Ringed Surfaces Research Articles

Rohdea rotiformis (Asparagaceae), a new species from Northern Sichuan, China

Rohdea rotiformis is described and illustrated as a new species from Sichuan province of China. The new species possesses a cushionlike, ringed, fleshy appendage above the stamens in perianth throat. In these features, the new species is morphologically similar to R. urotepala but differs in its ringed appendage surface smooth without verruculose.

Design and Experiment of a Hollow Beam Electron Optics System for Ka-Band Extended Interaction Klystrons

In this article, the design and experiment of a hollow electron beam (HEB) electron optics system (EOS) for Ka-band extended interaction klystrons (EIKs) are presented. To realize the fast switch of the current emission at a low cutoff voltage, an electron gun with two focus electrodes (FEs) is chosen. The emitting surface of cathode is a ringed spherical surface. The first focus electrode (FE1) is around the cathode, and the second focus electrode (FE2) is set in the central hole of the cathode. The current emitted from the cathode surface is about 2A, when the operating voltage is 25 kV. Three simulation codes, Egun, Superfish, and CST, are used for designing the electron gun and the permanent magnet focusing system (PMFS). Some sensitivity analyses with respect to the PMFS’s critical parameters are presented. The simulation shows the dc beam transmission is 100%. The electron beam, confined by 6000 Gauss uniform field, propagates through the drift tube of 37 mm with good laminarity and small ripple. The emission current can be suppressed when the voltage of the FEs is −3.4 kV. According to the results of simulation, the Ka-band EIK had been constructed and the hot-test experiments had been fulfilled in succession. The experimental results exhibit that the dc beam transmission is about 99% and the beam can sustain 98% transmission at high frequency (HF) operation. In addition, the output power is higher than 8 kW in the bandwidth of over 100 MHz.

Eulerian formulation of elastic rods.

In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface. This reformulation hinges on describing the rod-deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this re-parametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two-points boundary-value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment.

Geometry of the ringed surfaces in [formula omitted] that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curver(t)=(x(t),y(t),z(t)) has the distinctive property that the components of its derivative r′(t) satisfy x′2(t)+y′2(t)+z′2(t)=σ2(t) for some polynomial σ(t). Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x′(t),y′(t),z′(t) in terms of polynomials u(t),v(t),p(t),q(t) through a construct that can be interpreted as a mapping from R4 to R3 defined by a quaternion product or the Hopf map. Under this map, r′(t) is the image of a ringed surface S(t,ϕ) in R4, whose geometrical properties are investigated herein. The generation of S(t,ϕ) through a family of four-dimensional rotations of a “base curve” is described, and the first fundamental form, Gaussian curvature, total area, and total curvature of S(t,ϕ) are derived. Furthermore, if r′(t) is non-degenerate, S(t,ϕ) is not developable (a non-trivial fact in R4). It is also shown that the pre-images of spatial PH curves equipped with a rotation-minimizing orthonormal frame (comprising the tangent and normal-plane vectors with no instantaneous rotation about the tangent) are geodesics on the surface S(t,ϕ). Finally, a geometrical interpretation of the algebraic condition characterizing the simplest non-trivial instances of rational rotation-minimizing frames on polynomial space curves is derived.

On modeling with rational ringed surfaces

A surface in Euclidean space is called ringed (or cyclic) if there exists a one-parameter family of planes that intersects this surface in circles. Well-known examples of ringed surfaces are the surfaces of revolution, (not only rotational) quadrics, canal surfaces, or Darboux cyclides. This paper focuses on modeling with rational ringed surfaces, mainly for blending purposes. We will deal with the question of rationality of ringed surfaces and discuss the usefulness of the so called P-curves for constructing rational ringed-surface-blends. The method of constructing blending surfaces that satisfy certain prescribed constraints, e.g. a necessity to avoid some obstacles, will be presented. The designed approach can be easily modified also for computing n-way blends. In addition, we will study the contour curves on ringed surfaces and use them for computing approximate parameterizations of implicitly given blends by ringed surfaces. The designed techniques and their implementations are verified on several examples.

