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In the present paper geometric locus of points (GLP) equidistant to a sphere and a plane is considered; the properties of the acquired surfaces are studied. Four possible cases of mutual location of a sphere and a plane are considered: the plane passing through the center of the sphere, the plane intersecting the sphere, the plane tangent to the sphere and the plane passing outside the sphere. GLP equidistant to a sphere and a plane constitutes two co-axial co-focused paraboloids of revolution. General properties of the acquired paraboloids were studied: the location of foci, vertices, axis and directing planes, distance between the sphere center and the vertices, the distance between the vertices. GLP for each case of mutual location of a plane and a sphere constitutes: in case one passes through the center of other, two co-axial co-focused oppositely directed paraboloids of revolution symmetrical with respect to the given plane; in case they intersect each other, two co-axial co-focused oppositely directed non-symmetrical paraboloids; in case they are tangent to each other, a paraboloid and a straight line passing through the tangency point; in case they have no common points, a pair of co-axial co-focused mutually directed paraboloids of revolution.

Origami offers an avenue to program three-dimensional shapes via scale-independent and non-destructive fabrication. While such programming has focused on the geometry of a tessellation in a single transient state, here we provide a complete description of folding smooth saddle shapes from concentrically pleated squares. When the offset between square creases of the pattern is uniform, it is known as the pleated hyperbolic paraboloid (hypar) origami. Despite its popularity, much remains unknown about the mechanism that produces such aesthetic shapes. We show that the mathematical limit of the elegant shape folded from concentrically pleated squares, with either uniform or non-uniform (e.g. functionally graded, random) offsets, is invariantly a hyperbolic paraboloid. Using our theoretical model, which connects geometry to mechanics, we prove that a folded hypar origami exhibits bistability between two symmetric configurations. Further, we tessellate the hypar origami and harness its bistability to encode multi-stable metasurfaces with programmable non-Euclidean geometries. The mechanisms behind origami having saddle shapes made from concentrically pleated squares remain elusive. Here, the authors connect geometry and mechanics to show that this type of origami is invariantly a hyperbolic paraboloid that exhibits bistability between two symmetric configurations.

Resolvent analysis of the linearized Navier–Stokes equations provides useful insight into the dynamics of transitional and turbulent flows and can provide a model for the dominant coherent structures within the flow, particularly for flows where the linear operator selectively amplifies one particular force component, known as the optimal force mode. Force and response modes are typically obtained from a singular-value decomposition of the resolvent operator. Despite recent progress, the cost of resolvent analysis for complex flows remains considerable, and explicit construction of the resolvent operator is feasible only for simplified problems with a small number of degrees of freedom. In this paper we propose two new matrix-free methods for computing resolvent modes based on the integration of the linearized equations and the corresponding adjoint system in the time domain. Our approach achieves an order of magnitude speedup when compared with previous matrix-free time-stepping methods by enabling all frequencies of interest to be computed simultaneously. Two different methods are presented: one based on analysis of the transient response, providing leading modes with fine frequency discretization; and another based on the steady-state response to periodic forcing, providing optimal and suboptimal modes for a discrete set of frequencies. The methods are validated using a linearized Ginzburg–Landau equation and applied to the three-dimensional flow around a parabolic body.

In response to the emerging demands of next-generation space communications and deep space exploration for high-precision, high-gain, and wide-coverage antenna systems, the development of large deployable mesh reflectors has become a key technological direction. For large-aperture offset antennas, surface accuracy is typically evaluated via a best-fit paraboloid. However, limited structural stiffness and harsh thermal conditions in orbit often cause significant rigid-body displacements, which traditional best-fit methods—based on small-deformation assumptions and fixed reference frames—fail to account for, leading to notable surface error inaccuracies. To address this issue, a novel best-fit paraboloid calculation method based on a floating coordinate system is proposed. The method does not require constructing a best-fit paraboloid equation; instead, it introduces a floating coordinate system attached to the deformed reflector. Within this coordinate system, an objective function is defined to quantify the root-mean-square deviation between the reflector nodes and the ideal paraboloid. A differential evolution algorithm is employed to globally search for the optimal spatial configuration of the floating coordinate system, thereby determining the best-fit paraboloid position in space. Case studies on 2-meter and 100-meter AstroMesh-type deployable mesh reflectors show that the proposed method provides significantly more accurate surface evaluations compared to conventional methods, confirming its effectiveness and applicability under large-deformation conditions.

A team of civil and environmental engineers at Princeton University is investigating the possibility of using what they call concrete umbrellas to hold back the sea--an attractive addition that would provide shade for pedestrians most days, but with a roof that can swivel toward the ocean, as a barrier against an incoming storm. The design uses a hyperbolic paraboloid shape that looks like a saddle, curving inward along one axis and outward along the other.

We obtain a sharp bilinear restriction estimate for the paraboloid in R 3 for q > 13 / 4 .

We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

Hyperbolic paraboloid molecules, as systems with remarkable negative Gaussian curvature, have garnered considerable attention in chemistry and materials science due to their unique physicochemical properties. However, controlled organization of these curvature building blocks into well-defined superstructures remains challenging, primarily attributed to the constraints of their inherently contorted structures. Herein, we report the successful formation of hyperbolic-paraboloid-based crystalline two-dimensional (2D) superstructures in a controlled hierarchical manner. Employing a rigid yet flexible molecular design strategy, we guide the hyperbolic-paraboloid molecules through hierarchical self-assembly method from dimers to one-dimensional (1D) supramolecular columns and ultimately to 2D superstructures. Remarkably, these 2D superstructures exhibit strong second harmonic generation (SHG) with excellent polarization dependence. The strongest SHG signals were recorded at 1.5 × 10 5 for 2D HPS-MT and 1.1 × 10 5 for 2D HPS-Z, with polarization dependence of the SHG responses reaching 87.5% for 2D HPS-MT and 81% for 2D HPS-Z. Hyperbolic paraboloid molecules possess unique physicochemical properties. However, their controlled organization into well-defined superstructures remains challenging. Here, the authors report a macrocycle with hyperbolic paraboloid topology as building blocks to self-assemble a series of two-dimensional superstructures with four-level hierarchy in a controlled hierarchical manner.

We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.

Spacetime wormholes in isotropic spacetimes are represented traditionally by embedding diagrams which were symmetric paraboloids. This mirror symmetry, however, can be broken by considering different sources on different sides of the throat. This gives rise to an asymmetric thin-shell wormhole, whose stability is studied here in the framework of the linear stability analysis. Having constructed a general formulation, using a variable equation of state and related junction conditions, the results are tested for some examples of diverse geometries such as the cosmic string, Schwarzschild, Reissner–Nordström and Minkowski spacetimes. Based on our chosen spacetimes as examples, our finding suggests that symmetry is an important factor to make a wormhole more stable. Furthermore, the parameter γ, which corresponds to the radius dependency of the pressure on the wormholes’s throat, can affect the stability in a great extent.