Recent zbMATH articles in MSC 51N https://www.zbmath.org/atom/cc/51N 2021-04-16T16:22:00+00:00 Werkzeug Projections and distances in $$\mathbb{R}^7$$, double vector cross product and associated hyperplanes. https://www.zbmath.org/1456.51018 2021-04-16T16:22:00+00:00 "Catarino, Paula" https://www.zbmath.org/authors/?q=ai:catarino.paula-m-m-c "Vitória, José" https://www.zbmath.org/authors/?q=ai:vitoria.jose Summary: The distance between two skew lines in $$\mathbb{R}^7$$ is expressed both in terms of a inner product and in terms of a double vector cross product. The best approximation pair of two skew lines in $$\mathbb{R}^7$$ is expressed with inner product and double vector cross product, as well. The obtained formulas using a double vector cross product are not valid for $$\mathbb{R}^n$$, with $$n\neq 3,7$$. Another feature of this paper is to show that the distance from a point to a line and the distance between skew lines are essentially the distance from a point to a hyperplane. Thickness design for ambiguous cylinder illusion. https://www.zbmath.org/1456.00104 2021-04-16T16:22:00+00:00 "Sugihara, Kokichi" https://www.zbmath.org/authors/?q=ai:sugihara.kokichi Summary: This paper proposes methods for giving as uniform a thickness as possible to a class of illusion solids called ambiguous cylinders. Ambiguous cylinders are solids that have two quite different appearances when seen from two specific viewpoints, and thus create the impression of impossible objects. In order to realize them as physical objects, we have to give them thickness. However, it is impossible to give a completely uniform thickness despite this being desirable. Instead we have to content ourselves with second-best methods. For this purpose, this paper proposes three alternative strategies for creating objects as uniform as possible. Each strategy has its own merits and demerits, and hence users can choose their method according to their priorities for the visual effects which they want to emphasize. Constructions in the locus of isogonal conjugates for a quadrilateral. https://www.zbmath.org/1456.51019 2021-04-16T16:22:00+00:00 "Hu, Daniel" https://www.zbmath.org/authors/?q=ai:hu.daniel For any quadrilateral $$ABCD$$ in the Euclidean plane, there is an associated cubic plane curve $$\mathcal{C}$$ which consists of points $$X$$ such that the pairs of lines $$( XA,XC ), \; ( XB,XD )$$ have the same pair of angle bisectors. This curve is called the isogonal cubic of $$ABCD$$ and the author studies it in general case when $$A$$, $$B$$, $$C$$, $$D$$ are distinct points. In the first half of the paper, geometric properties of $$\mathcal{C}$$ are discussed. In the second half of the paper, all isogonal cubics are characterized either by embedding of $$\mathcal{C}$$ in the complex projective plane, or by a specific cubic polynomial defined on the Euclidean plane. Reviewer: Georgi Hristov Georgiev (Shumen)