Frontal Curves Research Articles

On the equiform geometry of special curves in hyperbolic and de Sitter planes

<abstract><p>In this paper, we aim to investigate the equiform differential geometric properties of the evolute and involute frontal curves in the hyperbolic and de Sitter planes. We inspect the relevance between evolute and involute frontal curves that relate to symmetry properties. Also, under the viewpoint of symmetry, we expand these notions to the frontal curves. Moreover, we look at the classification of these curves and introduce the notion of frontalisation for its singularities. Finally, we provide two numerical examples with drawing as an application, through which we authenticate our theoretical results.</p></abstract>

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.

Lorentz–Darboux rectifying surfaces and osculating surfaces along frontal curves on timelike surfaces

In this paper, we consider two kinds of developable surfaces along a timelike frontal curve lying in a timelike surface in Minkowski 3-space, the Lorentz–Darboux rectifying surfaces and the Lorentz–Darboux osculating surfaces. Meanwhile, we also consider two curves generated by such a timelike frontal curve. We give two new invariants of the frontal curve which characterize singularities of the Lorentz–Darboux developable surfaces and Lorentz–Darboux rectifying and osculating curves. Unlike the regular curves, the frontal curves may have singular points. Using the methods of the unfolding theory in singularity theory, we complete the classifications of the singular points of these two surfaces and two curves. The main results indicate that compared with developable surfaces along a regular curve, there exists a more complicated construction for the singularities of the developable surfaces along a timelike frontal curve, there will appear an extra locus, arising by the singular point of the timelike frontal curve, for the singularities of the Lorentz–Darboux rectifying surface, whereas the Lorentz–Darboux osculating surface did not. In addition, we investigate the geometric properties of the timelike frontal curve, it is shown that the timelike frontal curve can be regarded as the envelope of a family of timelike frontal rectifying curves. Finally, we provide several examples to illustrate the theoretical results.

Primitivoids of curves in Minkowski plane

<abstract><p>In this work, we investigate the differential geometric characteristics of pedal and primitive curves in a Minkowski plane. A primitive is specified by the opposite structure for creating the pedal, and primitivoids are known as comparatives of the primitive of a plane curve. We inspect the relevance between primitivoids and pedals of plane curves that relate with symmetry properties. Furthermore, under the viewpoint of symmetry, we expand these notions to the frontal curves in the Minkowski plane. Then, we present the relationships and properties of the frontal curves in this category. Numerical examples are presented here in support of our main results.</p></abstract>

Classification of Evolutoids and Pedaloids in Minkowski Space-time Plane

In this paper, we study the families of relatives of pedals and evolutes in the Minkowski spacetime plane R 2 1 . We obtain some relationships between these families which turn out to be different from Euclidean plane. Also, we classify and generalize these notions to the category of frontal curves in R 2 1 . Finally, some computational examples in support of our main results are given and plotted.

Developable surfaces along frontal curves on embedded surfaces

We consider two types of developable surfaces along a frontal curve on an embedded surface in the Euclidean 3-space. One is called the osculating developable surface, and the other is called the normal developable surface. The frontal curve may have singular points. We give new invariants of the frontal curve which characterize singularities of the developable surfaces. Moreover, a frontal curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants is constantly equal to zero.

A hierarchical classification of adolescent idiopathic scoliosis: Identifying the distinguishing features in 3D spinal deformities.

This study aimed to identify the differentiating parameters of the spinal curves’ 2D projections through a hierarchical classification of the 3D spinal curve in adolescent idiopathic scoliosis (AIS). A total number of 103 right thoracic left lumbar pre-operative AIS patients were included retrospectively and consecutively. A total number of 20 non-scoliotic adolescents were included as the control group. All patients had biplanar X-rays and 3D reconstructions of the spine. The 3D spinal curve was calculated by interpolating the center of vertebrae and was isotropically normalized. A hierarchical classification of the normalized spinal curves was developed to group the patients based on the similarity of their 3D spinal curve. The spinal curves’ 2D projections and clinical spinal measurements in the three anatomical planes were then statistically compared between these groups and between the scoliotic subtypes and the non-scoliotic controls. A total of 5 patient groups of right thoracic left lumbar AIS patients were identified. The characteristics of the posterior-anterior and sagittal views of the spines were: Type 1: Normal sagittal profile and S shape axial view. T1 is leveled or tilted to the right in the posterior view. Type 2: Hypokyphotic and a V shape axial view. T1 is tilted to the left in the posterior view. Type 3: Hypokyphotic (only T5-T10) and frontal imbalance, S shape axial view. T1 is leveled or tilted to the right, and 3 frontal curves. Type 4: Flat sagittal profile (T1-L2), slight frontal imbalance with a V shape axial view, T1 tilted to the left. Type 5: flat sagittal profile and forward trunk shift with a proximal kyphosis and S shape axial view. T1 is leveled or tilted to the right. In conclusion, a hierarchical classification of the 3D scoliotic spine allowed identifying various distinguishing features of the spinal curves in patients with a right thoracic curve in an orderly fashion. The subtypes’ characteristics resulting from this 3D classification can be identified from the pairs of the frontal and sagittal spinal curves i.e. X-rays in right thoracic AIS patients.

