Spherical Curves Research Articles

The limestone spheroids of 'Ubeidiya: intentional imposition of symmetric geometry by early hominins?

Spheroids are one of the least understood lithic items yet are one of the most enduring, spanning from the Oldowan to the Middle Palaeolithic. Why and how they were made remains highly debated. We seek to address whether spheroids represent unintentional by-products of percussive tasks or if they were intentionally knapped tools with specific manufacturing goals. We apply novel three-dimensional analysis methods, including spherical harmonics and surface curvature, to 150 limestone spheroids from 'Ubeidiya (ca 1.4 Ma), presently the earliest Acheulean occurrence outside of Africa, to bring a new perspective to these enigmatic artefacts. We reconstruct the spheroid reduction sequence based on trends in their scar facets and geometry, finding that the spheroid makers at 'Ubeidiya followed a premeditated reduction strategy. During their manufacture, the spheroids do not become smoother, but they become markedly more spherical. They approach an ideal sphere, a feat that likely required skilful knapping and a preconceived goal. Acheulean bifaces are currently thought to represent the earliest evidence of hominins imposing a premeditated, symmetrical shape on stone. The intentional production of sphere-like objects at 'Ubeidiya similarly shows evidence of Acheulean hominins desiring and achieving intentional geometry and symmetry in stone.

Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid

Total torsion of three-dimensional lines of curvature

A curve gamma in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when gamma lies on an oriented hypersurface S of M, we say that gamma is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that gamma is three-dimensional and closed. We show that if gamma is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2pi ; and that, conversely, if the total torsion of gamma is an integer multiple of 2pi , then there exists an oriented hypersurface of M in which gamma is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of gamma vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

Framed Natural Mates of Framed Curves in Euclidean 3-Space

In this study, we consider framed curves as regular or singular space curves with an adapted frame in Euclidean 3-space. We define framed natural mates of a framed curve that are tangent to the generalized principal normal of the framed curve. Subsequently, we present the relationships between a framed curve and its framed natural mates. In particular, we establish some necessary and sufficient conditions for the framed natural mates of specific framed curves, such as framed spherical curves, framed helices, framed slant helices, and framed rectifying curves. Finally, we support the concept with some examples.

One-year results for myopia control with aspheric base curve orthokeratology lenses: A prospective randomised clinical trial.

To compare the effect of orthokeratology (ortho-k) using aspheric or spherical base curve (BCA vs. BCS) contact lenses on axial elongation and the relative peripheral refraction change (RPRC) in Chinese children. Children aged 8-12 years with myopia between -0.75 and -4.00 D and astigmatism ≤1.00 D were randomly assigned to the BCA or BCS group. Peripheral refraction was assessed at 10°, 20° and 30° along the temporal and nasal retina at baseline and at the 12-month visit. Axial length (AL) was measured under cycloplegia at baseline and at the 6- and 12-month visits. Only right eye data were analysed. Repeated-measures analysis of covariance was performed to examine the differences in axial elongation and the RPRC between the BCA and BCS groups. The 1-year results from 31 BCA and 32 BCS subjects were analysed. No significant between-group differences were found at baseline (p ≥ 0.28). At the 12-month visit, the BCA lens produced a greater absolute RPRC along the horizontal meridian than the BCS lens (p < 0.001). Axial elongation was slower in the BCA group (0.19 ± 0.20 mm) than in the BCS group (0.29 ± 0.14 mm; p = 0.03). Axial elongation was correlated with the RPRC at 10° (r = 0.43, p = 0.02) and 20° (r = 0.39, p = 0.03) along the temporal retina in the BCA group; however, these correlations were not observed in the BCS group. The BCA ortho-k lens could improve the efficacy of slowing axial elongation in children. The improved myopia control observed in the BCA group may be the result of a larger myopic shift in relative peripheral refraction within 20° along the temporal retina.

The critical curvature degree of an algebraic variety

In this article we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing them.

Concircular helices and concircular surfaces in Euclidean 3-space $\mathbb{R}^{3}$

In this paper we characterize concircular helices in $\mathbb{R}^{3}$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $\mathbb{R}^{3}$ as a special family of ruled surfaces, and we show that $M\subset\mathbb{R}^{3}$ is a proper concircular surface if and only if either $M$ is parallel to a conical surface or $M$ is the normal surface to a spherical curve. Finally, we characterize the concircular helices as geodesics of concircular surfaces.

