Geometry

12 pages, 281 KB

Open AccessFeature PaperArticle

Geometric Reinterpretation of Partial Differential Equations and Applications

by Satyanad Kichenassamy

Geometry 2026, 3(1), 6; https://doi.org/10.3390/geometry3010006 - 13 Mar 2026

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We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations. Full article

10 pages, 299 KB

Open AccessArticle

Focal Circular Cubic and Complex Numbers

by Jiří Blažek

Geometry 2026, 3(1), 5; https://doi.org/10.3390/geometry3010005 - 10 Mar 2026

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Abstract

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of pairs of foci satisfy a quadratic equation. This equation can be expressed as a linear combination of two basic quadratic equations, which form a basis of a vector space. Furthermore, we give a nonstandard analytical condition, expressed in complex numbers, under which a circle can be inscribed in a quadrilateral. Finally, we leave the complex plane and show the construction of an arbitrary pair of foci of the curve by Euclidean means. Full article

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14 pages, 438 KB

Open AccessArticle

Properties of Elliptic Cycloids

by Matthew A. Pons and Nicholas D. White

Geometry 2026, 3(1), 4; https://doi.org/10.3390/geometry3010004 - 9 Feb 2026

Viewed by 556

Abstract

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic [...] Read more.

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

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13 pages, 577 KB

Open AccessFeature PaperArticle

Family of Non-Minkowski Measurable Fractals in

R^{2}

by Uta Freiberg and Jonas Lippold

Geometry 2026, 3(1), 3; https://doi.org/10.3390/geometry3010003 - 2 Feb 2026

Viewed by 411

Abstract

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it [...] Read more.

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in

R^{2}

that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results. Full article

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

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15 pages, 302 KB

Open AccessArticle

Witten Deformation and Divergence-Free Symmetric Killing 2-Tensors

by Kwangho Choi and Junho Lee

Geometry 2026, 3(1), 2; https://doi.org/10.3390/geometry3010002 - 13 Jan 2026

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Abstract

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for

p = 2

. Full article

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

13 pages, 1033 KB

Open AccessArticle

A New Characterization of the Symmedian Point

by Yu Chen and Robert J. Fisher

Geometry 2026, 3(1), 1; https://doi.org/10.3390/geometry3010001 - 8 Jan 2026

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Abstract

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, [...] Read more.

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, it does not lie on the medians of

▵ A B C

, and the lines

ℓ_{A P}

ℓ_{B P}

, and

ℓ_{C P}

intersect

ℓ_{B C}

ℓ_{C A}

, and

ℓ_{A B}

at points

P_{a}

P_{b}

, and

P_{c}

, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates

Q_{a}

Q_{b}

, and

Q_{c}

P_{a}

P_{b}

, and

P_{c}

\bar{B C}

\bar{C A}

, and

\bar{A B}

, respectively, are collinear. We prove that K lies on the line through

Q_{a}

Q_{b}

, and

Q_{c}

, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle

C

onto the set

L_{K}

of all the lines in

E

that pass through K. Full article

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Geometry, Volume 3, Issue 1 (March 2026) – 6 articles

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