Parallels plane projection and its geometric features

A new equivalent map projection called the parallels plane projection is proposed in this paper. The transverse axis of the parallels plane projection is the expansion of the equator and its vertical axis equals half the length of the central meridian. On the parallels plane projection, meridians are projected as sine curves and parallels are a series of straight, parallel lines. No distortion of length occurs along the central meridian or on any parallels of this projection. Angular distortion and the proportion of length along meridians (except the central meridian) introduced by the projection transformation increase with increasing longitude and latitude. A potential application of the parallels plane projection is that it can provide an efficient projection transformation for global discrete grid systems.

A discrete square global grid system based on the parallels plane projection

We developed a direct partitioning method to construct a seamless discrete global grid system (DGGS) with any resolution based on a two-dimensional projected plane and the earth ellipsoid. This DGGS is composed of congruent square grids over the projected plane and irregular ellipsoidal quadrilaterals on the ellipsoidal surface. A new equal area projection named the parallels plane (PP) projection derived from the expansion of the central meridian and parallels has been employed to perform the transformation between the planar squares and the corresponding ellipsoidal grids. The horizontal sides of the grids are parts of the parallel circles and the vertical sides are complex ellipsoidal curves, which can be obtained by the inverse expression of the PP projection. The partition strategies, transformation equations, geometric characteristics and distortions for this DGGS have been discussed. Our analysis proves that the DGGS is area-preserving while length distortions only occur on the vertical sides off the central meridian. Angular and length distortions positively correlate to the increase in latitudes and the spanning of longitudes away from a chosen central meridian. This direct partition only generates a small number of broken grids that can be treated individually.

Plane curve diagrams and geometrical applications

We look at plane curve diagrams (f,alpha), which are given by a plane curve multigerm alpha : (R, S) -> R(2) and a function on it f : (R, S) -> R. We obtain a classification of all such diagrams, where alpha has e-codimension <= 2 and f has finite order. Then we define an equivalence between plane curves which we call Ah(alpha)-equivalence and which is determined by the class of the diagram (h(alpha), alpha). Here, h alpha denotes the height function of alpha with respect to its normal vector. This is an equivalence which not only takes into account the topology of the singularity of alpha, but also its flat geometry. Finally, we apply our results in order to obtain a classification of all the plane projections of a generic space curve gamma embedded in R(3).

A Parameterization Technique for the Continuation Power Flow Developed from the Analysis of Power Flow Curves

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Anomalous metamagnetic-like transition in a FeRh/Fe3Pt interface occurring at T approximate to 120 K in the field-cooled-cooling curves for low magnetic fields

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Double point curves for corank 2 map germs from C-2 to C-3

We characterize finite determinacy of map germs f : (C-2, 0) -> (C-3, 0) in terms of the Milnor number mu(D(f)) of the double point curve D(f) in (C-2, 0) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs f(t) : (C-2, 0) -> (C-3, 0) is equivalent to the constancy of both mu(D(f(t))) and mu(f(t)(C-2)boolean AND H) with respect to t, where H subset of C-3 is a generic plane. (C) 2011 Elsevier B.V. All rights reserved.

Topologically correct intersection curves of tori and natural quadrics

This thesis provides efficient and robust algorithms for the computation of the intersection curve between a torus and a simple surface (e.g. a plane, a natural quadric or another torus), based on algebraic and numeric methods. The algebraic part includes the classification of the topological type of the intersection curve and the detection of degenerate situations like embedded conic sections and singularities. Moreover, reference points for each connected intersection curve component are determined. The required computations are realised efficiently by solving quartic polynomials at most and exactly by using exact arithmetic. The numeric part includes algorithms for the tracing of each intersection curve component, starting from the previously computed reference points. Using interval arithmetic, accidental incorrectness like jumping between branches or the skipping of parts are prevented. Furthermore, the environments of singularities are correctly treated. Our algorithms are complete in the sense that any kind of input can be handled including degenerate and singular configurations. They are verified, since the results are topologically correct and approximate the real intersection curve up to any arbitrary given error bound. The algorithms are robust, since no human intervention is required and they are efficient in the way that the treatment of algebraic equations of high degree is avoided.