Geometric Properties of Planar and Spherical Interception Curves

Geometric Properties of Planar and Spherical Interception Curves

The Intersection of Two Ruled Surfaces Corresponding to Spherical Indicatrix Curves on the Unit Dual Sphere

In this study, we first investigate the intersection of two different ruled surfaces in R^3 for two different tangential spherical indicatrix curves on DS^2 using the E. Study mapping. The conditions for the intersection of these ruled surfaces in R^3 are expressed by theorems with bivariate functions. Secondly, considering two different principal normal spherical indicatrix curves on DS^2, we examine the intersection of two different ruled surfaces in R^3 by using E. Study mapping. Similarly, the conditions for the intersection of these ruled surfaces in R^3 are indicated by theorems with bivariate functions. Thirdly, using E. Study mapping, we explore the intersection of two different ruled surfaces in R^3 by considering two different binormal spherical indicatrix curves on DS^2. Likewise, the conditions for the intersection of these ruled surfaces in R^3 are denoted by theorems with bivariate functions. Fourthly, considering two different pole spherical indicatrix curves on DS^2, we study the intersection of two different ruled surfaces in R^3 by using E. Study mapping. In the same way, the conditions for the intersection of these ruled surfaces in R^3 are specified by theorems with bivariate functions. Finally, we provide some examples that support the main results.

Some Characteristic Properties of Non-Null Curves in Minkowski 3-Space 𝔼13

This paper gives new characteristic properties of non-null spherical and rectifying curves in Minkowski 3-space E13. In the light of the causal characteristics, we give some representations of rectifying non-null curves. Additionally, we proved that the tangential function of every non-null curve fulfills a third-order differential equation. Then, a number of well-known characteristic properties of rectifying, Lorentzian, and hyperbolic spherical curves are consequences of this differential equation.

Research on Determining Elastic-Plastic Constitutive Parameters of Materials from Load Depth Curves Based on Nanoindentation Technology.

It is of great significance for structural design and engineering evaluation to obtain the elastic-plastic parameters of materials. The inverse estimation of elastic-plastic parameters of materials based on nanoindentation technology has been applied in many pieces of research, but it has proved to be difficult to determine the elastic-plastic properties of materials by only using a single indentation curve. A new optimal inversion strategy based on a spherical indentation curve was proposed to obtain the elastoplastic parameters (the Young's modulus E, yield strength σy, and hardening exponent n) of materials in this study. A high-precision finite element model of indentation with a spherical indenter (radius R = 20 µm) was established, and the relationship between the three parameters and indentation response was analyzed using the design of experiment (DOE) method. The well-posed problem of inverse estimation under different maximum indentation depths (hmax1 = 0.06 R, hmax2 = 0.1 R, hmax3 = 0.2 R, hmax4 = 0.3 R) was explored based on numerical simulations. The results show that the unique solution with high accuracy can be obtained under different maximum press-in depths (the minimum error was within 0.2% and the maximum error was up to 1.5%). Next, the load-depth curves of Q355 were obtained by a cyclic loading nanoindentation experiment, and the elastic-plastic parameters of Q355 were determined by the proposed inverse-estimation strategy based on the average indentation load-depth curve. The results showed that the optimized load-depth curve was in good agreement with the experimental curve, and the optimized stress-strain curve was slightly different from the tensile test, and the obtained parameters were basically consistent with the existing research.

Measurement of gain-saturation properties of a Q-switched Nd:YAG laser oscillator with a three-pass amplifier and corresponding beam diagnosis

Here, a homemade Q-switched Nd: YAG laser oscillator alongside a three-pass amplifier has been designed and fabricated. The oscillator operates nearly in the fundamental mode and the spot size/phase front curvature is determined by making use of ABCD law regarding the thermal lensing effect. To scale-up the pulse energy in a compact volume, a three-pass amplifier arrangement is coupled to the oscillator. The systematical measurements to obtain the small-signal gain and saturation properties are carried out in terms of various pumping energies. Simultaneously, the output beam from the oscillator and triple-pass amplifier are well diagnosed to determine output energy, pulse duration, spectral width, M2 factor and the radius of curvature. Despite the oscillator pulse giving out 60 mJ, 13 ns however, the output pulse of 800 mJ, 1 Hz, 9 ns at 1064 nm with FWHM 0.024 nm and beam quality factor M2= 2.21 are determined in favor of multi-pass amplifier. Scaling up the pulse energy up to 800 mJ it is shown that the output beam alters its phase front from the initial planar wave to the spherical curvature during the propagation through a multi-pass amplifier due to the elevation of a radial temperature gradient. Eventually, the novelty of this manuscript arises from the effect of energy scale-up on the gain and saturation in laser amplifier and their correlation with the output beam properties.