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.

Gn-blending of multiple parametric normal ringed surfaces by adding implicit closings Gn-continuous with the surfaces

The functional spline is a highly effective tool for blending multiple implicit surfaces, but the pipes to be blended are generally defined as parametric normal ringed surfaces. To bridge the gap between the equation forms, Hartmann presented a closing-based algorithm (constructing a closing for each pipe and blending the closings), where a closing is an implicit surface sealing a pipe end (inlet or outlet). So far, however, the closing construction remains imperfect: the initial normal ringed surfaces must satisfy a certain geometric requirement and the obtained closings are only G1-continuous with these surfaces (as is the resulting blending surface). This paper proposes a new algorithm that finds the desired closings without restricting its scope of application. The procedure involves the rational parametrization of the initial surfaces and seeking the closings, interpolating rational surfaces with higher-order continuity through constrained optimization. Three practical examples illustrate the strong performance of this algorithm. In particular, the last example addresses the normal elliptic surfaces.

Blending multiple parametric normal ringed surfaces using implicit functional splines and auxiliary spheres

First presented by Hartmann, closings (implicit surfaces sealing the inlets or outlets of pipes) can bridge the gap between parametric pipe surfaces and implicit functional splines (a powerful tool for blending several implicit surfaces). This paper proposes auxiliary spheres instead of the initial pipe surfaces as the base surfaces in constructing closings, so that the closing based algorithm of two steps (constructing a closing for each pipe and blending the closings) can G 1-continuously connect multiple parametric normal ringed surfaces with freeform directrices and variable radii. The basic theory of an auxiliary sphere tangent to the normal ringed surface is addressed. Either one or two (yielding more design parameters) auxiliary spheres can be added. How the parameters influence the closing configuration is discussed. In addition, the blending shape can be optimized by genetic algorithm after assigning some fiducial points on the blend. The enhanced algorithm is illustrated with four practical examples.

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.

Intersecting a freeform surface with a general swept surface

We present efficient and robust algorithms for intersecting a rational parametric freeform surface with a general swept surface. A swept surface is given as a one-parameter family of cross-sectional curves. By computing the intersection between a freeform surface and each cross-sectional curve in the family, we can solve the intersection problem. We propose two approaches, which are closely related to each other. The first approach detects certain critical points on the intersection curve, and then connects them in a correct topology. The second approach converts the intersection problem to that of finding the zero-set of polynomial equations in the parameter space. We first present these algorithms for the special case of intersecting a freeform surface with a ruled surface or a ringed surface. We then consider the intersection with a general swept surface, where each cross-sectional curve may be defined as a rational parametric curve or as an implicit algebraic curve.

Gn-continuous connections between normal ringed surfaces

A ringed surface is a surface generated by sweeping a circle with non constant radius along a curve. The ringed surface is called normal if the circles are contained in the normal planes of the curve. The paper introduces a method for constructing G n -continuous transition surfaces between two given normal ringed surfaces based on a recent G n -blending method for parametric curves. Various design parameters give user the possibility to manipulate the solution. An advantage of the method is that there is no need for calculation of derivatives of the given ringed surfaces. For special cases implicit G n -connection surfaces of several cylinders/cones are given. The implicit method is based on results on functional splines and can be extended to pipe surfaces with parametrically defined directrices.

The Intersection of Two Ringed Surfaces and Some Related Problems

We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ?uCu generated by a moving circle. Given two ringed surfaces ?uCu1 and ?vCv2, we formulate the condition Cu1 ? Cv2 ? ? (i.e., that the intersection of the two circles Cu1 and Cv2 is nonempty) as a bivariate equation ?(u, v)=0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution of ?(u, v)=0 to the intersection point Cu1 ? Cv2. Thus it is trivial to construct the intersection curve once we have computed the zero-set of ?(u, v)=0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation ?(u, v)=0 is reduced to a decomposable form, f(u)=g(v) or ?f(u)?g(v)?=|r(u)|, which can be solved more efficiently than the general case.

