliable and delivers a 21% decrease of the simulation time, compared to the duration of a high level of detail simulation, with an acceptable performance deviation and the result is robust across the analyzed scenarios. A peak 39% decrease is obtained too, but with a considerable performance deviation. The outcomes also show a high dependency of the performance deviation on WCR, especially for simulations with very few polygons. Useful insights on calibration of green modeling accuracy for lighting performance simulation can be drawn from the results of this work.

The TPN-USP Maneuvering Simulation Center is the largest Ship Maneuvering Simulation Center in Brazil. The simulator is used to evaluate new ports and operations, risk analysis, pilots and captains training. The center has 6 cabins, in which of them, 3 are classified as full-mission. The simulation can be executed in single or multiplayer mode. This article gives an overview of graphics technologies in TPN-USP simulator center. Its visualization system aims to immerse the pilot into a realistic virtual environment, providing a high definition visual information to make an operation decision. This article describes some of technologies developed and applied in the simulation center. A full mission simulator with 32 projectors filling 282° of dome screen with 3.6 m height, using 11 render computers cluster is presented. An overview of modeling process, model textures that utilizes PBR (Physically Based Rendering) approach is also presented. This technique generates a photorealistic render. A wave GPU optimized rendering was also integrated. The technique allows to adapt various wave spectrums and spreading effect in modular way. Water spray and foam effects was also included. Additionally, advanced shader methods were used to ensure realistic ocean rendering, including light of sun reflection and sub-surface light scattering.

Scientific Pre-Initiation Program (Pre-IC) is an activity that aims the participation of high school students in scientific research activities of the Brazilian universities. It is also a social welfare activity as its main focus is to motivate the teenagers in their studies, especially those good students from public schools living in the suburb of the metropolis surrounded by poverty and social degradation, and to make them active agents of social transformation. The aim of this paper is to report the experience of the modules on Computational Geometric Modeling in Engineering Design in a Pre-IC program. To meet the limited time available for the student to master the concepts and techniques of computer-based geometrical modeling, the teaching method relied on active learning with hands-on activities, where a problem is presented to the students, which in turn must propose a solution, as well as carry out its modeling, prototyping and test. The basic concepts and techniques are shown and/or provided in parallel to the development of the projects when they are demanded.

In this paper, the authors present a panoramic view of the teaching and learning of graphical geometry contents at Fine Arts School of the Federal University of Rio de Janeiro, along its 203 years of existence. At the end, the authors show two disciplines currently taught in the Specialization Course in Graphic Representation Techniques that allow mastering those subjects in line with the current demands of graphic communication in several working areas.

Architectural Geometry has two main targets: to provide the three-dimensional geometry for creating forms and structures as well as to introduce and to acquaint the students with projection and visualization methods, working between 2D and 3D. To mediate the ways of spatial imaging and thinking remains the most important part. Besides physical models, digital models and animations are able to support those learning processes. We developed 3D models and animations and used the platform Sketchfab® for publishing and sharing, because there the models can be used in a browser, independent from a program and even in an app for smartphones and tablets. Those models and animations will be shown and explained in their various purposes. They will be related to the respective learning aim in architectural geometry. Such a multimedia approach with physical and digital models, animations, drawings with the help of different projection methods revealed as a promising strategy in the learning processes.

Mechanical Drawing is a basic technical course for engineering majors, which is of great significance to the development of specialized courses, curriculum design and graduation design. With the rapid development of network courses, the traditional teaching methods must be reformed to meet the needs of curriculum development in the current educational environment. In this paper, SPOC teaching mode is applied in the course of teaching, so that high-quality MOOC resources and classroom teaching can be integrated organically. The construction of online courses, off-line classroom design and teaching evaluation form are explored and reformed. Practical application shows that this mixed teaching mode enhances the learning experience, enhances learning enthusiasm and improves the teaching effect.

The article discusses approaches to diagnosing the level and structure of students’ knowledge of engineering graphics on the basis of the developed methodical suggestions. In order to determine the level of knowledge of a particular student the quantitative indicators are proposed. The article provides a definition of the qualimetric assessment of a particular student to determine his degree of closeness to the “ideal” student. The Delphi method is used as one of the options for expert assessment. The educational trajectory of the student during the period of study is analyzed. A comparative analysis of the graphs for determining the qualimetric assessment during the semester has been carried out. The conclusions of the statistical processing of the results of continuous monitoring are given.

