ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics

According to Robert E. Schofield, looking back to the golden age of Scientific Societies we discover that, from the middle of XVII to the XIX century, rather than academic institutions they were considered as the proper alma mater by scientists [1]. Over time, the general reform of the university has gradually reversed this state of things, with few exceptions. This paper proposes some brief reflections on being a Scientific Society (of Geometry and Graphics) nowadays (in its 30th year), including a glance at the present COVID-19 pandemic impact.

A billiard is the trajectory of a mass point in a domain with ideal physical reflections in the boundary e. If e is an ellipse, then the billiard’s sides are tangents of a confocal conic called caustic c. The variation of billiards in e with caustic c is called billiard motion. We recall and extend a classical result of Poncelet on the diagonals of billiards which envelope motion-invariant conics. Each billiard in e with caustic c is the flat pose of a Henrici framework. Its spatial poses define focal billiards in an ellipsoid with a fixed focal conic $$c\,$$ c . We prove that for even j the j-th diagonals are located on a motion-invariant one-sheeted hyperboloid.

We explore properties and loci of a Poncelet family of polygons – called here Steiner-Soddy – whose vertices are centers of circles in the Steiner porism, including conserved quantities, loci, and its relationship to other Poncelet families.

We study the two-parameter manifold of parabolas circumscribed to triangles in a Poncelet porism between two circles (Chapple’s porism). It turns out that the focal points of the parabolas in a certain one-parameter subfamily trace a straight line. The vertices of these parabolas move on rational cubic curves whose acnodes trace an ellipse centered at the poristic stationary triangle center which is the midpoint of the common incenter and the common circumcenter. The axes of the circumparabolas envelop a Steiner hypocycloid over the course of a porism. Varying the pivot point of the circumparabola, the one-parameter family of Steiner cycloids envelops two ellipses, one with the fixed common incenter and circumcenter as foci, the other one carrying the cusps of all cycloids.

The permutation of the trilinear coordinates of a point yield the six permutation points which are conconic. This idea leads to the following generalization: The six permutations of the trilinears of a point together with the six permutations of the trilinears of the image of that particular point under a certain quadratic Cremona transformation yield twelve points which lie on a single cubic curve. This note is devoted to the study of the thus defined cubic curves, especially those defined by some triangle center and its isogonal or isotomic conjugate.

The nine-point conic n contains the three diagonal points and the midpoints of the six sides of a complete quadrangle. We show that for any quadrilateral $${\mathcal Q}=P_1P_2P_3P_4$$ Q = P 1 P 2 P 3 P 4 and an arbitrarily chosen point P there exists a conic l passing through ten points: P, the three diagonal points of $${\mathcal Q}$$ Q , and the six inverses of the poles of the lines $$[P_i,P_j]$$ [ P i , P j ] with respect to any circumconic k of $${\mathcal Q}$$ Q and the inversion center P. The circumconic k can be any conic from the pencil circumscribed to $${\mathcal Q}$$ Q . The nine-point conic shows up as a special case of the conic l. The projective nature of the definition of the conic l has implications on a certain normal problem of asymptotic quadrilaterals in the hyperbolic plane.

A geometrical problem is to determine the location of the image equidistant from four points that do not coincide in three-dimensional space. A practical example of this problem is to determine the location of the lamp equidistant from four lamps, arbitrarily located in the hall of an art gallery, a sports hall, on a site in a park and other objects. A method has been developed for the graphical determination of the geometrical place of an image equidistant from four points that do not coinciding in three-dimensional space. A methodology for the graphical solution of engineering geometric problems is proposed.

We study rational circular billiards. By viewing the trajectory formed after each reflection point to another inside the circle as the number of circle divisions into regions we derive a general formula for the number of division regions after each reflection. This will give rise to an integer division sequence. Restricting to the special cases $$\vartheta =\frac{p}{2p+1}\cdot 2\pi $$ ϑ = p 2 p + 1 · 2 π we show that the number of regions after each reflection n is beautifully related to Gauss’s arithmetic series.

The article deals with the study of surfaces defined by a point frame. Methods of general topology are involved, since it is these methods that make it possible to study the general essential properties of an arbitrary surface. A new definition of a topographic surface (TS) is given as a set of points in T1-T2-T3-T4 space. The concepts of the degree of curvature of a point and the type of a point on a TS are introduced. The concepts of the maximum neighborhood of a point and its order are introduced. Definitions of the trivial and nontrivial boundaries of a domain on a TS are given. Relationships between the number of points in a certain TS area and its boundary are shown. It is shown that the presence of different order vertices is a feature of the TS, in contrast to regular surfaces. The properties of the TS section have been proved to have an arbitrary large curvature. It is shown that the inclusion of new points in the TS region increases the curvature of the region, but does not increase the length of the boundary. Considering the TS as a topological set makes it possible to reveal new properties of topographic surfaces.

A. Kokotsakis studied the following problem in 1932: Given is a rigid closed polygonal line (planar or non-planar), which is surrounded by a polyhedral strip, where at each polygon vertex three faces meet. Determine the geometries of these closed strips with a continuous mobility. On the one side, we generalize this problem by allowing the faces, which are adjacent to polygon line-segments, to be skew; i.e. to be non-planar. But on the other side, we restrict to the case where the four angles associated with each polygon vertex fulfill the so-called isogonality condition that both pairs of opposite angles are equal or supplementary. In more detail, we study the case where the polygonal line is a skew quad, as this corresponds to a $$(3\times 3)$$ ( 3 × 3 ) building block of a so-called V-hedra composed of skew quads. The latter also gives a positive answer to a question posed by R. Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist.

The main objective of this work is focused on classifying the families of curves defined by the intersection of an arbitrary ellipsoid with an arbitrary torus, sharing the same center, based on the number of their connected components and on the number of their auto-intersection points. The graphic geometric representation of these curves, in GeoGebra, and the respective algebraic descriptions, supported from a theoretical and computational point of view, were of fundamental importance for the development of this work. In this paper, we describe a procedure and the necessary implementations to achieve the objective outlined.

A straight line intersects a circle in two, one, or no real points. In the last case, they have two complex conjugate intersecting points. We present their construction by tracing the circle with all lines. To visualize these points, the real plane is extended with the imaginary dimensions to four-dimensional real space. The surface generated by all complex points is orthogonally projected into two three-dimensional subspaces generated by both real and one of the imaginary dimensions. The same method is used to trace and visualize other real and imaginary conics and a cubic curve. Furthermore, we describe a graphical representation of complex lines in the four-dimensional space and discuss the elementary incidence properties of points and lines. This paper provides an accessible method of visualization of the complex number plane.

Voronoi diagrams belong to frequently used structures in computational geometry with application in many fields of science. The properties of Voronoi diagram are already studied in various metric spaces – Euclidean, Manhattan, Minkowski, Hausdorff, or Karlsruhe and also in the hyperbolic metric. In this paper, we focus on the Voronoi diagram and its dual in the Poincaré ball model of the three-dimensional hyperbolic space. We first present some basic tools from the Poincaré ball model needed to construct a Voronoi diagram and for a closer observation of its properties. We have determined certain conditions for the position of the generators controlling the behavior of the hyperbolic Voronoi diagram. In the last section, we demonstrate this effect on the dual graph of a hyperbolic Voronoi diagram, i. e., on hyperbolic Delaunay tessellations.

A new and sharp construction for twins in arbelos, foreseen as solutions of two Apollonius problem is given. To perform it, we use polar reciprocity, the method forged by Poncelet to prove his Porism. In the process, other circles, displaying archimedean affinities, came into scene.

Let $$O=(0,0)$$ O = ( 0 , 0 ) be the intersection point of two plane algebraic curves F and G. According to existing results, we know that their intersection multiplicity $$I_O$$ I O at O satisfies the inequality $$I_O(F,G) \ge mn+t$$ I O ( F , G ) ≥ m n + t , where m and n are the multiplicities of O on F and G respectively, and t is the number of their common tangents at O (counted with multiplicity). The aim of this paper is to investigate under which conditions the equality occurs. These conditions are given in terms of individual common tangents of F and G at O and their relations to the polynomials defining these curves.

