Theorem 1 is to prove angle sum conditions for a skew quadrilateral to be planar. Theorem 2 is about the angle sum of a non-planar skew quadrilateral. Theorem 3 proves that a tetrahedron must have a vertex with all three angles acute. A skew quadrilateral with pairwise equal opposite edges is called reversible. A tetrahedron that contains a reversible skew quadrilateral is reversible. An equal-angled skew quadrilateral may not be reversible. However, Theorem 4 states that if a tetrahedron contains an equal-angled skew quadrilateral, then the tetrahedron must be reversible. Our last Theorem 5 is on an angle condition of an isosceles tetrahedron.
When a surface undergoes bending, the length of any curve on it remains unchanged. In the general case, a curve on a surface can be defined by a functional relationship between curvilinear coordinates that is, through an intrinsic equation. The length of such a curve can be computed using the first fundamental form. Since this length remains invariant under bending, the expression of the first fundamental form also remains unchanged. This invariance forms the foundation of the theory of surface bending. Bending of surfaces occurs under certain constraints on their deformation. For ruled surfaces whether developable or non-developable a typical constraint is the preservation of straight-line generatrices. A clear geometric example is the bending of developable surfaces while keeping their generatrices unchanged. In the case of non-developable surfaces, a non-ruled surface can be bent into a ruled one. A classic example is the bending of a surface of revolution, such as the catenoid, into a helical surface, such as the helicoid. The helicoid is a ruled surface; however, when its pitch is gradually decreased during the bending process, the surface becomes non-ruled. By continuously reducing the pitch, one can construct a one-parameter set of intermediate surfaces, making the bending process continuous. When the pitch reaches zero, the helicoid transforms into a catenoid. This example illustrates the bending of helical surfaces into surfaces of revolution. According to Bour's theorem, when a helicoidal surface is bent into a surface of revolution, the helical lines correspond to parallels, and their orthogonal trajectories correspond to meridians. The present work explores the inverse process the bending of a surface of revolution into a helicoidal surface. The surface of revolution is defined via the explicit equation of its meridian. Parametric equations describing the one-parameter set of intermediate surfaces are derived, and several of these surfaces are constructed. The paper also considers the continuous bending of the catenoid into the helicoid.
Visualizing a scene of objects in 4-D space faces several challenges. Mere projections into 3- or less-dimensional spaces usually contain overlapping parts, making them difficult to comprehend or study. Illuminating the scene can enhance intuition about its "dimensionality." Our contribution describes a geometric approach to creating visualizations of 4-D hypersurfaces represented by implicit algebraic equations without their parametrization. By geometric, we mean methods using constructions of geometric objects without their approximation, for example, by polyhedral meshes. Therefore, instead of sets of many points and operating with meshes, we work with implicitly represented hypersurfaces, their projections, contours, intersections, etc. We provide a general algorithm to find shadow boundaries in an arbitrary dimension and apply it in a 4-D space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The results of our algorithm are components of the 3-D model of a scene image represented by polynomial equations and inequalities prepared for plotting by standard computer algebra systems with visualization tools. The method is presented on three 4-D scenes with gradual many properties of the visualized shapes, they are suitable for precise mathematical or scientific visualization. On the other hand, processing higher-degree time.
Geometric optimization has been frequently studied in a recurrent way, where it has been fundamental to developing more complete algorithms. Regions of interest can be obtained as user-defined polygons as a first step toward many practical applications. This article focuses on lattice polygons defined on a regular partition and presents an efficient method for computing all possible polygons contained within regions of interest bounded by arbitrary obstacles such as points, segments, and holes. The developed algorithm calculates all the simple polygons with the maximum area or perimeter contained within the region of interest with O(n5k) computational time. The user can define the polygon to be calculated (triangle, quadrilateral, pentagon, hexagon, etc.) as well as the desired solution: maximum area or maximum perimeter. The paper presents several practical applications that demonstrate the efficiency and versatility of the algorithm. The pseudocode for the algorithm is presented, as well as the source code (Java and Python) in a GitHub repository for research purposes.