The purpose of this study is to discuss how the teaching of Descriptive Geometry (DG) for freshman students of architecture at the State University of Londrina, PR, Brazil has evolved from 1990 to the present. This study examines the problems faced during these thirty years of teaching of DG and the ways they have been solved. It also involved conducting a survey of the different tasks carried out with the architecture students during this period, as well as examining the different educational resources that have been used. These include for example: (a) the use of concrete materials to help students visualize points, straight lines and planes in three-dimensional space; (b) the use of card games that have been purposely designed to enable students to learn lines and planes; (c) the designing of physical objects with a specific architectural purpose; (d) the use of a CAD tool to visualize 3D objects represented through orthographic projections; (e) the use of BIM authoring tools and (f) the creation of a website to make classes and exercises available to students, as well as for setting up the virtual learning environment (VLE) Moodle Platform. Finally, we provide a report on the changes that have been made to this course following the revision of the architecture curriculum in 2019 and the removal of the DG course from the current syllabus.

In this paper the basic concepts of education and teaching is discussed. The purpose of education is not to learn knowledge, but to learn a way of thinking. Education is a process of teaching and educating people, and teaching is a unique human talent training activity composed of teachers’ teaching and students’ learning. This paper expounds the concept of teaching students in accordance with their aptitude and expands it to make it clear that the choice of a teaching mode is based on multi-dimensional factors of teaching. Each teaching mode should point to a certain teaching goal, which is the core element of the teaching mode influencing the operation and implementation of the teaching mode and the combination of teachers and students, as well as the standard and scale of teaching evaluation. This paper reviews the typical teaching modes of graphics course in 2013–2019, including the traditional teaching mode represented by classroom teaching and the online teaching mode represented by online courses, carries out teaching research and practice with the hybrid teaching mode taken as an example, and forecasts the development trend of the teaching mode. It is believed that no matter how the teaching mode of graphics in colleges and universities changes, it must conform to the basic laws of graphics teaching. The classified research and multi-dimensional development should be carried out according to the different teaching objects, environment and content.

At the College of Arts and Sciences at the University of Tokyo, graphic presentation education through the use of 3D-CAD/CG is offered in the 2nd semester and descriptive geometry based on hand drawing is in the 3rd. Our interest is whether there is a difference between the spatial abilities of students who take each course. Whether each course influences the improvement of students’ spatial ability is also our concern. To get some insight into these questions, an investigation using the Mental Cutting Test (MCT) to evaluate students’ spatial ability was conducted. The results of an unpaired t-test revealed a weak significant difference between the mean scores in the MCT given to students on the first day of each course. It is suggested that the difference is due not to the improvement of students’ spatial ability during the course in the 2nd semester but rather that students with high spatial abilities tend to take the course in the 3rd semester.

Representing an architectural shape, mediating design/formal/semantic needs, means respecting its specificity according to the purposes with which one operates; therefore, teaching how to represent an architectural shape is a complex operation, especially if this happens in the first year of the degree course in Architecture where the heterogeneity of students’ background requires a preliminary definition of a common language. Students are firstly introduced to theoretical geometries which underlie architectural shapes. So, they have to know the basis of Geometry (both Descriptive and Analytical) in order to proceed within these issues. This process requires to underline the two ‘souls’ of architectural shapes: the theoretical and the build one. Moreover, it also leads to investigate two different types of theoretical shapes: the one that lies behind the design idea and the other one which underlies the built. We propose teaching examples focused on reading architectural shapes as a result of intersections of surfaces.

