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

Table of Contents

1. Frontmatter

2. Theoretical Graphics and Geometry

   1. Frontmatter

   2. String Constructions of Quadrics Revisited

      Hellmuth Stachel

      Abstract

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

   3. Space Kinematics and Projective Differential Geometry over the Ring of Dual Numbers

      Hans-Peter Schröcker, Martin Pfurner, Johannes Siegele

      Abstract

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

   4. Examples of Isoptic Ruled Surfaces

      Boris Odehnal

      Abstract

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

   5. A One-Parameter Family of Triangle Cubics

      Boris Odehnal

      Abstract

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

   6. Vermeer’s Specific Ratio

      Noriko Sato

      Abstract

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

   7. Perspective and Illusion in Four-Dimension: Droste-Effect in Four-Dimension Based on Escher’s Work

      Ikko Yokoyama

      Abstract

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

   8. Hidden Structures in Tessellations of Convex Uniform Honeycombs

      László Vörös

      Abstract

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

   9. Geometry and Proportion: Materialization of an Architectural Carpentry Project

      Cristiana Bartolomei, Cecilia Mazzoli, Caterina Morganti, Giorgia Predari

      Abstract

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

   10. Factorization of Locus Polynomials Using DGS

       Pavel Pech

       Abstract

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

   11. A Spatial Generalization of Wallace–Simson Theorem on Four Lines

       Jiří Blažek, Pavel Pech

       Abstract

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

   12. Interactive 4-D Visualization of Stereographic Images from the Double Orthogonal Projection

       Michal Zamboj

       Abstract

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

   13. The M-Points Related to the Perfect Circles in Any Triangle ABC as the Next Points Lying on the Generalized Soddy-Line and About “Square Root Angle”

       Michael Sejfried

       Abstract

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