Conic curved creases with reflected rule lines is a style of curved origami design, first explored by David Huffman, that is attractive in that it gives one-DOF folding motions with rigid rule lines (i.e., the rule lines remain the same throughout the motion). We show how to discretize any such curved crease pattern into a similar straight-line crease pattern that has a one-DOF rigid folding motion. We develop two general methods for such discretization, where each curve is replaced by an inscribing or circumscribing polygonal line, respectively, and show in both cases that the resulting discretized crease patterns are rigidly foldable. In the case of the circumscribed discretization, the crease pattern is also locally flat foldable. On the other hand, only careful sampling in the inscribed method results in locally flat-foldable crease patterns.

In this paper, we present methods of dissection and transformation between developments and perspective models of three and four dimensional cubes. In the three dimensional case, several types of developments of a cube are divided into polygonal pieces and transformed to the perspective model with equal area, where some transformations are possible with hinged dissection. In the four dimensional case, two methods are presented for dissection and transformation between the development of a hypercube and its perspective model, where one method is based on hinged dissection using subcubes and the other is based on dissection using tetrahedra. We materialized some of the dissections using planar sheets of EVA resin for the cube, and paper and ABS resin for the hypercube. We finally confirmed that transformations based on these dissections are done smoothly by physical manipulation.

The family of perfect circles begins at the Fermat point, contains an incircle, and ends at the circumcircle of the reference triangle ABC. The relevant drawing shows this structure having many surprising properties. On the other hand, when looking for loci of vertices of bundles of equal angles (two or more), we find interesting curves. This, in conjunction with the perfect circles and amicable triangles, allows to determine two new constants in any triangle. One of them is the product of the three proportions s, t and u (like in Routh’s theorem), and the other - the angle occurring in each triangle at least 18 times.

Geometry has impacted human being in different dimensions not only in the creation of meta-objects but also in a deep analysis of them. From this perspective it seems that the object of study focuses on the properties of meta-objects., However it is important to analyze the interaction processes that people carry out for the construction of the geometric space of shape. The process varies on each person depending on their mental models, dominant sensory mechanisms among other factors. In this sense, people that have vision problems do not carry out the same processes in the construction of space compared with people with normal vision. In this sense, the way in which a Weak-Visual (Dv) person interacts with space and with a Work System (WS) like a medium., provides useful information to study the processes of creation and construction of the space on this kind of meta- objects. It was detected from the use of networks that spatial construction of geometric shapes requires different human factors - 2 spatial components, Temporary Interaction Micro-Processes (MPT-I), human interaction cycles, the interrelation of two systems, the interrelation of two typology of meta-representations (MRs), and Focal Attention Processes (PAFs).

This article introduces the concept of a geometric basis for algebraic surfaces of the second and third orders, which allows to write their extensive Grassman equations. Points of the surface of third degree are double points of projective correspondences, which are established on the rays of the bundle of lines with the center at one of the points of the geometric basis. Coordinates of these double points are calculated using programs developed by the authors. Some questions related to the common elements of an algebraic surface and their geometric basis are considered. This work is a continuation of the article [8] presented at the conference in Milan in 2018.

Loci of Poncelet triangle families are explored with a bespoke web-based, interactive tool. Hundreds of beautiful loci have been compiled into a web “gallery”. Some loci aesthetically resemble, or seem to have a common motif with, wrought iron gate designs of the early 20th century. Are Poncelet loci attractive and surprising enough to inspire new artwork, metalwork and/or architectural designs?

The reconstruction of 3D objects from photographs is well-known and several methods are implemented in various software. The vast majority of reconstruction software uses statistical methods and rather less geometric knowledge. At hand of an example, we shall illustrate that the reconstruction of a 3D object from a single edgy photo can be done with a small portion of geometric knowledge about conics. We first collect the basic facts about central projections, perspective images, and their rectification. Subsequently, we use some basic properties of conics in order to rectify an image and to reconstruct an object without previously undistorting or deskewing the image itself. The constructive approach is preferred because of its simplicity and shall also be understood as a plea for Descriptive Geometry and Constructive Geometry.

The aim of point cloud upsampling is to generate dense and regular point sets from the sparse and usually irregular inputs. Numerous approaches based on deep learning have been recently introduced, since they are able to outperform the most of the optimization-based methods. In this paper, we introduce novel upsampling method PU-QF, which is based on local description of the small areas of the input by quadratic surfaces. These are obtained by algebraic fitting to the neighbourhood of each point-normal of the input set. For each pair of point-normals we introduce new point-normal by placing it close to their respective quadrics. The proposed method is compared with current state-of-the-art methods and outperforms them in terms of the preservation of geometric details of the reference point clouds. Also, we achieved comparable results in terms of the regularity of the upsampled sets.

Images and objects depend on the perception and the point of view. The focus of our research had been on exploring these relationships with experiments in various projects with a small group of students. Theory on perception and gestalt psychology as well as publications on anamorphoses formed the background of this research and determined the creations. The theme included also reflections on the extent to which perception can be included in a design or the designed space or object is transformed by the point of view. Possibilities arise in the expansion of space through images or the creation of objects that enable different perceptions depending on the point of view. The experiments studied effective spatial representations, merging of image and real object as well as the creation of images through elements in space. The anamorphoses are not just artistic gimmicks, but they are created by optical-geometric rules and thus provide thought-provoking impulses for reflection on perception and design.

A means of increasing the level of solar energy use for energy supply of buildings is geometric modeling of real conditions of solar irradiation. The process of modeling solar irradiation is a complex system problem in defining many variables in time and space. The aim of the presented study is to formulate the theoretical basis for the geometric modeling of solar irradiation in dynamics based on the diurnal cone of variable geometry. The model of the diurnal cone presented in the article makes it possible to determine the variable coordinates of the sun, the duration of irradiation of surfaces for different latitudes, hours and days of the year.The proposed method makes it possible to simulate the interaction of solar irradiation with the surface of an architectural object in accordance with its shape, position, and shadow-forming influence of the environment. At the same time, thanks to the geometric model, there is no need to use modeling and calculation of solar irradiation conditions with astronomical data or the results of instrumental observations or graphic materials created on this basis.

In ancient times, blowgun was used as a simple ranged weapon to hunt small animals. Today it is mainly used for sport wellness in Japan and is called Fukiya. The main part of a blowgun is a long narrow pipe for shootingdarts. Its underlying principles, especially the dynamic behaviours of the dart and airflow in the blowpipe, are important for blowgun design, but have not been studied sufficiently. This paper presents the results of the simulation of a blowgun by using a moving particle semi-implicit (MPS) software called Particleworks. The experimental study to verify the simulation results is also presented.

Folded surfaces of origami tessellations sometimes exhibit non-trivial behaviors, which have attracted much attention. The oscillation of tubular waterbomb tessellation is one example. Recently, the authors reported that the kinematics of waterbomb tube depends on the discrete dynamical system that arises from the geometric constraints between modules and quasi-periodic solutions of the dynamical system generate oscillating configurations. Although the quasi-periodic behavior is the characteristic of conservative systems, whether the system is conservative has been unknown. In this paper, we decompose the dynamical system of waterbomb tube into three steps and represent the one-step using the two kinds of mappings between zigzag polygonal linkages. By changing parameters of the mappings and composite them, we generalize the dynamical system of waterbomb tube to that of various tubular origami tessellations and show their oscillating configurations. Furthermore, by analyzing the mapping, we give proof of the conservation of the dynamical system.

Karauchi, one of the traditional braided cords, has a circular cross-section in which the constituent strips follow the geodesic of the cylinder, spiraling from one end to the other. In this paper, we propose a structure in which the bi-axial braids of Karauchi are merged and branched through a tetrahedral curved surface composed of a tri-axial Kagome pattern. Each strip entering the tetrahedron from one vertex follows a geodesic of the tetrahedron in a spiral and always exits at another vertex. The behavior of the strips is corroborated by the geometry of the geodesics and triangular tessellations on the tetrahedron. We created an art installation by using a flat braided cord as the strips to construct the proposed curved surface structure. The geometry of this structure and the design and production of the installation are described. We also computationally model the proposed structure under the equilibrium of tension force.

