Geometry

11 pages, 326 KB

Open AccessArticle

The Sum of the Solid Angles of an n-Simplex

by Harold R. Parks and Dean C. Wills

Geometry 2025, 2(3), 15; https://doi.org/10.3390/geometry2030015 - 19 Sep 2025

It was known in antiquity that the sum of the three angles of a triangle equals

π

. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of [...] Read more.

It was known in antiquity that the sum of the three angles of a triangle equals

π

. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid angles in a tetrahedron lies within the interval of

[0, 2 π]

and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that

2 π

was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in

R^{n}

with

n \geq 3

, the solid angles at the vertices add up to a positive number that is less than one-half the

(n - 1)

-dimensional area of the unit sphere in

R^{n}

. We also show that there are examples for which the sum can be made arbitrarily close to the extreme values of 0 and one-half the

(n - 1)

-dimensional area of the unit sphere in

R^{n}

. Full article

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19 pages, 345 KB

Open AccessArticle

On d and M Problems for Newtonian Potentials in Euclidean n Space

by John Lewis

Geometry 2025, 2(3), 14; https://doi.org/10.3390/geometry2030014 - 2 Sep 2025

Abstract

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides these [...] Read more.

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides these potentials into subclasses whose criteria for membership is that a given member has its maximum on the closed unit ball at most M and its minimum at least d. It then lists the extremal potential in each subclass, which is conjectured to solve certain extremal problems. In Theorem 1, we show the existence of these extremal potentials. In Theorem 2, we prove an integral inequality on spheres about the origin, involving so-called extremal potentials, which lends credence to the conjecture. Full article

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

8 pages, 284 KB

Open AccessArticle

Generalization of Napoleon–Barlotti Theorem

by Jiří Blažek and Pavel Pech

Geometry 2025, 2(3), 13; https://doi.org/10.3390/geometry2030013 - 19 Aug 2025

Abstract

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the centers of these regular n-gons form a regular n-gon. [...] Read more.

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the centers of these regular n-gons form a regular n-gon. In the paper we give a generalization of this theorem. Full article

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17 pages, 1159 KB

Open AccessFeature PaperArticle

The Largest Circle Enclosing n Interior Lattice Points

by Jianqiang Zhao

Geometry 2025, 2(3), 12; https://doi.org/10.3390/geometry2030012 - 11 Aug 2025

Abstract

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains [...] Read more.

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the

x y

-plane in its interior. In this paper, we are mainly interested in when the largest n-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all

n < 1100

with the aid of a computer. We find that frequently such a circle does not exist, e.g., when

n = 5, 6

. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when the largest n-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of non-negative integers. Throughout this paper, we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher-dimensional generalizations at the end of the last two sections. Full article

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17 pages, 1278 KB

Open AccessReview

The Multiple Utility of Kelvin’s Inversion

by Eleftherios Protopapas

Geometry 2025, 2(3), 11; https://doi.org/10.3390/geometry2030011 - 9 Jul 2025

Abstract

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

for

n > 2 .

Lord Kelvin overcame this [...] Read more.

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

for

n > 2 .

Lord Kelvin overcame this problem by defining a new (at the time) inversion, the so-called Kelvin’s inversion (or transformation). This inversion has many good properties, making it extremely useful in each case where the geometry of the original problem raises issues. But by using Kelvin’s inversion, these issues are transformed into easier ones, due to a simpler geometry. In this review paper, we study Kelvin’s inversion, deploying its basic properties. Moreover, we present some applications, where its use enables scientists to solve difficult problems in scattering, electrostaticity, thermoelasticity, potential theory and bioengineering. Full article

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

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9 pages, 609 KB

Open AccessFeature PaperArticle

On Yiu’s Equilateral Triangles Associated with a Kiepert Hyperbola

by Cherng-tiao Perng

Geometry 2025, 2(3), 10; https://doi.org/10.3390/geometry2030010 - 1 Jul 2025

Abstract

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are [...] Read more.

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle. Full article

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43 pages, 497 KB

Open AccessArticle

Problems in Invariant Differential Operators on Homogeneous Manifolds

by Jae-Hyun Yang

Geometry 2025, 2(2), 9; https://doi.org/10.3390/geometry2020009 - 9 Jun 2025

Abstract

In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

,

S L (n, R) / S O (n, R)

, [...] Read more.