Creating open online resources containing didactic materials verified in terms of substantive correctness, which support independent learning, becomes a huge need and a challenge that contemporary scientists, teachers, and educators try to meet. The need to create this type of material became particularly evident in the extreme situation in which educational systems found themselves in almost all countries of the world, in the era of the global COVID-19 pandemic, in 2020. The article presents the open Internet platform “DIAD-tools” created as part of the project No 2017-1-LT01-KA2020-035177 “Development of interactive and animated teaching tools - DIAD-tools”, which has been developed from October 1, 2017, to June 30, 2020, in the framework of the Erasmus + project, Action 2, Strategic Partnership. The platform, which is dedicated to pupils of technical high schools and students of technical universities, contains interactive and animated teaching materials supporting the study of technical drawing. The subject matter of the developed materials includes the geometric basis of the methods of projecting of spatial objects on the drawing plane which are used in technical drawing as well as issues from technical drawing, technical drawing - mechanical and architectural - construction drawing. Materials posted on the platform have been prepared in five national languages of the project partners (Estonian, Lithuanian, Latvian, Polish, Slovak) and in English.

Establishing a new drawing system demands critical attention from the user in order to make the correlation between objects that are being represented, and the instruments used for their depiction. The successful use of a technique requires taking into consideration the knowledge implied in that same procedure. In the field of graphic representation, this expertise includes a mental process that reflects the understanding of the relationship between the geometric elements of a figure and the experience with the tools used in the process of creating that same figure. The fundamental basis of all this knowledge is theoretically grounded in geometry, in the way geometry operates, or in geometric thinking. For a long period of time the teaching of this science, as graphic education, has been linked to the use of multi-view drawings. As a result, a pattern of preconceived solutions for such a practice has been largely adopted. However, many of these procedures have become obsolete with the use of digital technology. Observing that technique guides the way of thinking, but does not change the implied knowledge, this paper considers the idea of solving classical problems of geometry through new paths that contemplate space representing technology. A classic solution is presented in two orthographic projections. Then, the same reasoning is applied to a 3D modeling software but adapted to the respective tool. Finally, a new script is elaborated while taking under consideration the peculiarity of the selected program. The goal of this paper is to encourage teaching geometry with an emphasis on geometric thinking and problem solving, making use of the specificity of three-dimensional digital programs rather than training students using old and obsolete methods.

We started to develop an Android application which implements tasks in Augmented Reality, analogous to the classic Mental Cutting Test. Our goal is to offer practicing exercises to our users with the support of gamified elements and AR technology. Thus, users can switch their 2D scenarios into the AR, rotate and scale them to determine the current answer of the scenario. As a motivation we also implemented the most important and popular elements of gamification principles. Thus, students complete their missions, claim rewards, get experimental points and achieve levels. Our application offers different surfaces and tools to users with different roles; thus, it is capable to measure the efficiency of gamification and AR with the use of four user groups such as users who can only access the 2D scenarios, users who can access the AR technology or the gamified elements and users who can access both of them. Besides, we also developed a repository which contains the 3D models (GLB files) and 2D images (SVG files) of the scenarios and various exercises can be composed or generated via its interface. As the result of our development, a prototype of application is ready to use and collect data about the activities of the users in June 2020.

Enabling students the ability of solving complex problems related to their major and cultivating talents with comprehensive qualities are of most important objectives of university education. Under the spirit of the quality criteria of talent cultivating in Engineering Education Professional Certification (EEPC) and the nature of Engineering Graphics course, the objective of this course was adjusted from the traditional knowledge imparting to abilities cultivating. A series of participatory sections were designed in the process of Engineering Graphics course. Basically, students were divided into groups of about five of each, and typical topics were designed and assigned to individual students or study groups, and the group members discussed and completed the tasks. Making solids practice was designed as making real objects according to the dimensions students gave, which is used to deepen the understanding of the relationship between knowledge and real application, and train the practical ability of students. Representation scheme practice was designed to cultivate the ability of using knowledge that the students learned to solve major related complex problems. Retrieving extension materials practice was designed to train the ability of students to retrieve, select and induct needed materials. Creative modeling section was designed to develop innovative spirit of students by virtually constructing complex models.

Based on the research background of mixed teaching, focusing on the construction and development strategy “New Engineering”, and taking the basic college engineering course, Engineering Drawing, as the research object, this article explores a new method suitable for offline teaching in the mixed teaching. It makes full use of the complementary advantages of online teaching and classroom teaching, and adopts the three-stage guided discussion teaching mode, which breaks the traditional teaching concept. The purpose of this research is to improve the teaching quality, make sure students understand the course objectives, stimulate students’ enthusiasm for learning, and improve their application ability.