The arrangement of buildings and their forms in houses, such as roofs, eaves, and walls, in the Important Preservation Districts for Groups of Traditional Buildings were analyzed to determine the composition of historic Japanese streetscapes. Then, based on the form data and publicly provided map data, simplified three-dimensional models of the streetscapes, depicting their images, were represented using the automatic form generation method. The characteristics of the historic Japanese streetscapes were explored using the simplified models.

The research carried out in the Surveying of Architecture and the City course in the SAAD at the University of Camerino studied the squares in smaller historical centres. These urban spaces were created between the twelfth and fourteenth centuries and are located in the Marche and Abruzzo regions, areas that contain interesting examples. For each square, a surveying campaign was conducted using direct and indirect methods. The acquired data were processed using orthographic views in the cardinal directions. The photogrammetry survey led to the creation of three-dimensional models for an understanding of the relationships between the parts, the state of conservation of the surrounding buildings, and the square itself. Through the representation, the ‘spatial configuration’ of each square was examined, and the primary elements, architectural aspects, axes, foci, and main profiles were shown. Following this, the analyses were compared to discover the similarities and differences. This research investigates questions related to the use of various survey methods applied to real case studies. It also reflects on the conservation and use of squares in small urban centres. Depopulation of these towns has led to a drop in the ‘physical and identifying’ nature of these places. As a result, the squares no longer perform the role assigned to them over the centuries. This research therefore aims to identify new functions capable of enhancing the squares and also make them attractive in order to overcome this problem.

Ramps are elements of vertical circulation adopted in modern architecture and that allow for the creation of a fluid and continuous spaces, both visually and formally. Among the design strategies employed by Vilanova Artigas, one that stands out is the adoption of ramps to bring these concepts into his projects. This text presents the results of the research that has analyzed the projects and works with ramps designed by Vilanova Artigas through the study of texts written by the architect, visits and photographs, investigative drawings, geometric and graphic analysis, physical and digital models, relating theory and design in the modern period of Brazilian São Paulo architecture. The research has found that the ramps are presented not only as vertical circulation elements but as a part of the architectural program, as an additional environment in Vilanova Artigas’ work that promotes and encourages meetings and coexistence, in addition to materializing concepts inherent to Modern Architecture. The geometry of the ramps in the residential work of Artigas materializes concepts of his architecture and follows in constructive harmony with the adoption of reinforced concrete.

3D surface reconstruction plays a vital role in augmented reality, virtual reality, and robotics. Previous work can obtain high-quality mesh when the input data is dense and evenly distributed. Although many reconstruction algorithms that can get high-quality meshes have been proposed, the quality of the obtained meshes will be degraded when encountering some particular circumstances, such as noise and data missing. This article introduces a surface reconstruction method with good performance despite non-uniformly distributed and missing data. We supplement the input with the sketch and estimate the normal of the sketch globally based on Duchon energy which makes the estimated normal accurate and smooth. Then we use the signed distance function, whose zero iso-surface represents the surface, to combine the sketch with the original input and compute the implicit function as the solution of a sparse linear system. The experimental results show that this method can generate a high-quality mesh. The use of smoothness energy allows our approach to be much more resilient to sampling imperfections than existing methods. In addition, the technique can add sketches to existing models to obtain 3D models that are more in line with the creator’s intent.

This paper presents a graphic methodology for structural analysis of domes and other surfaces of revolution, based on the combined use of funicular and projective geometry. The dome is considered as a network of lines of latitude and longitude, and the equilibrium of this network is analyzed in both horizontal and vertical projection. The equilibrium in the vertical plane is assured by fitting the weight forces polygon to the polar rays, so the dome geometry is a closed antifunicular polygon of the force system. The equilibrium in the horizontal plane is assured by considering the values of the hoop forces in the dual figure, where they form a closed polygon. The new methodology is presented through its application to a hemispherical brick dome of small thickness, considering three structural situations, hemisphere complete, hemisphere with oculus, and hemisphere with lantern; considering different slopes of support reaction; and considering different hypotheses regarding the structural behavior of the haunch filling, in accordance with its morphological characterization. Finally, the minimum thickness required to cancel the tension forces in the dome, is calculated.

Different branches of science and technics use mathematics, including geometry. For example, researches on the positive effects of “green” structures use mathematic models and geometry. Sometimes, indeterminate forms and infinities are happening. Intermediate infinite results can be correct. The following calculation can give 1/∞= 0. A superconductor conductance [Sm] is infinite. Thus, different specialists should understand the concept of infinity. The best possibility is the graphical representation. We define telescopic mapping on a metric space as a bijection by a function that maps the space to a finite figure. If the figure is a unit hypercube, there is a normalized telescopic mapping. We defined the mapping in Cartesian coordinates, which preserves the continuity, differentiability and integrability. The functions of type y = 1/|x| after mapping behave as continuous, which allow to pick out such discontinuities. The mapping visualizes the difference between the divergence of improper integrals to infinity and by oscillations, which are mixed in some sources. An example of a velocity field factor in a duct with a valve shows the principle of building such graphs.

The use of models generated by 3D scanning in Design projects, particularly in the production of packaging, has been gaining relevance as scanning systems become popular. However, the high costs of these devices are an obstacle to their large-scale use. This article is incorporated into the proposal of a low-cost method, by photogrammetry using smartphones and open source software, aimed to 3D scanning of fresh foods. Experiments presented allow the comparison among results provided by different camera resolutions and variations in textures, colors and food morphology. Three scenarios for capturing images are evaluated, demonstrating that the simplest of them, with a fixed base, produces good results, faster than those performed with a rotating base or a mini studio. Some structural packaging projects are carried out from the images generated by the use of the proposed low-cost method.

Forest thinning generates a lot of “waste” in the form of irregular tree branches. Their unpredictable shape and properties are a challenge for the construction industry; however, this research sees this as an opportunity and proposes a novel way to utilise them as structural members of grid shells suited for shelters, roofs, etc. The properties of trees at micro-level informed the generation process. There, chains of cellulose and hemicellulose bundle together to form microfibrils, that on the other hand are the building blocks of tree cells. Peculiar observation is the fact that microfibrils possess self-assembling power that allows them to form structures by themselves resulting in geometry and form with a specific pattern, as a function of the forces at the micro-level. There, reducing the free energy of the surfaces is of paramount importance to ensure form stability. In attempt to correlate to, but not imitate this process, the author proposes rules for morphogenesis based on reciprocal connections that allow the branches to self-form structures based on their original unmodified shape and properties, thus creating geometry from the available material. Architects and engineers can interfere in this assembly process by setting higher level conditions that guide the result but never define it explicitly.

Today, as always throughout the history of nature, it is an inexhaustible source of inspiration. One of the main goals of creating new biophilic/biomimetic forms inspired by natural structures and patterns is to create a positive psychological impact on people (especially in urban areas) by using geometrical methods and the graphical representation of the plants should convey the symbolism of life. With the progress of civilization and the development of digital technologies, significant biodesign is growing rapidly and becoming necessary in the humanization of today's sustainable way of life. This paper deals with leaf of dandelion (Leontodon taraxacum L.) as bio-inspiration to gain land-art installation as result by using digital modelling throughout Blender, SketchUp and Lumion software tools.

This paper investigates the properties of Voronoi diagrams in which the generators are on an Archimedean spiral curve with various divergence angles. We apply geometrical properties to construct algorithms to recognize whether the given Voronoi diagram is the Archimedean Voronoi diagram from linear parastichy patterns.

The paper considers the aesthetic characteristics of architectural objects formed using media facades, features of the created artistic images on the examples of a number of implemented projects of the Nur-Sultan city. Another area of ​ ​media design research is aimed at highly developed structures of cognition and their regularities: the tendency to completeness, harmony, symmetry, simplicity. The socio-psychological mechanism for the formation and assimilation of stable aesthetic stereotypes - “archetypes” can be interpreted as fixing social experience in the subconscious. This should include the formation of style, artistic image of media design, taste. The image of the city develops in the process of its everyday perception. The authors revealed the conditions for visual perception of the media design of the architectural and spatial environment of the city.

