Line Congruence Research Articles

Bone morphotypes of the varus and valgus knee.

Coronal deformity correction with total knee arthroplasty (TKA) is an important feature in the treatment of osteoarthritis (OA). The hypothesis of this study was that bone morphology would be different in varus and valgus deformity, both before osteoarthritis development as well as during and after the disease process of OA. Retrospective study with measurements on preoperative and postoperative full leg standing radiographs of 96 patients who underwent TKA. The included patients were selected for this study because they had an OA knee on one side and a non-arthritic knee on the contralateral side presenting the same type of alignment as the to-be-operated knee (varus or valgus alignment on both sides). The control group of 46 subjects was a group of patients with neutral mechanical alignment who presented for ligamentous problems. A single observer measured mechanical alignment, anatomical alignment, anatomical-mechanical femoral angle and intra-articular bone morphology parameters with an accuracy of 1°. Varus OA group has less distal femoral valgus (mLDFA 89°) than control group (87°) and valgus OA group (mLDFA 85°). Varus OA group has same varus obliquity as control group (MPTA 87°) but more than valgus OA group (MPTA 90°). Joint Line Congruency Angle (JLCA) is 3°open on lateral side in varus and medially open in valgus OA group (2°). The non-arthritic valgus group presents a constitutional mechanical valgus of 184° Hip-Knee-Ankle (HKA) angle. Varus deformity in OA as measured with an HKA angle (HKA) <177° is a combination of distal femoral wear, tibial varus obliquity and lateral joint line opening. Valgus deformity in OA with an HKA > 183° is a combination of femoral distal joint line obliquity and wear combined with medial opening due to medial collateral ligament stretching. The clinical importance of bone morphotype analysis is that it shows the intra-articular potential of alignment correction when mechanical axis cuts are performed. Bone morphology in varus and valgus deformity is different before and after osteoarthritis. Perpendicular cuts to mechanical axes do not necessarily lead to neutral mechanical axis. Constitutional mechanical valgus was observed as 184° HKA angle before the development of OA. Level IV study.

Fano congruences of index 3 and alternating 3-forms

We study congruences of lines $X_\omega$ defined by a sufficiently general choice of an alternating 3-form $\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$. We compute the degree of $X_\omega$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $\omega$ bijectively corresponds to $X_\omega$ except when $n=5$. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for $n=8$ and the Peskine variety for $n=9$. The residual congruence $Y$ of $X_\omega$ with respect to a general linear congruence containing $X_\omega$ is analysed in terms of the quadrics containing the linear span of $X_\omega$. We prove that $Y$ is Cohen-Macaulay but non-Gorenstein in codimension $4$. We also examine the fundamental locus $G$ of $Y$ of which we determine the singularities and the irreducible components.

Gradual Deformity Correction in Blount Disease.

Blount disease is a disorder of the posteromedial proximal tibial physis which causes a progressive varus, procurvatum, and internal rotation deformity of the tibia. Untreated, it can cause significant limb malalignment. The goal of this study is to evaluate the results of correction of Blount disease using types of external fixation. We conducted a retrospective review of 41 patients (51 limbs) who underwent correction of Blount disease with an Ilizarov external fixator or a Taylor spatial frame (TSF) by a single surgeon. The medial proximal tibial angle (MPTA), mean axis deviation (MAD), posterior proximal tibial angle, and joint line congruence angle (JLCA) were measured on radiographs preoperatively, at frame removal and at final follow-up. The average age at treatment was 9.6 years old, with a mean follow-up time of 34 months. Mean preoperative MPTA, MAD, and JLCA were significantly improved at the time of frame removal as well as at final follow-up with no significant changes in correction between the time of frame removal and final follow-up. There was no difference in MPTA and MAD in patients treated with an Ilizarov frame versus a TSF. MPTA, MAD, and JLCA all significantly improved regardless of the underlying diagnosis (infantile vs. adolescent Blount disease) or history of prior surgical intervention. The most common complication was superficial pin-site infection. Both Iliazarov and TSF are viable treatment options for infantile and adolescent Blount disease, with the ability to significantly improve both the limb mechanical axis and the mechanical axis of the affected tibia. Correction can be attained regardless of whether patients have previously failed surgical intervention. Level III-retrospective comparative study.

Defining the space in a general spacetime

A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].

On the blaschke trihedrons of a line congruence

On the blaschke trihedrons of a line congruence

(2+1)-dimensional gravity in Weyl integrable spacetime

We investigate (2+1)-dimensional gravity in a Weyl integrable spacetime (WIST). We show that, unlike general relativity, this scalar-tensor theory has a Newtonian limit for any dimension and that in three dimensions the congruence of world lines of particles of a pressureless fluid has a non-vanishing geodesic deviation. We present and discuss a class of static vacuum solutions generated by a circularly symmetric matter distribution that for certain values of the parameter ω corresponds to a spacetime with a naked singularity at the center of the matter distribution. We interpret all these results as being a direct consequence of the spacetime geometry.

