In Memoriam
Ksenija Horvatić-Baldasar (1929 - 2019)

Lattice Coverings by Congruent Translation Balls Using Translation-like Bisector Surfaces in Nil Geometry

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geometries.
We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given translation tetrahedron.
A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius and the volume of the circumscribed translation sphere of a given translation tetrahedron. The found minimal upper bound density of the translation ball coverings Δ≈1:42783. In our work we will use for computations and visualizations the projective model of Nil described by E. Molnár in [6].

Key words: Thurston geometries, Nil geometry, translation-like bisector surface of two points, circumscribed sphere of Nil tetrahedron, Dirichlet-Voronoi cell

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Focus Stacking from a Purely Geometrical Point of View

In order to create extremely sharp photographs, focus stacking has become a widely used method nowadays, mainly in macro or micro photography. Whereas the aspect of computationally detecting sharp regions of an image has been dealt with in many publications, there is only little published about the geometric background. This paper analyzes the process from a purely geometrical point of view, revealing some non-trivial aspects that may potentially also lead to improvements in a variety of applications such as the 3D scanning of small objects. It is shown that - under calibrated conditions and with certain restrictions - focus stacking of a scene leads to arbitrarily many geometrically correct perspective images of this scene, even including normal projections. The reason for this is that the process of photography never leads to purely two-dimensional images, but collinearly distorted spatial images.

Key words: ray optics, focus stacking, Depth of Field, computational photography, image registration, light field camera

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Covertex Inscribed Triangles of Parabola in Isotropic Plane

In the paper the concept of a covertex inscribed triangle of a parabola in an isotropic plane is introduced. It is a triangle inscribed to the parabola that has the centroid on the axis of parabola, i.e. whose circumcircle passes through the vertex of the parabola. We determine the coordinates of the triangle centers, and the equations of the lines, circles and conics related to the triangle.

Key words: isotropic plane, triangle, parabola

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Bouvaist Cubic of a Triangle in an Isotropic Plane

The cubic in an isotropic plane which passes through the intersections of the sides of an orthic triangle with the sides of a complementary triangle of a given triangle, and through the point which is complementary to the Steiner point of triangle is studied in this paper. It is proved that its non-isotropic asymptote is parallel to Lemoine line of a given triangle.

Key words: isotropic plane, Bouvaist cubic, point com- plementary to the Steiner point

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Use of a Game Engine Artificial Intelligence to Represent People Flows in Architectural Spaces via Geometry and Graphics

The digital and the related technological evolution in recent years have shifted to words such as Virtual Reality and Artificial Intelligence. The wider use of these and other technologies in architecture has been far limited by a lack of IT tools with which architects could interface since they have been made available in the last few years only. The evolution of the tools used by the architect can be condensed and simplified into a sequence of three stages: Drawing Board, CAD systems, Game Engines. The frames of this sequence, in addition to indicating instruments, are representative of the historical context in which they have been or are still being used. This study, based on a Master thesis recently discussed at the Politecnico di Milano [14], examines the role that Game Engines can play in the graphic representation and design processes. More specifically, it takes a closer look at the Unreal Engine as a tool for creating a real-time design environment and using Artificial Intelligence (AI) technologies to represent user flows in the space as valuable support and a relevant part of the design strategies aiming at implementing and evaluating design options. For this purpose, various simulations have been carried out both considering users' flows based on assigned spaces, and generating spaces based on the users' flows.

Key words: artificial intelligence, flows, parametric modeling, dynamic environment, game engines, simulations, unreal engine

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