Historically, architecture gradually became controlled by modulation, proportion, rhythm, harmony, and orthogonality of regular forms. In his investigation, Le Corbusier renews this fact maintaining the precise orthogonality of the right angle and modulation as a means to achieve his mathematical ‘vérités réconfortantes’. In recent years parametric modelling allows us to combine a great number of parameters, conducting us to explore and discover new possibilities to plan architectural forms. In this paper we study corbusian window panels using PM. The methodological procedures were: i) drawing of the grid of proportions; ii) parametric redrawing of the red and blue series; iii) geometric study of the so-called “jeu des panneaux”, with the reconstruction of the 2.26 m square and its golden section; iv) development of algorithms based on the values of the red and blue series; v) generation of combinations among parameters; vi) discussion of results and analysis. Using Grasshopper, the aim of this article is to report the parametric study of the red and blue series of the corbusian Modulor.

The architects had been attracted by grid systems and applied grids into realized architectures. We published Grid on Architecture in 2018 and discussed the relationship between the geometry of the grid and architectural works. In this study, we reckon the meaning of the grid in city and architecture by comparing game and toys with the grid. First, on reconsidering contemporary architectural theories, the grid, regarded as the “sea”, is associated with architectural monumentality, isolated but inside the city, regarded as an “island”, which makes us recognize the city and keep in the memory. We discuss the gameboard of the grid and brick toy. We can say that the game board grid is a “closed grid”, for its isolated world inside from contexts outside. On the other hand, brick toy like Lego is regarded as an “open grid”, for its multiplying square or cube system. Then, we discuss that these two ideas can also be seen in the idea of the city and architecture. In this sense, closed grids are seen in the ideal city of India and China, and open grids in an expansion of urban infrastructure, for example, Cardo and Decumanus, and Barcelona plan of Cerdà. After that, it treats how contemporary architects translated urban grid to their works on the real map.

Stencils focus on what ought to be painted, printed or sprayed on other surfaces by omitting it. Hence, a plus-minus ambivalence is introduced where a negative (cutout) allows for a positive (message). This yin and yang resonates in areas such as information design with airbrushed words and numbers as well as stencil art, ranging from political messages to abstract graffiti. With the advent of cost-efficient laser cutting for model making, cutout form templates – in contrast to stencil carriers – emerge in great numbers in dumpsters that can be considered illegible leftover waste (the cutout substance missing) or as off-cuts that spur imagination. We see visible and therefore substantial contour forms that are fundamentally empty in terms of the immateriality of the enclosed areas, oxymorons of ‘substantial’ omissions in 2D. Likewise 3D. 3D plastic printing of sophisticated objects requires interim support structures. Once exposed in the cleaning process, these support structures reveal impressive 3D balances of geometric reasonability and visual arbitrariness. Such 2D and 3D templates inspire to view geometry and graphics differently, to flip through the pages of art history, and – and most intriguingly – consider artistic reuse.

This paper introduces some considerations regarding the geometric principles that lead the configuration of precious gems. Geometry crosses and characterises different aspects of precious gems: from the atomic composition of its raw structure, to the classification of the crystalline system, up to defining shapes and configurations of possible cuts, as well as to control the functioning of the instruments that allow its execution. From age to age, geometry is leading the lapidary art that bases its foundations and rules on it, mainly in order to cut and manufacture the stones, and also to identify the rules that could enhance the brightness or minimize the wastefulness of precious material during the processing. Starting from research still in progress by the authors, this paper will outline the role of geometry in the fields of crystallographic classification, the cuts classification, the methods for design and execution, also through the conspicuous iconography and the digital modelling.

This contribution aims at presenting the current progress on the definition of automated procedures able to test different types of basic visual interactions within the built environment. Following the example of past thinkers in the field, the set of proposed operations aims at collecting different kinds of data to better support multi-scale analysis about the impact of transformations processes in different types of contexts.The proposed work will also serve as a base for further elaboration on the topic.

Urban atoll islands are most susceptible to rising sea levels due to low altitudes. The constant immigration of the local population to the water surface affects the growth of the existing spatial configurations of floating settlements and calls into question their stability. The aim of the research in this paper is to find out which spatial configuration would correspond to the unhindered expansion and dynamic geography of a floating settlement for an unlimited number of inhabitants. Hexagonal modular forms, branch structures and principles of generative design will be helpful. By defining 100 inhabitants as a sample and representing 3 different types of floating houses in percentages, a generative design process can be started. The aim of the research is to find a solution with the least number of different types of modular platforms and to show the possibility of unhindered expansion after generating possible combinations of floating houses on one platform and combinations of platforms necessary for a floating settlement.

This paper aims to introduce the implementation parameters of the layered infill patterns in additive manufacturing to evaluate their effects on the actual density in the fabricated parts. To achieve this goal, parameters involved in shaping different infill patterns are presented and the significance of different input parameters on the output (actual) infill density is evaluated. The reference models for infill implementation are generic cubic and cylindrical models as their specific geometries lead to fewer geometric parameters affecting infill density error and provide informative case studies.

The inter-institutional university research between the Politecnico di Milano and the Accademia di Brera (2021–2022), to which this survey has participated, has collected, classified and shared in digital formats the various archival and documentary sources about Camillo Boito: one of the most significant architects in the historical and cultural period of the Unification of Italy. Some of my previous publications on spherical photo-cameras, and about the devices for displaying these images through AR/VR viewers, are here contextualised in particular around the case of research on the meticulous spatiality conceived and realized by the architect Camillo Boito (1836–1914) in some significant projects. This further experience on the field made it possible to verify the functionality and effectiveness of the visualisation of spherical views, not only in technological terms, but in a comparative manner with respect to the usual methods of representing architecture, implying some methodological deductions. In this short essay I therefore briefly propose some methods and results from the survey on two case studies of 19th century interior spaces, through the graphic aspects that can be visualised by the geometric algorithms for their treatment as spherical images, and some considerations deduced from these experiments in the process of geometric visualisation of the designed space.

This study investigates the relationships between textile design and geometry, focusing to this end on the concepts of structure and symmetry. These aspects, which affect both the graphic systems of the decorations and the textures of the fabrics, explain with great effectiveness the compositional logic of textile patterns whose complexity is often attributable to simple flat isometric transformations. The goal is to reveal geometric principles and construction rules underlying the design and the textile structure. These aspects are generally difficult to interpret for those who deal with textile design and who frequently follow a practical and intuitive approach and are not always supported by a real geometric awareness of graphic constructs. The paper thus intends to offer a tool with which to convey a scientific approach to the graphic design of fabrics, stimulating reflection on teaching and research in the discipline.

Computational models and statistics are becoming more important in professional sports, particularly in soccer. With the advance in the Computer Vision area, information such as pose recognition and player orientation are useful for several applications: team statistics, player tracking, etc. Analyzing the data achieved can improve team performance. This article presents a technique to determine the pose of every player on a soccer field throughout the match. Specific hand-held devices for soccer players can determine and track the position of the player only for their own team. However, it fails in determining the posture of the player. Conventional pose determination of the player does not work well when there is a large number of people in the image. This work overcomes such gaps by combining the person detector YOLO and the pose recognizer OpenPose. The proposed solution utilizes regular transmission recordings of previous matches. The developed approach can successfully determine the player’s posture through these videos. However, this is only the beginning of Computer Vision applied to sports. For future work, player orientation and automatic event detection and classification are expected to be developed.

In BIM technology design, compared with other kinds of buildings, wood structure buildings possess a large number of holes, slots, embedded connections and other structures. In order to realize the parametric modeling of wood structure architectural design and solve a series of problems such as a large number of hole and groove modeling, material information setting and complex operation process, this paper interacts with external C# programming technology based on the API program commands provided by Revit, and designs the program code to receive or use data. It avoids inefficient methods such as manual drawing model and manual drawing data that rely on third-party plug-ins for model processing, provides automatic modeling of complex structures such as grooves and holes, reception and application of parameter variables, and adopts automation ideas such as automatic material matching, distributed modeling and visual process simulation for wood structure design, Gradually realizes the parametric modeling of wood structure roof structure. At the same time, this paper designs the corresponding algorithm according to the specification, designs the values of some program parameters, and further provides an example to prove the executable ability of the program.

