Collision-free Piecewise Quadratic spline with Regular Quadratic Obstacles

We classify mutual position of a quadratic Bézier curve and a regular quadric in three dimensional Euclidean space. For given first and last control point, we found the set of all quadratic Bézier curves having no common point with a regular quadric. This system of such quadratic Bézier curves is represented by the set of their admissible middle control points. The spatial problem is reduced to a planar problem where the regular quadric is represented by a conic section. Then, the set of all middle control points is found for each type of conic section separately. The key issue is to find the boundary of this set. It is formed from the middle control points of the Bézier curves touching the given conic section. Our results are applicable in collision-free paths computation for virtual agents where the obstacles are represented or bounded by regular quadrics. Another application can be found in searching for pointwise space-like curves in Minkowski space.

Key words: Bézier quadratic curve, regular quadric, intersection, collision-free paths

On the Metrics Induced by Icosidodecahedron and Rhombic Triacontahedron

The theory of convex sets is a vibrant and classical field of modern mathematics with rich applications. If every points of a line segment that connects any two points of the set are in the set, then it is convex. The more geometric aspects of convex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in R^n. Some examples of convex subsets of Euclidean 3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. In this study, we give two new metrics to be their spheres an archimedean solid icosidodecahedron and its archimedean dual rhombic triacontahedron.

Key words: Archimedean solids, Catalan solids, metric, Chinese Checkers metric, Icosidodecahedron, Rhombic triacontahedron
circles

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Elementary Constructions for Conics in Hyperbolic and Elliptic Planes

In the Euclidean plane there are well-known constructions of points and tangents of e.g. an ellipse c based on the given axes of c, which make use of the Apollonius definition of c via its focal points or via two perspective affinities (de la Hire's construction). Even there is no parallel relation neither in a hyperbolic plane nor in an elliptic plane it is still possible to modify many of the elementary geometric constructions for conics, such that they also hold for those (regular) non-Euclidean planes. Some of these modifications just replace Euclidean straight lines by non- Euclidean circles. Furthermore we also study properties of Thales conics, which are generated by two pencils of (non-Euclidean) orthogonal lines.

Key words: hyperbolic plane, elliptic plane, conic sections, de la Hire, Apollonius, Thales

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On Some Regular Polygons in the Taxicab 3-Space

In this study, it has been researched which Euclidean regular polygons are also taxicab regular and which are not. The existence of non-Euclidean taxicab regular polygons in the taxicab 3-space has also been investigated.

Key words: taxicab geometry, Euclidean geometry, regular polygons

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Sectrix Curves on the Sphere

In this paper we introduce a class of curves derived from a geometrical construction. These planar curves are the generalization of the less-known sectrix of Ceva. We also present three variations of the sectrix curves on the sphere with using the geometrical construction on the sphere, with the stereographic projection and with a so-called \rolled" transformation.

Key words: sectrix, folium, Chebyshev polynomial, curves on sphere

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Geometry for Learning Analytics

Learning analytics is focused on the educational challenge of optimizing opportunities for meaningful learning.
Assessment deeply influences learning, but at the same time data about assessment are rarely considered and utilized by learning analytics. Current approaches to analysis and reasoning about peer- assessment lack rigor and appropriate measures of reliability assessment. Our paper addresses these issues with a geometrical model based on the taxicab geometry and the use of the scoring rubrics.
We propose and justify measures for calculation of the final grade in peer-assessment and related inter-rater and intra-rater reliability measures. We present and discuss a geometrical model for two important peer-assessment scenarios.

Key words: taxicab geometry, metrics, learning analytics

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Family of Surfaces Heltocat

In this paper the Heltocat family of surfaces is defined according to [5]. It is shown that the surfaces de ned by the family are minimal and form an isometric deformation from the helicoid to the catenoid. The visualizations and computations were made by using the programs Sage and Mathematica.

Key words: minimal surface, Gaussian curvature, mean curvature, Heltocat family of surfaces

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When Image sets Reality Perspectival alchemy in Velázquez's Las Meninas

There are images in History of Art, Science, Technique, Humanities, which are milestones. Las Meninas is one of these. Several levels of reading and deepening have been proposed by historians and theoreticians, and many interpretations have been made. Yet the very sophisticated construction of this masterpiece seems to escape any univocal hypothesis, making it as well astonishing as enigmatic.
Our interest in this extraordinary opera stems from the fact that it belongs to that special category of paintings whose meaning is inextricably based on and linked to their projective structure; therefore, aware of the wideness of the implications, we will mainly focus on its geometric and graphic feature.

Key words: Diego Velázquez, Alcázar de Madrid, perspective, geometry and graphics, photogrammetry, projective geometry, descriptive geometry, optics, catoptrics

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