We extend distance-power identities for regular figures by deriving formulas for the sum of fourth powers of distances from an arbitrary point to the vertices of regular polygons and polyhedra. The method employs embeddings of these configurations into higher-dimensional Euclidean spaces and systematic use of the Pythagorean theorem. Our results unify and generalize known formulas in lower dimensions, providing a broader framework for distance relations in regular polytopes.
The object of the present paper is to construct a characterization of a three-dimensional Walker-Poisson manifolds with Walker metric; in other words, study the compatibility between Walker and Poisson structures. Some examples are given.
Each regular polygon P is clearly tangential, has commensurable sides and every choice of three consecutive vertices among those of P determines a triangle whose interior angles are pairwise commensurable. In this article we prove that these three conditions are also sufficient for a convex polygon to be equilateral. It turns out a new characterization of convex regular polygons with an odd number of sides.
We consider the following configuration. Let ABCD be a cyclic quadrilateral with circumcenter O, and for each vertex X, let H-X be the orthocenter of the triangle formed by the other three. Then A, B, C, D, H-A, H-B, H-C, H-D all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point P-D on the Euler line of DABC, we define corresponding points P-A, P-B, P-C on the respective Euler lines such that the ratio PXHX : PXO is constant for all X. We show that the four vertices A, B, C, D and the four isogonal conjugates Q(A), Q(B), Q(C), Q(D) of the points P-X all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points PX within the list of triangle centers.
The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal, then the corresponding sides have equal lengths. In this paper, we consider the triangle ABC whose all angles are less than 120 degrees, F is its Fermat point, and per(ABC), [ABC] stand for its perimeter and area, respectively. In Theorem 1, we prove the Fermat analogue of Steiner-Lehmus Theorem that states that if the cevians from B and C through the Fermat point F meet AC and AB at B ' and C ' respectively, then BB ' = CC ' is equivalent to AB = AC. More stronger forms are also proved such as AB > AC is equivalent to each of BB ' > CC ' and per(C ' BC > per(B ' CB). More variations on Fermat analogue of Steiner-Lehmus Theorem are proved in Theorems 3 and 4. In Theorem 3, the cevians through F from B and C meet the external angle bisectors of C and B at D and E respectively, and it is proved that, for example, AB = AC is equivalent to each of CE = BD, per(EC ' B) = per(DB ' C), and [EC ' B] = [DB ' C] and more stronger forms are also proved such as AB > AC is equivalent to each of CE > BD, per(EC ' B) > per(DB ' C), and [EC ' B] > [DB ' C]. In Theorem 4, we prove that if the angle A of the triangle ABC is not equal to 60 degrees and the circumcevians BK and CL of the Fermat point F, that meet the circumcircle of triangle ABC at K and L, are equal, then the triangle triangle ABC is isosceles with AB = AC.
Generalisations of mathematical problems often require innovative approaches and open up promising avenues for further research. In this article, we propose a generalisation of a geometry problem originally featured in the 1995 International Mathematical Olympiad contest. We employ projective geometric techniques to rigorously prove our generalisation, demonstrating the value of applying advanced mathematical tools to extend the boundaries of traditional problems. Even though this is a geometry problem originally intended for students, it opens up many interesting ideas for generalisation and the inclusion of more advanced tools to prove these generalisations. We define a special transformation with respect to two conic sections and a line intersecting the conics, and we prove several properties of the transformation that provide a solution to our generalised problem. Our main aim is to determine the invariant lines with respect to the transformation as a generalisation of the original IMO problem.
In this paper, the locus of the center of the top surface in an inverted cone was considered geometrically when the cone was tilted. When the area of the ellipse on the top surface of the volume is constant, the locus of the center of the top surface is found to be part of an ellipsoid. The locus of the center of the ellipse is also shown for the case where the volume between the top surface and the cone is held constant and the length of the major axis of the top surface of the volume is constant. Here, the behavior of the center of the top surface of a right circular cone cut in a plane is applicable to describe the motion of conduction electrons in single-layer graphene, which is being tremendously studied recently in the field of physics, since the behavior of conduction electrons in single-layer graphene is described by a right circular cone (Dirac cone).