There are numerous forming processes for complex surfaces, and different surface forming processes all may cause material changes, such as stretching or shrinking, etc., whether they are additive or subtractive manufacturing techniques. If the material wear caused by the forming process is not considered in the sheet metal stamping process, it will certainly produce a reduction in the forming size of the surface or an increase in the assembly error, and even lead to forming failure. This paper takes welding forming process as an example, based on welding deformation theory to analyze the influencing factors that cause deformation, build a welding deformation calculation model for linear and curved welds to propose the corresponding deformation correction calculation method to compensate for the amount of surface deformation caused by welding. Finally, on the basis of theoretical analysis, a method of calculating and correcting surface deformation based on welding theory is proposed, and the proposed method is verified by experiments. Through experimental data analysis, the surface deformation calculated by the method proposed in this paper is close to the experimentally measured deformation, and the error between the two is about 8%. As a result, the proposed method has good engineering application value.

The Variational Principle (VP) is designed to generate non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl. The solution pool of the original VP is based on an additive formulation and, consequently, is not invariant in the diffeomorphic Lie algebra. The original VP works well when the prescribed pair of JD and curl is calculated from a diffeomorphism, but not necessarily when the prescribed JD and curl are unknown to come from a diffeomorphism. In spite of that, the original VP works effectively in 2D grid generations. To resolve this issue, in this paper, we describe a new version of VP (revised VP), which is based on the composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems to have overcome the inaccuracy of the original VP in 3D grid generations. In the following sections, the mathematical derivations are presented. It is shown that the revised VP can calculate the inverse transformation of a known diffeomorphism. Its inverse consistency and transitivity of transformations are also demonstrated numerically. Finally, a new definition of averaging diffeomorphisms based on the revised VP is proposed.

The study presents some results of an experimental research on brick wall cladding, proposing methodological approaches for the creation of new geometric-compositional configurations. Citing innovative case studies of historical and contemporary architecture, the research has allowed to determine a different geometric shape of the brick, which takes its cue from the traditional wedge-shaped brick in majolica brick, widespread in Sicily since the sixteenth century for the realization of cusps and domes. Through the definition of procedural parametric algorithms for the analysis and geometric-spatial control, it is described the path that led to the digital prototype of a modular component of new design, able to adapt in shape and size to design patterns of cladding surfaces with simple and variable curvature free-form. The design of complex geometric and organic shapes, through the visual programming of digital algorithms (generative modelling, algorithmic modelling, computational modelling), as well as bringing a methodological and applicative renewal, has initiated interdisciplinary insights. Starting from a support grid for installation, we report several structural solutions of wall tessellations that highlight and validate its potential applications. The re-proposal of the building element with materials and techniques of the latest generation is in line with the goals of sustainable development OSS (Sustainable Development Goals SDGs) of the ONU 2030 Agenda.

How can we bring more flexibility into sitting? This was the initial question of the seminar “PerForm” of the department Descriptive Geometry and Perspective, Faculty of Architecture of the TU Kaiserslautern (TUK) in 2021. The course started with a workshop providing the students with crucial competencies in digital design, presentation and fabrication tools. After analyzing related studies and projects, the students then developed their own conceptual drafts based on their acquired knowledge and skills. As a central part of the seminar, the research involved designing kinetic and flexible furniture that encourages motion and agility using parametric design tools. In addition, the use of robotic fabrication techniques building 1:1 wooden prototypes was a part of the complex task. Finally, an exhibition was held on campus including the presentation of the final furniture, mockups, documentations and posters. This approach combining topics such as geometry of movement, construction techniques and graphic presentations shows possibilities for a future-proof teaching concept.

This paper describes the development of an algorithm to automatically generate seeds for lung CT (Computed Tomography) segmentation. The segmentation algorithm used is ROIFT (Relaxed Oriented Image Foresting Transform), a seed-based method for segmenting 3D images. Internal and external seeds are required for ROIFT. The internal and external seeds are automatically generated using the 2D watershed segmentation algorithm. Segmented images are transformed into a polyhedral model using the marching cubes algorithm. The segmented lungs will be used to create an anatomical atlas of the thoracic region. In this initial phase, 100 DICOM images were segmented. The anatomical atlas will be used as a regularization to solve the electrical impedance tomography of the human chest. Future work considers the segmentation of the ribs, skin, airways, and heart.

A pre-processing tool for fully-Lagrangian particle-based computational fluid dynamics methods is presented in this work. It adopts relatively simple strategies to generate particle models of the elements of the computational domain from boundaries represented by triangular meshes. Essentially, it is based on identification of the relative positions between cubic lattice nodes and the meshes to create particles that represent the geometry of the elements and assign the material type of the particles that are required for the simulations. Focusing on the recognition of the lattice nodes inside closed surfaces, variants of ray-casting and rasterization algorithms were proposed and evaluated. As a result, the rasterization considering three different orthogonal reference planes is the most accurate one with relatively low computational cost. Examples of the successful applications of the pre-processing tool are also provided.

A comprehensive knowledge of architectural and engineering assets can be implemented involving IT tools that allow, through interoperable 3D modeling, a more immediate visualization and interaction of shapes and structures in virtual space. The conception of such a process then provides for a close integration between creative thinking, scientific knowledge, and operational practice, as well as a series of checks and modifications. Through processes for parametric modeling of a specific case study – the Frontón Recoletos in Madrid – we intend to explore the perspective offered by current software regarding the development of geometric-structural concepts of architectural/engineering works.

This research focuses on the application of the definition of a convex set, which allows us to design and build the graph of bounded surfaces described by cylinders whose directrix curve is a polar curve that intersects with a quadric surface or a function defined in two variables and thus generates bounded surfaces to be constructed using the GeoGebra Suite Calculator. Three cases are presented where the importance of the convex set concept in the parameterization of this type of surface is highlighted. In the first case, we establish the surface that we have called the Flores surface and from this surface, we establish the construction of the Anthurium flower, in the second case we design the flower with 8 petals, and in the third case designed the leaf house modeled by Mareines and Patalano, whose structure is shaped like a flower with 6 petals. The construction of these surfaces is carried out using the GeoGebra Suite Calculator application for mobile devices, which through its GeoGebra AR tool allows these surfaces to be extended to an augmented reality environment and in this way visualize, manipulate, and understand the abstraction of these mathematical objects in a dynamic and interactive.

In this paper we tackle the challenge of visualizing large matrices, also called datacubes, in real time (RT) using Web technologies. The most critical issue is to achieve this for arrays of up to several billion floats. For this purpose, it is necessary to implement models running on CPU-based or dedicated graphic processor unit (GPU). We propose a convenient selection of volume-rendering models for the scientific community and especially data scientists. To this end we oriented our approach to an integration of these models in Jupyter Notebook. The rendering of these datacubes is made possible thanks to an improved raycasting algorithm supporting datacubes with sizes of $$16^3$$ 16 3 , $$64^3$$ 64 3 , $$256^3$$ 256 3 , and up to $$1024^3$$ 1024 3 , corresponding to a billion cells. We designed three different rendering models: (1) an X-Ray model for 3D datacubes in a (x, y, z) space; (2) a model simulating the implicit surfaces of 3D datacubes in a (x, y, z) space; (3) and an original model, we named Derived-rendering, for 3D datacubes in a (x, y, t) space where t represents time. We also introduce solutions to reduce the memory footprint and load on the GPU side. Tested with a recent hardware configuration, our proposal demonstrates we can even reach RT rendering for a billion-cell datacube.

This paper presents the first results of the project Ecolture. Ecology for Culture, an interdisciplinary research still underway at the University of Trento. The aim is to promote and disseminate knowledge of some documentary heritages preserved in archives, libraries, and museums in Trentino by means of visual communication based on open-source digital procedures. Biography is the tool used for selecting the materials to work on. Books, magazines, various kinds of manuscripts, drawings, and photographs, as well as objects and thematic collections, are linked to some characters who, from the sixteenth century onwards, have progressively built the cultural ecosystem of today’s territory in the artistic, humanistic, social and scientific fields. The translation of places, people, works, stories, and ideas into images can use simple or complex digital products such as cartoon drawings, photomontages, infographics, diagrams, maps, 3D models, interactive images, animated pictures, and multimedia videos. Contemporary visual languages make it possible to offer engaging cultural experiences to an audience of specialists and non-specialists both in the exhibition venues and remotely.