ROTATION MINIMIZING SPHERICAL MOTIONS AND HELICES

In this study we give the structure of the motion RMM with the spherical curve orbit by using rotation minimizing frames (RMF). Further, quaternionic helices and their characterizations of being CCR curve are given with the help of unit quaternions which are related spherical frame motion.

Mechanical properties of innovative prefabricated friction pendulum bearing with concrete-filled steel tube system

An innovative prefabricated seismic isolation system known as a friction pendulum bearing with a concrete-filled steel tube (FPB-CFST) was proposed to improve the mechanical performance and accelerate the construction process of the isolation layer. Unlike the traditional friction pendulum bearing-pier system, which consists of a cast FPB isolator and two cast-in-place reinforced concrete buttresses, the CFST was selected as the main supporting structure, and a spherical curvature surface (SCS) manufactured by a pressing process was attached to the CFST to slide the superstructure in the proposed FPB-CFST system. A construction procedure for the FPB-CFST was proposed, and a series of FPB-CFST specimens with various SCS radii were fabricated. The axial local compressive and compressive shear behaviors of the FPB-CFST were experimentally investigated. Friction coefficient tests were conducted to gain insight into the friction of the FPB-CFST. The mechanism of the proposed FPB-CFST subjected to axial local compression and compressive shear was discussed. The effects of sliding velocity and equivalent pressure of the bearing material on the friction coefficient were evaluated. The influences of the radius of the SCS and local compression ratio on the behavior of the FPB-CFST were addressed. The force model and design method of an FPB-CFST subjected to axial local compression and compressive shear are presented.

Measuring compound eye optics with microscope and microCT images

With a great variety of shapes and sizes, compound eye morphologies give insight into visual ecology, development, and evolution, and inspire novel engineering. In contrast to our own camera-type eyes, compound eyes reveal their resolution, sensitivity, and field of view externally, provided they have spherical curvature and orthogonal ommatidia. Non-spherical compound eyes with skewed ommatidia require measuring internal structures, such as with MicroCT (µCT). Thus far, there is no efficient tool to characterize compound eye optics, from either 2D or 3D data, automatically. Here we present two open-source programs: (1) the ommatidia detecting algorithm (ODA), which measures ommatidia count and diameter in 2D images, and (2) a µCT pipeline (ODA-3D), which calculates anatomical acuity, sensitivity, and field of view across the eye by applying the ODA to 3D data. We validate these algorithms on images, images of replicas, and µCT eye scans from ants, fruit flies, moths, and a bee.

Experimental study on positive corona discharge characteristics in plane-single/double earthed rods air gaps under negative DC voltage

In order to analyze the corona discharge characteristics at the grounding object tip under the thundercloud electric field environment, scaled simulation test has been conducted on corona discharge in the plane-single/double earthed rods air gap. Five gap configurations with three different grounding rods were built during experiments. Factors as rod tip curvature, voltage and gap configuration have been considered in detail. Corona voltage, corona current and the number of ultraviolet photons were all recorded. The grounding rod tip curvature radius can affect corona discharge processes at grounding rod tip. With increasing applied negative DC voltage, five typical corona processes of dark discharge, burst streamer corona, periodic streamer corona, glow corona and brush discharge occurred at conical and spherical grounding rod tip (r=0.75cm), but under d=5cm and d=10cm no glow corona and brush corona appeared at spherical grounding rod tip (r=1.25cm). It is much easier for grounding rod tip with smaller curvature to form glow corona phenomenon. The mean value Iav of periodic streamer corona current pulse amplitude decreases with grounding rod tip curvature radius, while the mean corona current pulse width and pulse interval increases. For grounding rod with conical tip, there is little difference of average current pulse amplitude between periodic streamer corona and brush corona; for grounding rod with spherical tip (r=0.75cm), the mean value Iav of brush corona pulse amplitude increases to 3595.8 μA which has increased by a factor of about 36. The photon number at conical grounding rod tip is larger than that at spherical grounding rod tip at low voltage but the relation reverses as the applied voltage exceeds a critical value. The shielding effect of two grounding rods in plane-double rods air gaps increased with spherical grounding rod curvature radius and decreases with the lateral gap distance between two grounding rods.