The climate of Ukraine, which is located in several climatic zones, provides hot periods in summer and autumn-spring times and periods of heating houses in winter and off-season. An important factor influencing the energy saving of buildings is solar energy, from which it is necessary to protect against overheating during periods of overheating and use its heat during the operation of the heating system. Green spaces are actively used in modern cities to form the micro climate of streets, protect facades from overheating, create shading for windows, terraces and balconies. This is aesthetic and ecological. In winter, plants that shed their leaves open access for solar radiation to enter the premises, which contributes to energy saving. Modern architecture of cities and individual buildings provides for a variety of uses of green spaces: individual trees, rows of trees along facades, plants on facades as overhangs, side edges or green curtains. This publication analyzes shading from edges, overhangs and curtains.
In this paper, we investigate the developability conditions of ruled surfaces in Euclidean 3-space according to the Darboux frame of an arbitrary regular surface along their common base curve. Moreover, the position vectors such surfaces are found.
This study focuses on the reconstruction and fitting of curves to 2D and 3D scanned point clouds in the context of HBIM applications. The paper begins by analyzing the distinctive characteristics of the input data, such as inherent noise, outliers, non-uniform sampling, and variable point density. Recognizing the critical impact of these factors on the choice of approximation methods, a classification system is proposed to evaluate both input data attributes and modeling objectives. The key aspect of selecting a method is defined by the modeling objectives, namely the required accuracy and level of detail of the reconstructed curves. The author emphasizes that aligning method selection with this classification enables optimization of computational resources and processing time. The methods proposed in this study varies depending on the specified criteria and includes the Normal Vector method, the Crawling method, the Trend method, and the Gravity method. Each technique is briefly described in terms of its core principles and applicability based on the proposed classification. This task-oriented approach aims to achieve a balance between reconstruction quality and computational efficiency. To validate the proposed methods, a comparative evaluation is conducted. The results demonstrate satisfactory accuracy in curve reconstruction combined with optimal processing time. Overall, the proposed approximation methods significantly enhance the processing and analysis of 2D and 3D scanned point clouds. Their ability to handle noise, outliers, and variable point densities along with their computational efficiency makes them valuable tools for applications in architecture and industrial design.
The tetrahedron having ex-centers of a tetrahedron T as vertices is said to be the ex-center-tetrahedron of T, and let us denote it by T*. Theorem 1 shows that the ex-center-tetrahedron of a tetrahedron T and the tetrahedron that tangles T with respect to the in-center of T are the same. So T* is also used to denote the tetrahedron that tangles T with respect to the in-center of T. We define that a tetrahedron is weakly reversible if the sum of a pair of two face areas is equal to the sum of the other pair of two face areas. Theorem 2 shows that T is weakly reversible if and only if T and T* have the same volume. While T* may not be weakly reversible when T is weakly reversible, Theorem 3 shows that T* is reversible when T is reversible.
This paper provides a derivation of A. Du r's necessary and sufficient condition for an axonometric reference system to be, up to a uniform scale factor, the parallel image of an orthonormal reference system. After that, simple formulas relating the uniform scale factor and the projection direction are given. Finally, A. Du r's condition is extended to the case of the Pohlke-Schwarz theorem.
This paper establishes hyperbolic analogues of ten inequalities for triangles in spherical geometry, originally presented in Mitrinovic, Pecaric, and Volenec (1989). For comparison, the corresponding Euclidean versions are also included. Unified formulations of these inequalities across spherical, Euclidean, and hyperbolic geometries are provided. This work contributes to the broader effort of translating and unifying geometric results across the three classical geometries.
Theorems of Viviani and Walser, Vargyas on sums of distances from a point to the sides of polygons are generalized for polyhedra in R-n(n >= 3).
We investigate a hierachical relation between minimum and maximum angle conditions widely used in the interpolation theory and finite element analysis over simplicial partitions. We prove that the minimum angle condition implies the maximum angle condition in arbitrary space dimension.
A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of each face. In the late 1970's Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersky, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.
