Geometry

13 pages, 313 KiB

Open AccessFeature PaperArticle

Rigidity of Holomorphically Projective Mappings of Kähler and Hyperbolic Kähler Spaces with Finite Complete Geodesics

by Josef Mikeš, Irena Hinterleitner, Patrik Peška and Lenka Vítková

Geometry 2025, 2(1), 3; https://doi.org/10.3390/geometry2010003 - 10 Mar 2025

Viewed by 335

In the paper, we consider holomorphically projective mappings of n-dimensional pseudo-Riemannian Kähler and hyperbolic Kähler spaces. We refined the fundamental linear equations of the above problems for metrics of differentiability class

C^{2}

. We have found the conditions for n complete [...] Read more.

C^{2}

. We have found the conditions for n complete geodesics and their image that must be satisfied for the holomorphically projective mappings to be trivial, i.e., these spaces are rigid with precision to affine mappings. Full article

17 pages, 7219 KiB

Open AccessArticle

A Laguerre-Type Action for the Solution of Geometric Constraint Problems

by Nefton Pali

Geometry 2025, 2(1), 2; https://doi.org/10.3390/geometry2010002 - 18 Feb 2025

Viewed by 215

Abstract

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space

R^{n}

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space

R^{n}

with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature

(n + 1, 2)

. In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in

R^{n}

are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

35 pages, 2057 KiB

Open AccessArticle

How Null Vector Performs in a Rational Bézier Curve with Mass Points

by Lionel Garnier, Jean-Paul Bécar and Laurent Fuchs

Geometry 2025, 2(1), 1; https://doi.org/10.3390/geometry2010001 - 20 Jan 2025

Cited by 1 | Viewed by 548

Abstract

This article points out the kinematics in tracing a Bézier curve defined by control mass points. A mass point is a point with a non-positive weight, a non-negative weight or a vector with a null weight. For any Bézier curve, the speeds at endpoints can be modified at the same time for both endpoints. The use of a homographic parameter change allows us to choose any arc of the curve without changing the degree but not offer to change the speeds at both endpoints independently. The homographic parameter change performs weighted points with any non-null real number as weight and also vectors. The curve is thus called a rational Bézier curve with control mass points. In order to build independent stationary points at endpoints, a quadratic parameter change is required. Adding null vectors in the Bézier representation is also an answer. Null vectors are obtained when converting any power function in a rational Bézier curve and their inverse. The authors propose a new approach on placing null vectors in the representation of the rational Bézier curve. It allows us to break free from projective geometry where there is no null vector. The paper ends with some examples of known curves and some perspectives. Full article

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Geometry, Volume 2, Issue 1 (March 2025) – 3 articles

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