In recent years, with the continuous development of information technology, information-based hybrid teaching method is accelerating into the classroom. As a link between two-dimensional (2D) books and three-dimensional (3D) entities, 3D models and 3D animations, Augmented Reality (AR) technology is playing an unparalleled advantage in the construction of informatization teaching materials. Because of the need for abstract spatial imagination to understand spatial structure and geometry knowledge, graphics course has always been difficult for some students, and some students’ interest in learning is relatively low. The application of AR technology in the creation and compilation of graphics teaching materials can solve this problem. The creation and compilation of Graphics Textbook Based on AR technology needs the cooperation of cartographers, software technicians and publishing editors. The content creation team composed by the teachers of cartography course is the knowledge-based main body of the drawing textbook production based on AR technology; the technology company, as the software support team, is the technical main body of the drawing textbook production based on AR technology; the publishing editor is responsible for the organization and implementation of the new form of teaching material, and is the organizational main body of drawing textbook production based on AR technology. AR technology enhances the 3D presentation of the content of the teaching material, interesting animation demonstration, the deconstruction reappearance and rotation change of the 3D structure. It also strengthens the knowledge of the graphics textbook, improves the ease of learning, enhances the interest of learning, and helps students understand. The technical team develops the supporting app and presents the resources in the terminal equipment. The updating and maintenance of resource data need to be completed in the background database, and the sharing of resources plays an important role in promoting the integration of media, promotion and use of graphics teaching materials.

This work aims to focus on the changes in Geometry & Graphics education addressed to the architects occurred in the Italian university in the last two decades, mainly involving three generations of academics, researchers, young prospective scholars, in this occasion represented by the authors listed within the work title. Reason for the choice is to stimulate a discussion among these generations, especially in the desirable light of the opportunity – despite the Covid-19 restrictions – to share issues with educators operating in other universities worldwide during the conference days, looking for ideas and hypotheses for updating our academic curricula and researches. In this story, it is to be mentioned the relevant role played in our country by the Unione Italiana Disegno (UID), founded in 1980, and since 2014 an institutional member of the International Society for Geometry and Graphics (ISGG), also thanks to the convinced determination of its former president, professor Vito Cardone, unforgotten member and leader of our Italian academic community. Several loved colleagues and friends are no longer with us today, without whose passion, teachings and wisdom we could not have been here to tell this story, as well as, maybe, we would not have had the lucky chance to spend our academic life enjoying Geometry and Graphics.

By the modernization of production media and processes in the labor market, new skills and qualifications are increasingly required, especially in the engineering area, as well as the need for new teaching methodologies that can adapt to this reality. Within this context, the active methodology arises to improve the teaching-learning processes, opening the opportunity for the student to be the protagonist of his education, developing critical sense in the face of what is learned. Project-Based Learning and Problem-Based Learning are types of active methodologies that have systemic approaches based on students’ involvement in the acquisition of knowledge and skills development, through a process that explores their ability to investigate, create a hypothesis and test it. Therefore, this article presents the quali-quantitative analysis of the application of these teaching-learning techniques in the classes of Graphic Expression of engineering of the University of Pernambuco. Teachers and monitors became mediating parts of this process, as long monitoring is an extension of the production of these activities, students were encouraged to research and discuss to find practical solutions in the elaboration of the proposed project. As a result, there was a recognized improvement in the performance of the students who had an assiduous presence in monitoring. By the overlooked aspects, it was possible to note that the active learning methodology presents a model of integration, training, and professional development of significant importance for the study of Graphic Expression, which is one of the most fundamental disciplines for the academic education of an engineer.