Coronal plane segmental flexibility in thoracic adolescent idiopathic scoliosis assessed by fulcrum-bending radiographs

Knowledge about segmental flexibility in adolescent idiopathic scoliosis is crucial for a better biomechanical understanding, particularly for the development of fusionless, growth-guiding techniques. Currently, there is lack of data in this field. The objective of this study was, therefore, to compute segmental flexibility indices (standing angle minus corrected angle/standing angle). We compared segmental disc angles in 76 preoperative sets of standing and fulcrum-bending radiographs of thoracic curves (paired, two-tailed t tests, p < 0.05). The mean standing Cobb angle was 59.7 degrees (range 41.3 degrees -95 degrees ) and the flexibility index of the curve was 48.6\% (range 16.6-78.8\%). The disc angles showed symmetric periapical distribution with significant decrease (all p values <0.0001) for every cephalad (+) and caudad (-) level change. The periapical levels +1 and -1 wedged at 8.3 degrees and 8.7 degrees (range 3.5 degrees -14.8 degrees ), respectively. All angles were significantly smaller on the-bending views (p values <0.0001). We noted mean periapical flexibility indices of 46\% (+1), 49\% (-1), 57\% (+2) and 81\% (-2), which were significantly less (p < 0.001) than for the group of remote levels 105\% (+3), 149\% (-3), 231\% (+4) and 300\% (-4). The discal and bony wedging was 60 and 40\%, respectively, and mean values 35 degrees and 24 degrees (p < 0.0001). Their relationship with the Cobb angle showed a moderate correlation (r = 0.56 and 0.45). Functional, radiographic analysis of idiopathic thoracic scoliosis revealed significant, homogenous segmental tethering confined to four periapical levels. Future research will aim at in vivo segmental measurements in three planes under defined load to provide in-depth data for novel therapeutic strategies.

Investigations in the theory of reflected ray-surfaces, and their relation to plane reflected caustics.

Mode of access: Internet.

Deformation of sand in plane strain and oxisymmetric compression

Small scale laboratory experiments, in which the specimen is considered to represent an element of soil in the soil mass, are essential to the evolution of fundamental theories of mechanical behaviour. In this thesis, plane strain and axisymmetric compression tests, performed on a fine sand, are reported and the results are compared with various theoretical predictions. A new apparatus is described in which cuboidal samples can be tested in either axisymmetric compression or plane strain. The plane strain condition is simulated either by rigid side platens, in the conventional manner, or by flexible side platens which also measure the intermediate principal stress. Close control of the initial porosity of the specimens is achieved by a vibratory method of sample preparation. The strength of sand is higher in plane strain than in axisymmetric compression, and the strains required to mobilize peak strength are much smaller. The difference between plane strain and axisymmetric compression behaviour is attributed to the restrictions on particle movement enforced by the plane strain condition; this results in an increase in the frictional component of shear strength. The stress conditions at failure in plane strain, including the intermediate principal stress, are accurately predicted by a theory based on the stress- dilatancy interpretation of Mohr's circles. Detailed observations of rupture modes are presented and measured rupture plane inclinations are predicted by the stress-dilatancy theory. Although good correlation with the stress-dilatancy theory is obtained during virgin loading, in both axisymmetric compression and plane strain, the stress-dilatancy rule is only obeyed during reloading if the specimen has been unloaded to approximate ambient stress conditions. The shape of the stress-strain curves during pre-peak deformation, in both plane strain and axisymmetric compression, is accurately described bv a combined parabolic-hyperbolic specification.

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow.

Projectively Normal Line Bundles on K-Gonal Curves and Rational Surfaces

2000 Mathematics Subject Classification: 14H50.

Piecewise Convex Curves and their Integral Representation

2000 Mathematics Subject Classification: 52A10.