Curvature continuous connections of cones and cylinders

Dupin cyclides are used to obtain connections of cones and cylinders with continuous tangent planes. However, it is impossible to obtain connections with continuous principal curvatures in the same way. In the paper, curvature continuous connections are formed by using so-called normal ringed surfaces. Like cyclides, these surfaces are generated by a circle of variable radius sweeping through space. Further, the curve of the centres is perpendicular to the planes of the circles. A parametric representation of normal ringed surfaces is given which allows a simple algorithm to be developed for the construction of curvature continuous connections of cones and/or cylinders.

Canal surfaces versus normal ringed surfaces

Canal surfaces versus normal ringed surfaces

A study of group 1b metal adsorption on iridium

Field emission probe hole microscopy has been used to study the changes in work function φ of annealed (100)-oriented iridium tips which are produced by vapour deposition of submonolayer copper, silver and gold in ultrahigh vacuum. The commonly observed bright ring around (100) lowers φ for clean Ir(100) from 6.00 eV to 5.8 eV but has little effect on metal adsorption, φ(100) changing progressively with addition of metal to that expected for (100) of the bulk metal. Measurement of binding energies of copper and silver by thermal desorption confirms the general similarity of the ringed and ring-free surfaces. The three metals differ sharply in their capacity to populate Ir(100). Gold is present on Ir(100) at all coverages, silver at > 1020 atoms m−2, and copper at > 4 × 1019 atoms m−2. The temperature dependence of copper invasion taken together with changes in the field emission image around (100) suggests a surface reaction with iridum atoms which commences at the plane edge. On the basis of this and other work it is tentatively suggested that the ringed plane has the (5 × 1) superstructure and the ring-free surface the (1 × 1) structure.

A new intersection algorithm for cyclides and swept surfaces using circle decomposition

The present vocabulary of a solid modeler is canonically the plane, (some subset of) the quadrics, and the torus. The class of cyclides is also becoming important. Quadrics and cyclides lie in the more general class of ringed surfaces: surfaces that can be swept out by a circle. This class also contains the important class of revolute surfaces. We will present a method for the exact intersection of any ringed surface with any quadric or cyclide. This algorithm shows that it is feasible to expand the vocabulary of solid modeling primitives to include all ringed surfaces. In solid modeling, surface intersection is crucial to the design of solids and their subsequent analysis. Our intersection algorithm is exact: that is, the intersection is computed symbolically rather than numerically. For exact intersection, we must reduce to degree-4 computations. We do this by concentrating on the decomposition of a surface into simpler components. Previous algorithmic development has centered around the degree of an algebraic surface. Two keys to our algorithm are circle decomposition and inversion. Solutions are provided for the inversion of a cyclide to a torus, a torus, a torus to a cyclide, and the inversion of any circle.

Regeneration of re-covered bark in Eucommia ulmoides.

The girdled trunk of Eucommia ulmoides, sometimes could not regenerate a new bark due to ringed surface necrosis. But, if an extension of 10cm of the bark on each end of the necrotic part was removed in time and re-covered with new bark taken from another tree, the re-covered bark and the newly girdled part could heal together one week after. Opposite to the necrotic surface the loose part of the re-covered bark regenerated callus from the phloem ray cells. If the re-covered part was very close to the necrotic surface, no callus was regenerated. However, many scattered tracheid-like elements derived from phloem were found. And then groups of cells including several tracheid-like elements were discernible. There was a ring of meristematic cells around that group of cells. Finally these meristems gradually became normal and had usual activities of vascular cambium.

Lunar global figure from mare surface elevations

Laser altimetry data from the Apollo 15, 16, and 17 missions show that the ringed maria surfaces lie on one particular reference surface and that the center of gravity is definitely displaced from the optical center. If these extensive surfaces are assumed to be near hydrostratic surfaces, then there must have existed a time in lunar history when lunar tides and/or internal processes were much different than they are today.