The intention of this work is to show a simple and efficient way to compute the deforestation area using Landsat 8 satellite images. It considers only its best band and segmentation based on thresholding. The described approach is simple to be used in public available images of this type ( http://www.dgi.inpe.br/catalogo/ ), can be easy implemented in any computed language or downloaded in this public repository: https://github.com/rmparanhos/analise-imagens . Maybe the most complex aspect on processing this type of imaging is due to the great numbers of pixels and band (or channel) that compose each mosaic and each real area to be studied. The presented material can be useful due the difficulty of finding benchmarks to allow comparison among results.

This paper addresses the issue of visualizing the right information among large data sets by proposing to represent raw data as a set of mathematically-based implicit curves. Implicit curves are proving to be a powerful yet underused tool. The methodology we propose not only allows a more relevant visualization of information, but also a faster and efficient access to it: (1) since curves are extracted and compressed during precomputation, real-time rendering is possible on the end-user’s computer, even for very large datasets; (2) this property can be extended by enabling real-time data access and transfer at the server level – i.e. simultaneously saving local storage costs and increasing raw data security. Our proposal also achieved a high compression ratio (3%) while maintaining the visual significance of the data and reducing discrete artifacts such as curve gaps and pixel aliasing. We based our tests using two-dimensional height maps, but extending it to more dimensions is not a problem since we can consider any two-dimensional slice in these data.

This paper describes a workflow to realize a Virtual tour with a 360° camera to be enjoyed with a new device, a 360° immersive theater. Since their inception, 360° cameras have been subjected to many applications that surpassed initial expectations: not only Virtual tours obtained by image stitching of panoramic photographs as a novel form of visualization but also photogrammetric projects for the creation of 3D Virtual environments too. In this paper we test a new immersive situation of a Virtual tour obtained by stitching of images, not only enjoyed through web application for a desktop or smartphone but through its projection inside an immersive theater of 7 m in diameter also. The use of equirectangular images in an immersive theater is a process already tested; the use of a Virtual tour is the implementation offered. To obtain it the geometry of the equirectangular images projected on the screen of the theater and the implementation of the activities related to its fruition have been investigated and made available through specific app carried out through Unity software.

The history of small villages in inland areas inevitably also passes through the history of its renovation because it leads to new stimuli and drives design approaches. In this context, the large-scale planning and redevelopment of historic centers lead to demanding research and reflection work on the part of the planner. Recently, generative adversarial networks (GAN) have shown how they can be helpful for image generation following certain imposed rules, including those coming from architecture. This research work, exploiting the potentialities of Computational Geometry and automated information management through Data Engineering processes, aims to offer a graphical decision-support tool for urban renewal based on GAN to elaborate new color scales respecting the pre-existing ones. The case study in which the proposed approach is tested is the historic center of Caggiano Municipality, a small inland village in the Campania Region (Italy). The color scale generated was evaluated and compared with the color plan adopted at the time, with results very close to the traditional approach. Therefore, GANs represent valuable graphic data management support for designers who have to deal with historical contexts, especially in the initial study and context analysis phase.

Design for Manufacture and Assembly (DfMA) based construction comprises of prefabrication of volumetric units in a factory control environment and its transportation for onsite installation. A discrete volumetric unit make a self-supporting structure which combine with other units to create a structure. The volumetric based construction various from conventional onsite construction in terms of the guidelines and requirements for construction. Unfulfillment of these guidelines in DfMA based construction can create intensive onsite work after installation of volumetric unit modules which leads to delay of a project. To highlight and raise awareness regarding these guidelines, this article proposed guidelines for DfMA oriented construction by reviewing the already developed guidelines. The guidelines developed are placed in a proper structural format which highlight the necessary steps to be taken during each stage of a construction project. The guidelines also incorporate the use of BIM for DfMA for consecutive stages of construction. A case study is used to validate some of the guidelines which are followed in the corresponding case study. This article contributes to the literature in consolidating the guidelines and requirements for DfMA based construction which will facilitate every stage of DfMA based construction.

In this paper, a new approach is presented for topology optimization (TO) of differential capacitive accelerometers for seismic applications. This method presents the most efficient topology of an accelerometer to have maximum sensitivity in measuring seismic motion. On the other hand, the bandwidth and transverse sensitivity are limited to reduce the non-linearity and errors in the measured acceleration. The results show that TO gives creative geometry to improve the performance of an accelerometer with respect to the application and design limits. A variety of optimized designs have been presented here with respect to the objective of performance and design constraints.

Engineering uses scientific knowledge to design or create and fabricate an object or a system. Contrary to that, reverse engineering starts from a product or process to know about or improve it. There are several reverse engineering techniques, however three of them are detailed and used in this work: 3D reconstruction from slice images, surface reconstruction from a point cloud, and component virtualization using images and measurements. The present work aims to create useful objects for shoulder surgical pre-planning, through the application of reverse engineering methods and the 3D printing technology. The 3D reconstruction from slice images is applied to reconstruct the scapula of as anonymous patient from computed tomography (CT), following the Digital Imaging and Communications in Medicine format (DICOM). The surface reconstruction from a point cloud is applied to recreate the curvature of the scapula’s glenoidal cavity using a Non-Uniform Rational Basis Spline (NURBS) surface. The 3D reconstruction from slice images is the source of the implant component representation, principal part of the arthroplasty implant. As contribution, the three objects described were digitally recreated and 3D printed with polylactic acid filament. The synthetic models served as the basis for research that will be published in other articles.

This document describes the design and development process of an App considering four factors to carry it out: the resources and elements of descriptive geometry, the algorithms, code programming, and the users. At the end of the application, each student will be able to see the successes, errors, and attempts they obtained in each phase. And they will see the time it takes to respond to each of the phases. Successes, errors, and attempts indicate the knowledge that each of the students has acquired about the fundamental concepts of descriptive geometry. And time indicates the observation and concentration that each of the students has to solve any geometric problem.

The article shows some of the results of a research project launched in 2021 in collaboration with the IUAV Project Archive, whose aim is to disseminate the heritage of drawings, documents and projects it contains. From the perspective of a transversal and multiple management of the archive document, the digital exploration investigated here aims at reconstructing the unbuilt project by Valeriano Pastor, presented in 1978 within the framework of an international competition organized by the Università IUAV di Venezia for the rethinking of the Cannaregio Ovest area. Interpretations and digital elaborations narrating new places and imaginaries starting from the production of unpublished images, obtained by the graphic-communicative possibilities of the digital model. This research area combines theoretical study with a mode of simultaneous knowledge, revealing multiple points of observation of the same object, interpreted according to different values of Venetian memory, from landscape to architecture.

The mind map is a revolutionary thinking tool in the 21st century. It can effectively systematize the scattered knowledge fragments in the learning process of students, help students grasp the context of knowledge, help students think deeply and summarize and organize courses. It can reveal the problems that arise in the learning process of students and improve the teaching effect. The effectiveness of the application of the mind map in the teaching process is discussed in this paper discussed to help students learn scientifically.

Humans and machines recognize or understand various 3D shapes through projections from different views. Descriptive Geometry (DG) constructs dimensional one-to-one map with unambiguity. The presentation and computing advantages of DG are presented from the perspective of modern computing, including the methods of dimension reduction with projection and dimension upgrade with geometric construction. The concept of graphic computational thinking (GCT) is then proposed, with an integration of graphic thinking and computational thinking. The characters and the hierarchical structure of GCT in DG are described, along with the potential educational values.

On March 12, 2020 the COVID-19 pandemic was confirmed and as a result, several schools around the world had to adapt their classes to the remote teaching model. The purpose of this study is to record an experience that involved teaching remote classes of Geometric Drawing (GD), Descriptive Geometry (DG) and Technical Drawing (TD) courses to undergraduate students in Civil Engineering, Graphical Design and Agronomy at the State University of Londrina (UEL). For this reason, this article address the way teaching was planned to take place in remote classes and the difficulties experienced by the teachers and students. The students completed two questionnaires in a survey, the first in 2020 and the second in 2021. Through an analysis of the answers, it was possible to evaluate the methodology employed for remote classes.