Kinematic geometry of a line trajectory in spatial motion

This paper derives the equations of line-trajectory in spatial motion by means of the E. Study dual-line coordinates. A special emphasis goes to the second-order motion properties for deriving a new proof of the Disteli formulae. As an application concise explicit expressions of the inflection line congruence are directly obtained. Also, a new metric is developed and used to investigate the geometrical properties and kinematics of line trajectory as well as Disteli axis. Finally, a theoretical expressions of point trajectories with special values of velocity and acceleration, which can be considered as a form Euler-Savary equation, for spherical and planar motions are discussed.

Forcing, damping and detuning for single and coupled Van der Pol oscillators

We use Melnikov function techniques together with geometric methods of bifurcation theory to study the interactions of forcing, damping and detuning on resonant periodic orbits for single and coupled forced Van der Pol oscillators. For a coupled pair the local bifurcation geometry is almost everywhere described in terms of the singularities of a line congruence in three dimensions.

A classification theorem on Fano bundles

In this paper we classify rank two Fano bundles ℰ on Fano manifolds satisfying H 2 (X,ℤ)≅H 4 (X,ℤ)≅ℤ. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization ℙ(ℰ), that allows us to obtain the cohomological invariants of X and ℰ. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Curvature Motion on Dual Hyperbolic Unit Sphere H&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;lt;sub style=&amp;quot;margin-left:-8px;&amp;quot;&amp;gt;0&amp;lt;/sub&amp;gt;

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere H20 in the dual Lorentzian space D31 with dual signature (+,+-) . We carry the obtained results to the Lorentzian line space R31 by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in R31.

Discrete Line Congruences for Shading and Lighting

AbstractTwo‐parameter families of straight lines (line congruences) are implicitly present in graphics and geometry processing in several important ways including lighting and shape analysis. In this paper we make them accessible to optimization and geometric computing, by introducing a general discrete version of congruences based on piecewise‐linear correspondences between triangle meshes. Our applications of congruences are based on the extraction of a so‐called torsion‐free support structure, which is a procedure analogous to remeshing a surface along its principal curvature lines. A particular application of such structures are freeform shading and lighting systems for architecture. We combine interactive design of such systems with global optimization in order to satisfy geometric constraints. In this way we explore a new area where architecture can greatly benefit from graphics.

Reliability of Digital Radiographs for Pediatric Lower Extremity Alignment

Digital radiography is the standard method for sharing and storing radiographs. The purpose of this study was to evaluate the interobserver and intraobserver reliability of computer-based and manual measurement methods in determining lower extremity alignment on digital images of pediatric patients. Thirty-two digital standing long leg radiographs of pediatric patients were evaluated with 9 varus, 11 valgus, and 12 neutral alignment films. Six evaluators measured the digital images with a standard computer-based measurement method twice and a manual paper print out method twice. Measurements included the lateral distal femoral angle (LDFA), the medial proximal tibia angle (MPTA), the joint line congruency angle, and the mechanical axis deviation (MAD). Interobserver and intraobserver reliability for computer-based and manual methods were calculated using intraclass correlation coefficients. The interobserver reliability for all angular measurements was found to be fair to good for both measurement methods. The MAD had excellent intraobserver and interobserver reliability. LDFA and MPTA interobserver reliabilities were better by the manual method than the computer-based method. Intraobserver reliability was higher in the computer-based LDFA than manual methods, whereas the MPTA measurements were more reliable by manual methods. Computer-based and manual methods for determining lower extremity alignment from digital radiographs are not dissimilar and both provide fair to good reliability. The MAD was a highly reliable measurement. Overall, measurement of the digital images was not as reliable by either method as measurement of traditional full-length teloroentgenograms. The observer should be familiar with the measurement program to minimize errors. Digital images can be measured reliably and then used for treatment decisions, however, time and care should be taken with measurements. Diagnostic level II.

Vorticity and entropy production in tilted Szekeres spacetimes

We analyze the properties of the tilted Szekeres spacetime, i.e. the version of such spacetime as seen by a congruence of observers with respect to which the fluid is moving. The imperfect fluid and the kinematical variables associated to the four-velocity of the fluid assigned by tilted observers are studied in detail. As it happens for the case of the Lemaitre--Tolman--Bondi spacetime, the fluid evolves nonreversibly (with nonvanishing entropy production) and is nongeodesic. However unlike that later case, the tilted observer detects vorticity in the congruence of the fluid world lines. Also, as for the nontilted congruence the magnetic part of the Weyl tensor vanishes, reinforcing the nonradiative character of this kind of spacetime. Possible physical implications of these results are discussed.

Analysis on Flow Behaviors of Stepped Horizontal Well through Multi-Independent Fault Blocks

Based on the fluid dynamics and seepage mechanics, a mathematical analysis model in which the wellbore flow has been coupled with reservoir flow is established for the stepped horizontal well through the oil layers in three independent fault blocks. In the model, the reservoir flow is considered to be unsteady-state and wellbore pressure drop calculation model is established considering the influence of the on-way influx by using mass conservation principle and the theory of infinitesimal line congruence. Taking different curvature radius into account, a pressure drop calculation method is put forward for the layer-layer connective segments. With the mathematical analysis model, the flow behaviors of stepped horizontal well through multi-independent fault blocks has been analyzed and suggested by the calculation results, the optimal arrangement with good oil layer close to the heel side and the poor close to the finger is recommended for the minimum wellbore friction and the highest production.