The present work aims to describe the experience and the use of active teaching methodologies of Project-Based Learning (ProjBL) and Problem-Based Learning (ProbBL) in the tutoring of the Graphic Expression II discipline, in the Polytechnic School of Pernambuco of the University of Pernambuco, Benfica campus, during the 2019/2 academic term. The discipline’s tutoring work aims to complement the active teaching methodology adopted by the teachers. The adopted methodology sought to expand the teaching-learning process by instigating the students to not develop projects automatically, but to improve their spatial visualization ability and critical thinking about the project and to expand the students’ skill in design, reading and interpreting architectural projects. Besides preparing the students for their professional life, the teaching methodology increases the self-confidence of the students when facing the problems encourages them to act as an active individual in the society in which they are inserted.

This study is part of a research on the geometric matrices analysis of eighteenth-century residential staircases in Naples (Italy). The research was conducted through the architectural survey of various examples, then cataloged through the planimetric schemes geometric analysis. From this comparison, the staircases examined are attributable to elementary geometric shapes, from which complex solutions derive. In the eighteenth-century, the Neapolitan architect who lends a scenic role to the noble residences’ staircases, thanks to the creative use of elementary geometric shapes, was Ferdinando Sanfelice (1675–1748). Among the many possible geometric configurations, the 45° rotation of two squares of equal size and their overlapping parts generates an irregular hexagon. This solution is the figurative matrix of the Palazzo Cassano Ayerbo d’Aragona staircase. We can find the same architectural experimentation in the treatise on civil architecture by Bernardo Antonio Vittone (1704–1770). In “Volume One” of the Treaty, Istruzioni diverse concernenti l’officio dell’architetto civile, Vittone publishes the project of a “City House” where the staircase design presents a shape surprisingly similar to that of Palazzo Cassano Ayerbo d’Aragona. Recent studies suggested that the Palazzo Cassano Ayerbo d’Aragona staircase design could be attributed to Ferdinando Sanfelice. In the absence of documentary sources, we wonder about the possible relations between Sanfelice and Vittone and/or a possible treaty writer visit to the Palazzo Cassano Ayerbo d’Aragona where Vittone, surprised by spatial complexity of the staircase, would have taken cue to replicate its geometric matrix in the project of the “City House”.

The Maxentius complex is located inside the Appia Antica Archaeological Park and was built in the reign of Emperor Maxentius (306–312 AD). It consists of the mausoleum, the villa and the circus of the same name at the III mile of the Regina Viarum. The research focuses on the geometric reconstruction of the Carceres of the Circus of Maxentius, which concerned not only the circumference arc underlying the arrangement of the Carceres with the towers, but also the geometric center of the circumference itself. The architectural composition of the circuses was conceived in function of the race, therefore each part assumed a precise role. The Carceres were the housing of the wagons before the race and were the place from which the race started.All the representations of the eighteenth and nineteenth centuries have returned the geometric construction of the side containing the Carceres. Their alignment, in fact, was not arranged according to a straight line, but according to an arc of circumference whose geometric center fell on the right side of the field.The analysis of the Ioppolo plant was carried out using vector graphics software that allows you to draw an arc of circumference through three known points and to subsequently visualize, once traced the arch, the center of the circumference. The study of the geometry and proportional correspondence of the artefacts allows us to understand their meaning, not only in relation to the compositional and structural principles, but in relation to the use that was made of them and the different ways of use.

The discerning of the visible part of an object from its invisible part, with respect to the centre of projection, is one of the problems that must be addressed in the representation of any object. The dividing line between these two parts is what we now call the apparent contour. In the history of representation, the study of this theme must be found in the analogies with the condition of illumination of bodies: when the light source is at a finite distance, this can be compared with a proper projection centre, as is the case of perspective. The research for the apparent contour is an easily solved problem in the case of a multifaceted object, otherwise, in curved or free surfaces its identification is more complex. The present study intends to investigate the important contribution provided by Piero della Francesca's work in De Prospectiva Pingendi to scientific theorizing on the central projection of the apparent contour. In this treatise, Piero addresses the theme in three cases: in the anamorphosis of the sphere and the goblet and in the perspective of the human head.