In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

,

S L (n, R) / S O (n, R)

,

S p (2 n, R) / U (n),

(G L (n, R) ⋉ R^{(m, n)}) / O (n, R)

,

(S L (n, R) ⋉ R^{(m, n)}) / S O (n, R),

(S p (2 n, R) ⋉ H_{R}^{(n, m)}) / (U (n) \times S (m, R)) .

They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory. Full article

10 pages, 304 KB

Open AccessArticle

On the Relation Between a Locus and Poncelet’s Closure Theorem

by Jiří Blažek

Geometry 2025, 2(2), 8; https://doi.org/10.3390/geometry2020008 - 9 Jun 2025

Abstract

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P [...] Read more.

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line

A B

lies on a certain circle (the line being the limiting case of the circle). To prove this proposition, we show how Poncelet’s closure theorem for quadrilaterals can be derived by elementary projective considerations only (without any computations, either in Cartesian or projective coordinates). Finally, the limiting case of the proposition, where the point P lies on the conic, is also mentioned. The problem can then be reduced to Frégier’s theorem and may represent a slightly different perspective on this theorem. Full article

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

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27 pages, 1329 KB

Open AccessFeature PaperArticle

Defining and Visualizing the Geometry of Relativistic Physics

by Yaakov Friedman and Tzvi Scarr

Geometry 2025, 2(2), 7; https://doi.org/10.3390/geometry2020007 - 14 May 2025

Abstract

We continue Riemann’s program of geometrizing physics, extending it to encompass gravitational and electromagnetic fields as well as media, all of which influence the geometry of spacetime. The motion of point-like objects—both massive and massless—follows geodesics in this modified geometry. To describe this [...] Read more.

We continue Riemann’s program of geometrizing physics, extending it to encompass gravitational and electromagnetic fields as well as media, all of which influence the geometry of spacetime. The motion of point-like objects—both massive and massless—follows geodesics in this modified geometry. To describe this geometry, we generalize the notion of a metric to local scaling functions which permit not only quadratic but also linear dependence on temporal and spatial separations. Our local scaling functions are defined on flat spacetime coordinates. We demonstrate how to construct various geometries directly from field sources, using symmetry and superposition, without relying on field equations. For each geometry, two key visualizations elucidate the connection between the geometry and the dynamics as follows: (1) the cross-sections of the ball of admissible velocities, and (2) the cross-sections of the local scaling function. Full article

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

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12 pages, 7101 KB

Open AccessFeature PaperArticle

Hyperbolic Cords and Wheels

by Andrew J. Simoson

Geometry 2025, 2(2), 6; https://doi.org/10.3390/geometry2020006 - 6 May 2025

Abstract

The cycloidal family of curves in

R^{2}

, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods in [...] Read more.

The cycloidal family of curves in

R^{2}

, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods in Euclidean space, outline pertinent hyperbolic geometry tools, using both the Klein and Poincaré models, and show that the two methods give distinct, yet similar, results in hyperbolic space. Full article

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

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15 pages, 802 KB

Open AccessArticle

A Theoretical Framework for Computing Generalized Weighted Voronoi Diagrams Based on Lower Envelopes

by Martin Held and Stefan de Lorenzo

Geometry 2025, 2(2), 5; https://doi.org/10.3390/geometry2020005 - 17 Apr 2025

Abstract

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimensional space. Central to our approach is an algebraic distance function that defines [...] Read more.

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimensional space. Central to our approach is an algebraic distance function that defines the minimum weighted distance from a point to a site. We also introduce a parameterization for the bisectors, ensuring a precise representation of Voronoi edges. The connection to lower envelopes allows us to derive (almost tight) bounds on the combinatorial complexity of a GWVD. We conclude with a short discussion of implementation strategies, ranging from leveraging computational geometry libraries to employing graphics hardware for approximate solutions. Full article

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25 pages, 368 KB

Open AccessArticle

LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

by G. F. Helminck and J. A. Weenink

Geometry 2025, 2(2), 4; https://doi.org/10.3390/geometry2020004 - 15 Apr 2025

Abstract

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of [...] Read more.

Let S be the

N \times N

-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra

k [S]

, together with the evolution equations of the deformations of the powers

{S^{i}, i ⩾ 1}

. They are called the

k [S]

-hierarchy and the strict

k [S]

-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in

GL (B)

can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup

G_{c p t} (B)

of

GL (B)

such that the procedure with LU factorizations works for each

g \in G_{c p t} (B)

. Full article

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

13 pages, 313 KB

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

Abstract

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 KB

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

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 KB

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

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 KB

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

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 KB

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

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 KB

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

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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Graphical abstract

13 pages, 303 KB

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

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

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