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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 297

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.

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 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 184

Abstract

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 [...] Read more.

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 502

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 [...] Read more.

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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16 pages, 304 KiB

Open AccessArticle

Trigonometric Polynomial Points in the Plane of a Triangle

by Clark Kimberling and Peter J. C. Moses

Geometry 2024, 1(1), 27-42; https://doi.org/10.3390/geometry1010005 - 23 Dec 2024

Viewed by 626

Abstract

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths

a, b,

and c of a triangle

A B C

. For example, the circumcenter is represented by [...] Read more.

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths

a, b,

and c of a triangle

A B C

. For example, the circumcenter is represented by the polynomial

a (b^{2} + c^{2} - a^{2})

. It is not so well known that triangle centers have barycentric coordinates, such as

\tan A : \tan B : \tan C

, that are also representable by polynomials, in this case, by

p (a, b, c) : p (b, c, a) : p (c, a, b)

, where

p (a, b, c) = a (a^{2} + b^{2} - c^{2}) (a^{2} + c^{2} - b^{2})

. This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form

f (a, b, c) : f (b, c, a) : f (c, a, b)

, where f depends on one or more of the functions in the set

{\cos, \sin, \tan, \sec, \csc, \cot}

. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution. Full article

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

4 pages, 404 KiB

Open AccessArticle

Hagge Configurations and a Projective Generalization of Inversion

by Zoltán Szilasi

Geometry 2024, 1(1), 23-26; https://doi.org/10.3390/geometry1010004 - 12 Nov 2024

Viewed by 802

Abstract

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion. Full article

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7 pages, 3051 KiB

Open AccessArticle

Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc

by Andrej Hasilik

Geometry 2024, 1(1), 16-22; https://doi.org/10.3390/geometry1010003 - 5 Aug 2024

Viewed by 1437

Abstract

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to [...] Read more.

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the former. These diagonals and the central angle of the pair of peripherals, the shortest diagonals, are used to construct n rhombi. The rhombi define the shape of lenses tangential to them. To construct the arcs of the lenses, beams perpendicular to the sides of each rhombus are drawn. Four beams radiating from the top and bottom vertices of each rhombus intersect in the centers of a pair of coaxal circles. Thus, the vertical axis of each rhombus coincides with the radical axis of the pair. The intersection of the pair represents the corresponding lens. All n lenses form a tangential sequence along the central diagonal. Their cusps circumscribe the polygon and the lenses themselves. The area covered by the lenses converges to (2/3) πr². Full article

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13 pages, 303 KiB

Open AccessArticle

Unary Operations on Homogeneous Coordinates in the Plane of a Triangle

by Peter J. C. Moses and Clark Kimberling

Geometry 2024, 1(1), 3-15; https://doi.org/10.3390/geometry1010002 - 8 Jul 2024

Cited by 1 | Viewed by 1306

Abstract

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear)

x : y : z

. Eight unary operations discussed in this paper include [...] Read more.

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear)

x : y : z

. Eight unary operations discussed in this paper include

u_{1} (X) = (y - z) / x : (z - x) / y : (x - y) / z

. For each

u_{i}

, there exist, formally, two points, P and U, such that

u_{i} (P) = u_{i} (U) = X

. To such pairs of inverses are applied nine binary operations, each resulting in a triangle center. If L is a line, then formally,

u_{i} (L)

is a cubic curve that passes through the vertices

A, B, C

. If L passes through the point

1 : 1 : 1

(the centroid or incenter, assuming that the coordinates are barycentric or trilinear), then the cubic is degenerate as the union of a parabola and the line at infinity. The methods in this work are largely algebraic and computer-dependent. Full article

2 pages, 297 KiB

Open AccessEditorial

Geometry: A Bridge Connecting All Things

by Yang-Hui He

Geometry 2024, 1(1), 1-2; https://doi.org/10.3390/geometry1010001 - 29 May 2024

Viewed by 1717

Abstract

In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...] Full article

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