The emergence of complex phenomena in our time has forced creators to connect knowledge with design. In architecture, this situation has provoked an interest in hidden or little-known historical academics who, for years, sought design answers in the interconnection with different dimensions of reality. In particular, we have been interested in how this happens in the architecture of the American northeast, more precisely the derivations of European theories in the mid-twentieth century. Above all, we were interested in the person of Anne Griswold Tyng,—who had a close professional and personal relationship with Louis Isadore Kahn, but specifically, in the way she materialized her seminal work between 1951 and 1953. In the text, this subject is addressed through her seminal project Elementary School in Bucks County P. A., which makes up the DNA of this architect’s work. The project is based on a prefabricated generative system of interconnected parts of tetrahedrons and octahedrons capable of producing a diversity of responses at different scales. Therefore, methodologically, we will penetrate A. G. Tyng’s imaginary and real-world of manufacture. An ideology conceived thanks to a search to integrate space and structure. There, geometry acted as an instrument to extract the structural codes underlying matter, what enabled her to defy gravity. The results of the study indicated that the ideal of progress, along with science and technology in the United States in the mid-twentieth century, fueled a flow of ideas between the professional and educational worlds. This led to a readjustment of the then-dominant reductive and totalizing architectural models. This fact guided Anne G. Tyng towards reflective inter- and transdisciplinary models close to generative systems and complex thinking, which helped this architect in her search to find the structures of tomorrow.

Today, geometrical studies are re-emerging and they are able of connecting and combining conceptual/philosophical/theoretical topics with the shape and configuration of the space, considered as the basic core of still existing or in fieri architecture. Thus, a graphic-documentary apparatus, organized and able to communicate the topics of curves and surfaces, is fundamental to facilitate the understanding of their historical perspective, and - through some exemplary case studies - to deal with their use in different areas of application. The purpose is therefore to broaden the range of critical tools available to any reader (such as researcher, architects, engineer), proposing multiple points of view. Then, an historic analysis about geometry can became the tool to achieve complete awareness through the actions of seeing, thinking, and judging critically. We then propose a tool to organize this knowledge for a better use in any VPL – Visual Programming Language – as an organization of points, lines and surfaces, conscious that not only architecture or design objects, but space in general, exist due to their structural/configurational properties, real or virtual.

This paper focuses on the geometric and graphic analysis of Table III among the four attached to the De Nola Treatise by Ambrogio Leone (1514), in which the author studied the relationship between the shape of the ancient city with the Renaissance one through geometric simplifications as polygonal regular surfaces (square, hexagon and dodecagon) and circular one. These assessments have been extensively described in Chapter II of the Treatise. Our analysis consisted in re-reading the text in which the author describes Table III, in the vectoral re-drawing of the table itself and in the algebraic and dimensional evaluations according to the original measurement units. The results achieved allowed a further verification with the shape of the contemporary city in GIS project, estimating the accuracy of the survey of the urban perimeter carried out by Leone, of its drawing in Table III, and of its geometric and algebraic analyzes.

The contribution focuses on Escher’s graphic production which includes images of polyhedra or complex geometric bodies generated by the combination of platonic solids and inspired by crystals. The Dutch artist was attracted by the sense of balance and perfection of the Platonic solids and the polyhedra derived from them. This article reviews, through conceptual graphic reworking, the geometric genesis of the compositional structure of some solids that recur in Escher’s works. The drawing allows to better understand the aggregation rules and the rigorous construction on which the sense of harmony of these solids depends. Through a graphic-geometric rereading, the authors intend to guide the observer towards a clearer interpretation of Escher’s imaginary configurations based on polyhedra. They therefore elaborate images of these solids to provide an immediate and visually effective reading of relationships and geometric correspondences. Finally, they show how extreme formal complexity equates to great simplicity in the generative matrix.

X-ray Computer Tomography systems (XCT) which can visualize internal structures of objects are very useful tool for detecting defects of parts. Their most common application is inspection of parts made by metal casting or by resin molding. These parts often contain voids, which are small empty regions inside the parts. XCT can scan these parts to generate the 3D CT images of their internal structure involving voids. The parts usually include both unacceptable and acceptable voids. What is practically needed for inspection is to detect unacceptable voids. In this paper we propose our unique classification method based on a concept so called golden part. We compute a golden part by averaging images of the parts containing only acceptable voids. Then the inspection of a test part can be made by comparing the CT image of the test part and the golden part. If the voids of the test part exist only in the void region of the golden part, the test part is passed while if some voids exist elsewhere, it is failed. We implemented a prototype inspection system and apply it to a test case. Its examples demonstrate the usefulness of our method.