The strategy of mutual recognition of credits for cross-college study is an important way to promote the sharing and co construction of first-class courses. Through the Internet and the construction of high-quality shared curriculum resources, students can learn the same course across time, space and regions. The cross-college credit mutual recognition mode is a new teaching implementation mode which is different from traditional classroom teaching. Based on the online and offline mixed teaching mode of the national excellent resource sharing course drawing geometry and engineering graphics courses, this paper explores and practices the cross-college study mode and credit mutual recognition for this course. Implementation experience and exploration conclusion can provide an effective reference for the cross-college teaching mode and credit mutual recognition reform of engineering courses.

The paper presents the exp-EIA© method applied to a Master of Science university architectural course at Politecnico di Milano for fostering a human-centered and evidence-based urban design approach. The method, coupling architectural and psychological perspectives, enables to investigate the people-environment relationship using Virtual/Augmented Reality technology and psychological assessment scales. The method adopts a tool designed ad hoc to collect data regarding the urban experience and represent the outcomes in various forms, including the cartographic one. The procedure includes a virtual exploration of an area through spherical panoramas and a survey investigating the area’s emotional and cognitive effects from specific points of view. An open discussion on the results with the 38 international students participating in the survey concluded the workshop. In particular, the collected individual reactions to the scenes were clustered, and the average emotional and cognitive results associated with the specific viewpoints were analyzed and discussed. Results show that some particular visual perspective constitutes robust social attractors characterized by intense emotional and cognitive reactions, whereas other points of view in the surroundings are socially negligible and emotionally neutral. This experiential approach favors practical considerations to inform the project phases from a sensory design perspective.

Geogebra® is open-source software with a graphical interface directed to math and calculus teaching. Besides that, it is frequent to figure out teaching applications in geometry. This paper aims to demonstrate two applications of 3D dynamic models developed in Geogebra® directed to descriptive geometry online teaching. Specifically, the applications regard concepts that students struggle to acquire the visual skills to comprehend: the inclusion of lines in planes and the changing of projection planes. To evaluate the quality of the 3D models, we applied a quantitative-qualitative survey to descriptive geometry professors in Brazilian undergraduate courses, in which they could interact with the 3D models. The results indicated that the developed 3D dynamic models are adequate for online teaching applications. Additionally, they contain the essential features to support the visual skills teaching-learning process required in architecture and engineering courses. Finally, some limitations and suggestions of improvement were pointed out by the participants.

This research focuses on the modeling of bounded surfaces using parameterizations based on cylindrical coordinates, which is one of the topics of the multivariable calculus course, where it seeks to develop modeling and problem-solving skills in real context, being essential for this to develop geometric skills that allow sketching graphs in space. Therefore, the main objective of the research was to present a modeling case of a sludge decanting tank, through cylindrical coordinates, using the GeoGebra software in its Suite version for mobile devices, in addition to interacting with augmented reality (GeoGebra AR). With this, a didactic, dynamic, and iterative form is proposed where the inter-action between mathematics, engineering and technology is evidenced, and seeks to improve the teaching-learning experience.

Building Information Modelling is being adopted worldwide and universities are thus expected to provide the market with new professionals with BIM knowledge and skills. However, introduction of this theme into the curriculum can be challenging to teaching staff. Having successful implementation examples can help carrying on this task. This paper presents the structure, syllabus, adopted tools and activities of an introductory BIM course offered to first-year engineering students. Implemented with only 2 credits, it covers BIM fundamental concepts and develops collaboration skills and abilities with BIM software tools. It was effectively deployed on big classes and successfully offered both in face-to-face and remote modes, adopting a practice focus. An innovative organization for student group projects was adopted, enabling student participation on several projects, performing a different role in each one. Perceived benefits to students’ development are reported. The covid-19 pandemics impact is discussed. Future improvements in the course are suggested. Overall results achieved were considered very good.

The objective of the paper is to share the experience on adoption of a Project-based Learning approach in the Engineering Design Graphics (EDG) course at the Escola Politécnica of the University of São Paulo. Flipped classroom reinforced by readiness assessments, and the intensive use of CAD systems for modeling, simulation and prototyping by laser cutting and 3D printing were also adopted. The background of the course reformulation, contents and activities are presented as well as discussions on the challenges and solutions implemented.

Mental Cutting Test is a widely used format to develop or measure the spatial skills of people in various situations, having different purposes. An exercise consists of a 2D projection of a 3D shape and an intersection plane, denoted by a frame. The task is to choose the shape of their intersection from the set of five shapes. In this paper, we investigate a great number of different shapes that are rendered with Blender, containing various errors that change the basic morphological features of the shapes. After a human-based validation process, we developed a post-processing Python script which detects and corrects the issues of the automatically generated intersections.

The study presents the results of graphic modelling and development of digital models for on-site and remote use. The case study is the church complex of Panagia Paraportiani on the island of Mykonos in southern Greece. The religious complex, built in the 15th Century A.D., is very popular with tourists and the curious who visit the Cyclades islands during the summer season. This activity determines the need for graphic and geometric knowledge and possible remote access in order to guarantee the conservation of the asset. Graphic and iconographic documentation is poor, and the study intervened to fill the gaps in graphic knowledge through 3D modelling. The survey process involved the architectural survey using drone photogrammetric technology and subsequent modelling of the architecture. The results were collected in a digital database, which contains the knowledge phases, point clouds and digital models. The activities allow an on-site and remote fruition of the asset, through online apps, for tourists who need to visit the closed spaces.

Developing or measuring the spatial skills of people is still an interesting research topic nowadays. Well-known assignments such as the exercises of Mental Cutting Test are still popular tools of performing measurements or validating the knowledge of our students. However, we found that researchers, instructors and students are not entirely supported with tools that can be used to enhance their processes. In this paper, we present a resource browser and a quiz application, that might be useful for people who have to deal with Mental Cutting Test exercises.

This article raises the question of the role of descriptive geometry in the training of technical specialists. We offer solutions for successful acquisition of the course of descriptive geometry for the purpose of high-quality training of technical specialists. To date, most technical universities have become part of the United (multi-disciplinary) universities, where the humanitarian and pedagogical direction prevails. According to the generally accepted world practice, at least 60% of the curriculum should take courses in the Humanities. Naturally, in technical specialty programs, special disciplines are combined in order to reduce their volume in order to free up hours (credits) to meet this requirement. It is inevitable that while some of the items involuntarily liquidated. And at this stage, the first victim usually becomes descriptive geometry as an object that is not pleasant to students and therefore unnecessary. The paradox is that the drawing itself remains, but under the name of “Engineering graphics” and in truncated form. But the course “Machine graphics” is inserted-formally, the total amount of loans for drawing is sufficient. It is obvious that with such training, the vast majority of graduates of technical specialties will not be able to technically correctly perform the drawing and understand a simple technical drawing.

This work presents the strategies used in the development of the teaching of the subject of Architecture Project III of the School of Architecture and Urbanism of the Federal Fluminense University from the return to classes during the period of the Covid-19 Coronavirus pandemic which started in 2020. The course was carried out in the online teaching system mediated by technology. Considering the complexity that the teaching of architecture design requires in the fourth semester of the course, and given the conditions imposed by the pandemic, the development of technology-mediated design teaching was a challenge faced by the teachers of the subject. The teaching project theme is complex and extensive and considers a mixed-use high-rise building, with residential and commercial units in a consolidated urban environment, oriented towards the design and development of open and closed spaces, as well as public, collective and private spaces. This article covers the experiences, the strategies adopted, the difficulties and the results obtained in the teaching of architectural design from September 2020 until February 10, 2022.

Mathematicians and computational designers from a School of Architecture, explore visual programming in the environment of a design software with exercises of a basic mathematics course to overcome the limitations of hand drawing and computerized representation of geometry. We link the learning of geometry, which was associated with abstract aspects of mathematics, with the possibilities of solving design problems in the same environment that students use to design with technologies. Visual programming allowed to create and reuse formulas that modeled variations of different exercises, preparing the model for visualization and digital fabrication with 3D printing and laser cutting. This research promotes a learning experience using algorithms and personalized computer graphics throughout design learning and to value the use of mathematics to explore geometric possibilities in design problems.