Analysis on Flow Behaviors of Stepped Horizontal Well through Multi-Independent Fault Blocks

Based on the fluid dynamics and seepage mechanics, a mathematical analysis model in which the wellbore flow has been coupled with reservoir flow is established for the stepped horizontal well through the oil layers in three independent fault blocks. In the model, the reservoir flow is considered to be unsteady-state and wellbore pressure drop calculation model is established considering the influence of the on-way influx by using mass conservation principle and the theory of infinitesimal line congruence. Taking different curvature radius into account, a pressure drop calculation method is put forward for the layer-layer connective segments. With the mathematical analysis model, the flow behaviors of stepped horizontal well through multi-independent fault blocks has been analyzed and suggested by the calculation results, the optimal arrangement with good oil layer close to the heel side and the poor close to the finger is recommended for the minimum wellbore friction and the highest production.

BILLIARD ALGEBRA, INTEGRABLE LINE CONGRUENCES, AND DOUBLE REFLECTION NETS

Billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrability condition of a line congruence. A new notion of the double reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properties and several examples are presented. Corresponding Yang–Baxter maps, associated with pencils of quadrics are defined and discussed.

On the ampleness of the normal bundle of line congruences

In this paper we study the normal bundle of the embedding of subvarieties of dimension n – 1 in the Grassmann variety of lines in . Making use of some results on the geometry of the focal loci of congruences ([Arrondo, Bertolini and Turrini, Asian J. of Math. 5: 535–560, 2001] and [Arrondo, Bertolini and Turrini, Asian J. of Math. 9: 449–472, 2005]), we give some criteria to decide whether the normal bundle of a congruence is ample or not. Finally we apply these criteria to the line congruences of small degree in .

On the one-parameter dual spherical motions

Based on E. Study's dual line coordinates, explicit expressions were developed for the differential properties of one-parameter dual spherical motions that are coordinate systems independent. This eliminates the need of laborious coordinate transformations necessary in the determination of the canonical systems. With the proposed technique, the Disteli formulae of the axodes were derived and the connections between kinematic geometry of a line trajectory were researched. Meanwhile, a ruled analogy of the curvature circle of a curve in planar kinematics was extended into general spatial kinematics. Then a lot of efforts have been directed to establish dual versions of the Euler–Savary equations of a line trajectory resulting in two different expressions depending on the axodes. As a result, theoretical expressions for the direction of a line congruence were revealed. Finally, an example illustrating the application of the obtained formulae was introduced.

Rank Classification of Linear Line Structures from Images by Trifocal Tensor Determinability

The problem we address is: Given line correspondences over three views, what is the condition of the line correspondences for the spatial relation of the three associated camera positions to be uniquely recoverable? The observed set of lines in space is called critical if there are multiple projectively nonequivalent configurations of the camera positions that can picture the same image triplet of the lines. We tackle the problem from the perspective of trifocal tensor, a quantity that captures the relative pose of the cameras in relation to the captured views. We show that the rank of a matrix that leads to the estimation of the tensor is reduced to 7, 11, 15 if the observed lines come from a line pencil, a line bundle, and a line field, respectively, which are line families belonging to linear line space; and 12, 19, 23 if the lines come from a general linear ruled surface, a general linear line congruence, and a general linear line complex, which are subclasses of linear line structures. We show that the above line structures, with the exception of linear line congruence and linear line complex, ought to be critical line structures. All of these structures are quite typical in reality, and thus, the findings are important to the validity and stability of practically all algorithms related to structure from motion and projective reconstruction using line correspondences.

On a class of first order congruences of lines

We study a class of new examples of congruences of lines of order one, i.e. the congruences associated to the completely exceptionalMonge-Ampere equations. We prove that they are in general not linear, and that through a general point of the focal locus there passes a planar pencil of lines of the congruence. In particular, the completely exceptional Monge-Ampere equations are of Temple type. Introduction In [AF01], Agafonov and Ferapontov introduce and study congruences of lines associated to hyperbolic systems of conservation laws. They prove that in PN, N ≤ 4, these families of lines, if the systems are of Temple type, are, in fact, linear congruences. Successively, they consider, for all N ∈ N, the completely exceptional MongeAmpere equations, studied in [Boi92], and state that these systems are of Temple type, and for N ≥ 5, the associated congruence of lines is not linear. In [DM07], we have studied the first interesting case, i.e. N = 5, giving a geometrical construction of the congruence of lines BMA in P 5 in this way obtained: it results to be a first order congruence and a smooth Fano 4-fold in P11 in the Plucker embedding; its focal locus is a sextic threefold X such that the lines of BMA through a general point of X form a (planar) pencil. This confirms the fact that the considered system is of Temple type. ∗This research was partially supported by MiUR, project “Geometria delle varieta algebriche e dei loro spazi di moduli” for both authors, and by Indam project “Birational geometry of projective varieties” for the second one 2000Mathematics Subject Classification : Primary 14M15; Secondary 53A25, 14M07.