Construction of spherical non-circular wheels formed by symmetrical arcs of loxodrome

Bevel gears are used to transmit torque between intersecting axes. They demonstrate high reliability and durability of work, as well as a constant gear ratio. The disadvantage of such a transmission is the mutual sliding of the surfaces of the teeth of the gears, which leads to the emergence of friction forces and wear of their working surfaces. In this regard, there is a task to design such bevel gears that would have no slip. Non-circular wheels are understood as a pair of closed curves that rotate around fixed centers and at the same time roll over each other without sliding. They can serve as centroids for the design of cylindrical gears between parallel axes. If the axes of rotation of the wheels intersect, then the gears are called conical. An analog of gears between parallel axes, in which centroids are flat closed curves, for gears with intersecting axes are spherical closed curves. For a bevel gear with a constant gear ratio, such spherical curves are circles on the surface of the sphere, and with a variable gear ratio, spatial spherical curves. This paper considers the construction of closed spherical curves that roll around each other without sliding when they rotate around the axes intersecting in the center of the spheres. These curves are formed from symmetrical arcs of the loxodrome, a curve that crosses all the meridians of the ball at a constant angle. This angle should be 45°, which ensures the intersection of the loxodrome at right angles. Analytical dependences have been derived underlying the calculations of profiles of spherical non-circular wheels and their visualization by means of computer graphics. The results could be used to design non-circular wheels for textile machines, hydraulic machine pumps, pump dispensers, etc.

Contact Path Method of Generating Spherical Conjugated Curves as Bevel Gearing Tooth Profiles

Abstract This paper proposes a general method to generate spherical conjugated curves based on their contact path. The main objective of generating conjugated curves is to use them as gearing tooth profiles. The derivation is based on the prescribed spherical contact path, the equation of meshing, and the rigid body motion between two bodies. This paper demonstrates the method by assigning great circle and small circle contact paths in circular pitch pair transmission to generate conjugated curves. With the great circle contact path, the conjugated curves are exact spherical involute, and with the small circle contact path, the conjugated curves are spherical cycloid. The contact properties of the conjugated curves, such as induced curvature and the sliding ratio, are also provided with mathematical equations. Based on the results, the procedures and the geometries between the planar and spherical contact path methods have corresponding relationships. In contrast to the Camus theorem, the exact spherical involute could be generated from a similar procedure of planar involute cases. The ultimate goal of this research topic is to propose a general approach to generating conjugated profiles in planar, spherical, and spatial cases. The successful derivation of the spherical cases in this paper paves the way for spatial cases in the future.

Robust spherical principal curves

Principal curves are a nonlinear generalization of principal components and go through the mean of data lying in Euclidean space. In this paper, we propose L1-type and Huber-type principal curves through the median of data to robustify the principal curves for a dataset that may contain outliers. We further investigate the stationarity of the proposed robust principal curves on S2. Results from numerical experiments on S2 and S4, including real data analysis, manifest promising empirical features of the proposed method.

The effects of base curve aspheric orthokeratology lenses on corneal topography and peripheral refraction: A randomized prospective trial

BackgroundTo investigate the effects of orthokeratology (ortho-k) lenses with aspheric and spherical base curve designs on corneal refractive power (CRP) and peripheral refraction. MethodsChildren aged 8 to 12 years with myopia between −0.75 D to −4.00 D, astigmatism ≤1.00 D, and corneal astigmatism ≤1.50 D were randomly assigned to the base curve aspheric (BCA) and base curve spherical (BCS) ortho-k lens groups. CRP was assessed for the central 8 mm cornea along horizontal and vertical meridians, and peripheral refraction was measured at 10°, 20°, and 30° along the nasal and temporal retina. Primary measurements included relative corneal refractive power change (RCRPC) and relative peripheral refraction change (RPRC). ResultsThe 3-month results of the 33 and 29 subjects (right eye only) in the BCA and BCS groups, respectively, were obtained. Nonsignificant differences were found in the baseline data between the two groups (p > 0.05). At the 3-month follow-up visit, the mean RCRPC in the BCA group (2.08 ± 0.65 D) was significantly greater than that in the BCS group (1.32 ± 0.81 D) (F1,51 = 25.25, p < 0.001). The BCA group (-1.82 ± 0.65 D) exhibited a larger absolute RPRC than the BCS group (-0.98 ± 0.54 D) (F1,57 = 33.73, p < 0.001). ConclusionsIt was found that the BCA ortho-k lens resulted in a more aspheric treatment zone and a more myopic relative peripheral refraction (RPR) along the horizontal meridian. The more myopic RPR was contributed by a more hyperopic central refraction and a more myopic peripheral refraction in the BCA group.