Considerations in sagittal evaluation of the scoliotic spine.

To predict the sagittal spinal parameters as measured in a 3D model of the spine using the 2D radiographic measurements. Bi-planar low-dose stereoradiography images of 73 right thoracic AIS patients were processed to generate 3D models of the spine and pelvis. T1-T12 kyphosis, L1-S1 lordosis, and pelvic rotation were calculated using these 3D models. With the same X-rays, T1-T12 kyphosis, L1-S1 lordosis, thoracic and lumbar frontal curves, and pelvic rotation (calculated from the frontal and sagittal distances between the femoral heads) were manually measured on the X-rays by two independent observers. 3D sagittal parameters were predicted from only 2D sagittal parameters (simple regression) and from 2D sagittal parameters, 2D frontal parameters, and pelvic rotation (multiple regression). The simple and multiple regression models were compared for efficiency and accuracy of prediction. Comparing single and multiple regression models, multiple regression improved the prediction of the 3D sagittal parameters for kyphosis (R2 = 0.78-0.86) and lordosis (R2 = 0.88-0.92) measurements when compared to simple regression. The impact of pelvic rotation was significant when 2D kyphosis was higher than 40° and thoracic curve was less than 60° or 2D kyphosis was less than 40° and thoracic curve was higher than 60°, p < 0.05. Lordosis of 60° and higher were more prone to measurement error when pelvic rotation was present, p < 0.05. Both pelvic rotation and frontal deformity affect the accuracy of the 2D sagittal measurements of the scoliotic spine. We suggest the importance of the 3D considerations in sagittal evaluation of AIS.

Comparison of scoliosis measurements based on three-dimensional vertebra vectors and conventional two-dimensional measurements: advantages in evaluation of prognosis and surgical results

A new concept of vertebra vectors based on spinal three-dimensional (3D) reconstructions of images from the EOS system, a new low-dose X-ray imaging device, was recently proposed to facilitate interpretation of EOS 3D data, especially with regard to horizontal plane images. This retrospective study was aimed at the evaluation of the spinal layout visualized by EOS 3D and vertebra vectors before and after surgical correction, the comparison of scoliotic spine measurement values based on 3D vertebra vectors with measurements using conventional two-dimensional (2D) methods, and an evaluation of horizontal plane vector parameters for their relationship with the magnitude of scoliotic deformity. 95 patients with adolescent idiopathic scoliosis operated according to the Cotrel-Dubousset principle were subjected to EOS X-ray examinations pre- and postoperatively, followed by 3D reconstructions and generation of vertebra vectors in a calibrated coordinate system to calculate vector coordinates and parameters, as published earlier. Differences in values of conventional 2D Cobb methods and methods based on vertebra vectors were evaluated by means comparison T test and relationship of corresponding parameters was analysed by bivariate correlation. Relationship of horizontal plane vector parameters with the magnitude of scoliotic deformities and results of surgical correction were analysed by Pearson correlation and linear regression. In comparison to manual 2D methods, a very close relationship was detectable in vertebra vector-based curvature data for coronal curves (preop r 0.950, postop r 0.935) and thoracic kyphosis (preop r 0.893, postop r 0.896), while the found small difference in L1-L5 lordosis values (preop r 0.763, postop r 0.809) was shown to be strongly related to the magnitude of corresponding L5 wedge. The correlation analysis results revealed strong correlation between the magnitude of scoliosis and the lateral translation of apical vertebra in horizontal plane. The horizontal plane coordinates of the terminal and initial points of apical vertebra vectors represent this (r 0.701; r 0.667). Less strong correlation was detected in the axial rotation of apical vertebras and the magnitudes of the frontal curves (r 0.459). Vertebra vectors provide a key opportunity to visualize spinal deformities in all three planes simultaneously. Measurement methods based on vertebral vectors proved to be just as accurate and reliable as conventional measurement methods for coronal and sagittal plane parameters. In addition, the horizontal plane display of the curves can be studied using the same vertebra vectors. Based on the vertebra vectors data, during the surgical treatment of spinal deformities, the diminution of the lateral translation of the vertebras seems to be more important in the results of the surgical correction than the correction of the axial rotation.