The purpose of the present research is to develop three-dimensional geometrical and analytical models to determine the required values of the parameters of an inaccessible point of an object.

This article conceptualizes a solid regular called GeoMotor capable of moving and directing the sediments of a mountain river and changing its geography. The GeoMotor manages to manipulate the directional growth of sediments in an artificial environment, unveiling emerging architectural structures. For this, an analog simulation of the mountain river flow was performed and provide data to understand the phenomenon. Subsequently, this data was used to train a neural network that recognizes the emerging architectural patterns. As future work, it is planned to improve the models to offer functionalities beyond the orthodox practices of traditional architectonic models.

The new technologies based on GIS and AR allow to make the urban space usable and focus attention on hidden hotspots. The research draws a GIS map of the paths identified by writers; redesigns the artist’s works in a 3D space using Blender3D; uses AR to project “on-site” virtual animations on the screen of mobile devices; extends this workflow to many other spontaneous arts within the city of Naples and in the major cities of Europe. NSN997 is one of the cases studied by the research. In this case, in addition to the dissemination of their art through technology, the research can establish “remote” connections between the works present in Naples and those of Madrid, leading the users to establish transversal reading keys and not strictly connected to the individual work. The second case studied is Gianluca Raro’s work, which is focused on peripheral areas of Naples and minor urban centres. The project can provide the opportunity to enhance the less frequented areas by tracing eco-sustainable alternative mobility paths. The research aims to create both physical and semantic connections, adapting to the urban and social contexts of those places that have now become real open-air canvases at the service of the artists and which are destined to become a huge cultural resource.

The implementation of the “reform plan of the teaching content and teaching system of higher engineering education”, the reform of the basic courses of engineering has been carried out in the Northeastern University, China. As one of the basic courses of engineering, the engineering graphics course is facing the education reform in China and the influence of the technology on the teaching mode of the course. So it is imperative to reform the teaching content, teaching methods and mode of the traditional classroom teaching. While exploring the complete theoretical system of engineering graphics course, the course content should strengthen the students' spatial imagination ability, and guide students to master the product information expression method to improve their practical and application ability.

A basic subject of computer graphics (CG) based on three-dimensional (3D) computer-aided design (CAD) was provided in a class of “Basic Media Design” for undergraduate students in their first years. The 3D-CAD used here was the web browser-based 3D modeling tool Autodesk Tinkercad®. During a three-hour class, students are introduced to the following tasks: in the first hour they learn basic functions and skills about 3D design through basic functions of Tinkercad®; in the next hour, they operate the 3D design just by executing the procedures; in the remaining, they create their own 3D model. At the end of the class, they show their 3D models in augmented reality (AR). AR is a technology that allow the visualization of 3D models in the real world adding new information that can also be available on a smartphone. In this way, the students made remarks for what were displayed.

Aiming a Project Based Learning process, it was proposed a challenge that involved the use of 2D and 3D open access computer-applications. The development of the 3D modeling reconstruction of a scene, extracting data from a simple photo emerged as a promising idea, once the challenge should have a large range of potential learning topics that contemplated, at least, Conic Projection, Descriptive Geometry, 3D modeling and 2D projection analysis. The method was developed based on the free version of the Sketchup and a bitmap image produced by a digital camera with the most common resolution. It consists roughly in recognizing in the image the three orthographic main axis (x, y, z) and his respective vanishing points (F). Establish the observer location (O), i.e. his distance from the projection plane, performing a tetrahedron with these four points (Fx, Fy, Fz, and O) and transferring all this data to a 3D environment. Trace all the important rays from O to the points in plane and extend it to the adequate plane. The result, after a number of iterations is the main representation in land plane (plan representation) of the objects. With the information about one single measure it is possible then to spread proportionally to all other objects. The work conclusion is that currently tools are powerful but with a suitable and even more powerful knowledge one can open the professional and creative abilities to a huge range of possibilities.

