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.