The end of the 18th century sets a significant date for the masonry bridges construction technique evolution. Studies regarding the thickness of the pillars and shape of the arches configure innovative theories towards the traditional practice that for centuries has produced masonry bridges characterized by large pillars and round arches by modest span. The masonry bridges construction technique evolution is linked to the design experimentation of the French engineer Jean Rodolphe Perronet (1708–1794), whose program on the construction of bridges includes not only the simultaneous construction of all the arches, but also a considerable reduction of the rise-clear span ratio which, through the adoption of polycentric curves, generates very low arches and the consequent reduction of the pillars number in the water. In this sense, this contribution aims to investigate the treatise sources on the geometric tracing modalities of the semi-oval arches ‘with n centers’ of the masonry bridges by Jean Rodolphe Perronet focusing attention on the analysis of the developments and applications in the 18th-20th centuries of the rule of semi-oval arches ‘with n centers and on the geometric analysis of the project for the Neuilly bridge described by Perronet in the volume Description des Projets et de la Construction des Ponts de Neuilly, de Mantes, d’Orléans, […], edites in Paris in the 1788.

This article is a study of the dimensions of a mausoleum built specifically for the founder of Sufism, a spiritual teacher of the Eastern Muslims and a cleric Hodja Ahmed Yassawi. During the construction of this building, the architects used a number of innovative architectural and construction solutions used in the construction of Samarkand, the capital of the Timurid Empire. Temirlan, the ruler of the Golden Horde, defeated Toktamys Khan in a great battle, and historians say that in honor of this victory he built a new, taller complex in place of the dilapidated mausoleum of Hodja Ahmed Yassawi. The ruler Temirlan himself determined the basic geometrical dimensions of the mausoleum. His first proposed geometric size was the diameter of a large dome. According to Temirlan’s plan, the mausoleum is religiously known in Central Asia and was built on the basis of surahs of the Holy Quran in order to propagate this religion. On the outer facade of the mausoleum, in addition to the verses from the Koran, the main dimensions of the building were studied based on the suras of the Koran. It is, in fact, a monument to Temirlan, consisting of eternal suras dedicated to Ahmet Yassawi. Today, the mausoleum is one of the most important and well-preserved buildings in Central Asia.

From 287 BC when Archimedes used the fulcrum to think about the statics equilibrium problem to 1864 when K. Culmann published Die Graphische Statik, the graphic statics gradually formed a system along the research direction of the funicular polygon method. For quite a long time, graphic statics has been explored and developed in the form of a single individual, with some scholars and engineers in local areas beginning to explore its continuation as a design tool, and new mechanical theories gradually arising in most parts of the world. Thus, there is a tendency for the graphic statics to be gradually replaced by new mechanical theories due to the shortcomings such as computational accuracy. However, it is necessary to revisit the historical value of graphic statics. In order to take advantage of graphical statics and to inherit and develop the theory of descriptive geometry, this paper proposes to combine the respective advantages of the principles of statics, Monge descriptive geometry, and computer graphics to construct descriptive statics, and some fundamental research work is carried out for this purpose.

The proposed study is part of a wider research concerning the analysis of indirect survey methodologies used in ancient times to measure the heights of slender architectures, that is, characterized by a strong development in height. For the definition of these techniques, whose geomechanical principles act as a foundation for the current instrumental survey methodologies, we have resorted to the detailed study of the main treatises written since the 15th century. This paper examines the treatise by Latino Orsini and commissioned by Egnazio Danti entitled “Radio Latino […], of 1583, which focuses on the analysis of the instrument of the radio latino, and its different case studies, through the use of which it is possible to obtain measurements of slender and inaccessible architectural elements, exploiting the principles of Euclidean geometry. The analysis carried out combines the study of the author’s writings with the realisation of graphic schemes for the understanding of the analogical techniques, allowing to: verify the degree of accuracy of the calculation methods proposed by the author, and of the measurement practices, in order to define and enrich with the different case studies the conditions for the determination of the heights of inaccessible architectures.

The paper aims to discuss the role of geometry within the Land Art movement, focusing on the ways in which the Euclidean geometry has been declined starting from the American movement that originated it, born at the end of the 1960s. The experience of the former discotheque, now abandoned, designed in the same period in Italy by the architect Filippo Monti is then analyzed: the Woodpecker discotheque in Milano Marittima, in the province of Ravenna. The designer is thought to have been influenced by this artistic movement, or in any case to all intents and purposes to present some of the typical characteristics of Land Art. In Monti’s work there is a geometry that transcends its own limits, going beyond the traditionally attributed characteristics and functions. It is a unique “geometry beyond geometry”. Starting from the recognition of the values of the Modern movement, Monti developed a new form of architecture for the “mathematical countryside of Romagna”. Monti experimented with functionalist language in the field of contrast-comparison between surfaces, moving on to the fluidity of space and compositional plasticity. Woodpecker is his greatest example. The area where this work was built, like all the works attributable to Land Art, is located in a desolate, deserted and isolated space and is an example of the application of Euclidean forms to architecture.

“Divers” by Hoyningen-Huene, maybe one of the most iconic fashion photos from the 1930s, finds still today a favor that needs to be explained. The historical context of this picture, shot after the exhaustion of the Pictorialist era and in the wake of Straight Photography, gives useful information, which, however, does not seem sufficient. An approach borrowed from the history of art, through the principles conceived by Heinrich Wölfflin, can highlight the key features of an image that is clearly inspired by classical tradition. The categories of linearity, planarity, closed form, absolute clarity and multiplicity thus restore iconic significance to this photo, suggesting, if not strong kinship with past masterpieces, at least evocative comparisons. These considerations can be combined with both the analysis of geometric proportions and the interpretation of space investigated according to the rules of perspective All these levels of interpretation invite us to reflect on the strategies implemented by a photographer with great sensitivity to spatial arrangement as well as to art tradition and can offer a more stratified and organic vision of a successful shot.

Physical mathematical models, conceived for distinctly didactic purposes, became the main protagonists in teaching mathematics and geometry between the end of the nineteenth century and the first half of the twentieth century. Long since abandoned in storerooms or the increasingly small display cabinets of mathematic halls, in recent years, they have been the topic of a movement of rediscovery. This study proposes the restoration of collections of mathematical models through rapid prototyping and the integration of explanatory content in Augmented Reality directly on the physical models. The main objective of the research aims at defining a methodology for the renewal of the teaching of geometry, proposing a multimedia didactic theater in which the physical model is once more the protagonist. Initially, an in-depth study is proposed on the mathematical model types representing polyhedra, which display characteristics that derive from the ancient fascination with the perfection of these geometric shapes, selecting for first experimentation the unique collection of Max Brückner’s paper polyhedra.

The truncated triangular trapezohedron depicted in the engraving “Melencolia” I by Albrecht Dürer is a rarely used polyhedron. In addition to the meaning of this engraving, various interpretations have been made about the meaning of this polyhedron. In the interpretations, the composition of the dimensions of the polyhedron and its relationship with the projection were mainly argued, and Dürer’s direct intention was not argued. This paper focuses on the oblique axis contained in the truncated triangular trapezohedron, consider the relationship with what is drawn around it and the historical background, and try to interpret the intention hidden in this polyhedron by Dürer. The meaning of the drawn sphere was also positioned as an extension of the intention.

Based on PBL (Project-based learning method), this paper explores the integrated teaching mode of theoretical teaching, experiment and curriculum design of engineering courses, and reverses the current teaching situation of emphasizing theory over practice to a certain extent. Taking engineering graphics as an example, through a series of targeted curriculum practice teaching method reforms, students can change from passive practice to active practice and training, so as to cultivate students’ innovative thinking and practical application ability. The application shows that the practical teaching link reform integrating PBL teaching method can better liberate students from the traditional passive learning mode and lay a foundation for cultivating compound talents with high innovation ability and practical application ability.

If different requirements of clients can be reflected on the platform, the DfMA (Design for Manufacturing and Assembly) for improving construction productivity can be effectively utilized. In this study, a design conformance evaluation model has been developed to recommend a design that satisfies users’ requirements on the platform. Thirty samples of modular design proposals that can be recommended to users have been collected, and items discussed with the owner when designing a detached house have been investigated. For each design, the number of items suitable for user requirements has been extracted. After dividing number suitable for user requirements from total number of evaluation items, it has been expressed as a percentage to calculate the degree of Conformance of the design based on user requirements. It is expected that this study will be able to recommend a design suitable for user requirements on the platform.