This work presents a teaching method developed within the discipline of Architecture Project III at the School of Architecture and Urbanism of the Universidade Federal Fluminense. It is based on knowledge and techniques related to Geometric Systems of Representation. This procedure aims at fostering the competence of abstraction in the student by means of exercises that involve spatial notions in the development of the language of drawing and architectural models. Topics of the flat and descriptive geometries are used for the development of spatial reasoning and for the understanding of the three-dimensional organization of space. It also aims to guide the practice of building volumes through reduced models, both for the study of shape and structure and for the visualization of orthogonal representations in different plans. The different ways of experimentation and the freedom with which the exercises are carried out result in a favorable environment for the students discover and develop their creative potential. The final presentation of the students’ projects and models show their development. Moreover, the initial difficulties almost disappear, and the circularity of the processes enables new discoveries and talents.

The report analyses how to proceed to communication between the triad components: concept (problem, engineering idea, existing object), teacher and student, from the semiotic point of view, using Peirce’s triadic model whose basic claim is that signs consist of three inter-related parts: object, interpretant and representamen. Successful application of representational images, i.e. engineering drawings (ED), is not a simple task; it is something that must be trained and used in professional practice, in order to prepare them for the assembly and manufacturing industry. A triadic pedagogical model is presented, where problem solving is focused on form: identified object-problem; interpretant (teacher, student), concept (teacher TC, student SC) and represent (teacher drawing TD, student drawing SD). At all stages of the learning process the teacher must rely on interpreting students’ representations as the evidence of their understanding.

In the Cayley – Klein model, the exterior domain of a projective plane $$P_2$$ P 2 with respect to the oval curve $$\gamma $$ γ , which is the ideal domain of the Lobachevski $$\breve{\i }$$ ı ˘ plane $$\varLambda _2$$ Λ 2 , is a hyperbolic plane $$\widehat{H}$$ H ^ of positive curvature. On this plane a nondegenerate cycle can be as follows: the hypercycle; the elliptic cycle; the hyperbolic cycle; the horocycle. Having investigated isoptic curves for cycles of the plane $$\widehat{H}$$ H ^ , we have proposed various algorithms for deriving the equations of these objects and one more scheme for their study. We consider the forming angles of isoptic curves with both real and imaginary conjugate sides. For each cycle, we find an isoptic curve that is the horizon of this cycle. The fact that among all the cycles of $$\widehat{H}$$ H ^ only an elliptic cycle has an orthoptic curve with a real forming angle is proved.

A new generation of digital visual devices is being available for representation research, with particular interest for the themes of architecture and urban environments. In fact spherical cameras and smartglasses offer advanced experimental opportunity also for some geometrical and iconographic considerations about the new features that digital image technologies made now possible. These digital optical devices are designed for the capture of the entire visual field from a single point of view. The picture of the whole sorrounding environment is so implemented into an equirectangular projection image, as a 1 × 2 format panoramic matrix file, available for further geometric algoritmic computation. This image is get ready to be interactively explored by eyepieces or smartglasses (in VR or AR) designed for giving back to the user the view of the specific optical cone relevant to the field of view to which it is oriented. Some specific capture devices integrate 3D scanning with the simultaneous optical capture of the surrounding space for 3D image/model. The features of such innovative devices and artifacts suggests an interesting case study of a technology, based on complex curvilinear geometry and algorithm, that offer an iconographical computational paradigm for the visual representation, up to social communication.

This paper presents partial results of the research developed over the Federal University of Pelotas original coat of arms design. The study supposes that the institution original coat of arms design, made in 1972, has been redesigned several times during early times of desktop publishing. Taking that in consideration, assumes that several tracing changes has taken place, generating diverse versions of the same graphic symbol. Searches for geometric locus and visual composition rules that defines elements tracing of the original design, as a way to achieve accuracy for a comparison between digital redesigned versions. Authentic graphic properties are identified as from a pattern sample of the original design. Through comparison between the original pattern and digital samples of UFPEL’s coat of arms gathered at the very institution, the research aims to verify mischaracterization events and evince possible losses of authenticity features of the original design. Finally, the study endorses the relevance of project documentation in order to achieve accurate representation of graphic properties in transposition processes of originally hand-drawing geometric designs to vector digital versions.