Three-dimensional mathematical reconstruction of the spinal shape, based on active contours

To reduce the amount of radiographs needed for patients with a scoliosis, a radiation-free method based on topographic images of the back was developed. An active contour model simulating spinal stiffness has been applied to video rasterstereographic (VRS) data. The aim of the present study is (a) to evaluate the applicability of active contours to improve the accuracy and the reliability of the three-dimensional (3D) spinal midline reconstruction from back surface data and (b) to design a more robust method to detect the spinal midline. To evaluate the reliability and accuracy, the active contour-based method is compared to a conventional procedure, which has been specifically developed for scoliosis; both methods produce a 3D curve of the spinal midline. The frontal projections and surface rotations of these spinal midlines are compared; r.m.s. deviations of 0.9 mm between the frontal curves and 0.4° between the surface rotations were obtained. Applying the active contour-based method does therefore not result in a substantial difference in accuracy to the conventional procedure. As a conclusion the active contour method is a valuable mathematical method that can accurately reconstruct the spinal midline based on back surface data. In addition, the method can be applied to various postures.

Isola spinal instrumentation system for idiopathic scoliosis.

Since the definition of three-dimensional components of the scoliotic deformity, there have been important improvements in the surgical treatment of the problem. A derotation maneuver was proposed as a treatment option with CD instrumentation, but the reports of imbalance and decompensation with this system repopularized sublaminar wiring and translation as a corrective maneuver. Isola spinal instrumentation is one of the modern systems that utilizes vertebral translation instead of rod rotation. This study analyzes the results of 24 patients with idiopathic scoliosis who had been followed up for at least 2 years, and were surgically treated with titanium Isola Spinal Instrumentation in the Department of Orthopaedics and Traumatology, Ankara Social Security Hospital. Patients were grouped according to the King-Moe classification. Patients with type III, IV or V curves received only posterior instrumentation while this procedure followed anterior release and discectomy in the same session in patients with type I or II curves. A translation maneuver was utilized in the correction of scoliotic curves using the cantilever technique, either alone or supplemented by sublaminar wiring with Songer multifilament titanium cables. This study aimed to elucidate the effects of this technique in the frontal and sagittal plane curves and the trunk balance. The balance was analyzed clinically and radiologically by measurement of the lateral trunk shift (LT), shift of stable vertebra (SS), and shift of head (SH) in vertebral units (VU). The postoperative correction was significant in the frontal plane for all types of curves (p < 0.05). The postoperative correction was 80.9% +/- 9.5% in type III curves. Overall, the mean Cobb angle of the major curve value in the frontal plane was 66.9 degrees +/- 18.8 degrees, and it was corrected by 62.8% +/- 20.1%. The correction loss of Cobb angles in the frontal plane was 5.4 degrees +/- 5.5 degrees at the last follow-up visit. A normal physiologic thoracic contour (30 degrees - 50 degrees) was achieved in 83.3% of the patients and normal lumbar contour (40 degrees - 60 degrees) in 66.7% of the patients in the sagittal plane. The correction was found to be significant in all balance values (p < 0.05). The postoperative correction in LT values correlated with the correction of the Cobb angle values in the frontal plane. All patients had complete balance (SH: 0 VU and SS: 0 VU) or balanced curves (0 VU < SH, SS < 0.5 VU).Finally, the study concluded that the translation maneuver, especially when used with the cantilever technique, resulted in high correction rates in the frontal plane. Additionally, the technique was also successful in obtaining normal sagittal contours and correcting balance values.

Simulation of Compression Effects during Scaleup of a Commercial Ion-Exchange Process

A modified version of the general nonlinear multicomponent rate equation chromatography model has been developed to simulate compression effects on system behavior. It accounts for the variations in porosity and particle deformation that occur within a packed bed during compression. This paper investigates the effect of compression on the scaleup of a commercial packed-bed ion-exchange process to manufacture a whey growth factor extract (WGFE). The resin employed in the ion-exchange process is Sepharose Big-Beads SP (Amrad-Pharmacia, Sydney, Australia). Compression-induced changes in packed-bed porosity and particle diameter for a laboratory- (2 cm) and a production-scale (20 cm) column were estimated from pressure-drop data by using a modified volume-averaged continuum theory. These were combined with model parameters from a previous experimental study for two major whey proteins, lactoperoxidase and lactoferrin. Model simulations were performed. First, model parameters were validated by replication of experimental frontal adsorption breakthrough and step-elution curves. Selection of numerical parameters and accurate adsorption equilibria were identified as critical steps in ensuring successful reconciliation of model predictions with experimental data. The effect of compression on frontal adsorption and elution steps during scaleup was then investigated. For both frontal adsorption and step elution, system behavior was found to be largely independent of compression. Increased compression created only minor and trivial variations in effluent concentration profiles. These were influenced by two competing mechanisms, namely, premature breakthrough and increased external film mass transfer. Further simulations with a smaller particle size and superficial velocity displayed no significant increase in compression effects. Compression effects are not important during scaleup of this system.

Analysis of steric hindrance effects on adsorption kinetics and equilibria

AbstractMost existing adsorption models do not properly consider steric hindrance effects of preadsorbed solutes. As a consequence, the models often fail to represent the adsorption kinetics and equilibria accurately. In this work, we extend the random sequential adsorption concept for irreversible adsorption to analyze reversible adsorption on a continuous surface and a random site surface. Based on simulation results of these processes, general kinetic equations for one‐component adsorption are developed. The equations are used to correlate chromatography frontal curves of lysozyme and isotherm data of ethane adsorption on activated carbon and ethylene adsorption on a molecular sieve. The significance of the equations, as compared with the Langmuir equation, lies not only in their ability to correlate the experimental data more accurately, but in the physical significance of the adsorption parameters such as the maximum adsorption capacity obtained from the correlation. Our study shows that steric hindrance effects alone result in nonlinear Scatchard and Hill plots with negative cooperativity.

Dynamics of physical adsorption for porous grains with a complex internal structure

The dynamics of physical adsorption for a porous medium consisting of porous grains having a complex internal structure is analysed. The analysis is based on a combined theoretical model describing the mass transfer of an adsorbate inside a porous adsorbent grain with a complex internal structure. The model takes into account the effect of the dead-end transport channels (pores) inside the porous adsorbent grain on the nature of the mass transfer of a substance. For frontal adsorption dynamics with convex isotherms equations have been developed for calculating the width of the zone corresponding to intensive mass exchange in the adsorber (the width of the stationary front). Equations were also obtained for calculating the times of relaxations due to different mechanisms of interphase mass transfer inside a porous grain with a complex internal structure and to mass transfer in the intergrain space of the porous medium. It is shown that a strong interaction can take place between the mass transfer within the zones of a porous grain and the dead-end transport channels. When this occurs the width of the stationary front and the relaxation times characteristic of mass transfer inside the porous grain are in quasilinear dependence on the geometrical dimension of the porous grain and the geometrical dimension of the microporous zone inside the porous grain. These dimensions are: radius R0 in the case of a porous grain of an adsorbent having a spherical form and radius R0 in the case of a microporous spherical zone. An analysis of the frontal dynamic curves and the measurement of the coefficients of diffusion inside the microporous zone by other physical methods (e.g., n.m.r. measurements) show a lack of coincidence of the values of the diffusion coefficients. The possible cause of this non-coincidence is indicated.

Topographical distribution of delta rhythms during sleep: Evolution with age

This study concerns the evolution of slow EEG activities (delta rhythms) and their topographical distribution during 31 night sleep records of 27 control subjects whose ages ranged from 1 month to 27 years. A system of data processing allows the representation of continuous changes in 8 derivations of the power of 0.5–4.5 c/sec waves. The obtained curves, whose respective magnitudes depend on the chosen montage, parallel the sleep cycles of the classical hypnogram. They do not clearly differentiate NREM sleep stages, but show a progressive increase of delta rhythm power with the deepening of NREM sleep. Moreover, measurement of the magnitude of these curves allows comparisons between the different channels in a given record and between different recordings. With the referential setting used in this study the topographical organization of the occipital, parietal and central curves proved remarkably consistent in all subjects. The general magnitude of these curves decreased with age while that of frontal curves slightly increased. The right parietal curves were of greater magnitude than the left in every record obtained from children aged 2–5 years. The changes with age, and the intra- and interindividual differences are discussed. The data suggest hypotheses about the importance of NREM sleep in developmental processes.

An experimental study of the frontal reference curves of binocular visual space.

An experimental study of the frontal reference curves of binocular visual space is presented, with several different control procedures. The basic design consists in a display of small points of light in a completely dark room, which the subject moves into position according to four basic criteria: apparent frontal curves (i.e. geodesics), apparent equidistant curves, apparent circular horizons, and apparent right angles. Differences were found between the curvature functions obtained in all four cases. Certain aspects of the classical horopter theory of Hering (1879) are thus refuted by the present finding of a clear difference between apparent frontal curves and apparent equidistant curves. Reversal points were not found, in many of the geodesic observations, in contrast to classical claims. This finding obviates those aspects of the classical Hillebrand (1893) theory and of the recent Luneburg (1947) theory which are predicated upon the existence of reversal points. Almost no reversal points occur with the equidistant horizons. Moreover, the equidistant horizon is an ellipse, and not a circle, as suggested by the Luneburg (1947) theory. The mobility of the ego-center, along the interpupillary line, is put forth as a possible explanation of this elliptical shape. This represents a further challenge to such theories as that of Luneburg (1947), which demand a fixed origin in the visual space.