Reference Plane Research Articles

A cross-sectional study on three-dimensional compensatory characteristics of maxillary teeth in patients with different types of skeletal Class III malocclusion with mandibular asymmetry.

The purpose of this study was to analyze three-dimensional dental compensation in patients with different types of skeletal Class III malocclusion with mandibular asymmetry, using cone-beam computed tomography (CBCT) and three-dimensional reconstruction measurement technology, thereby providing clinical guidance and reference for combined orthodontic and orthognathic treatment. 81 patients with skeletal Class III malocclusion with mandibular asymmetry were selected in accordance with the inclusion criteria. According to a new classification method based on the direction and amount of menton deviation relative to ramus deviation, patients were divided into three groups called Type 1, Type 2, and Type 3. In Type 1, the direction of menton deviation was consistent with that of ramus deviation and the amount of menton deviation was greater than that of ramus deviation. In Type 2, the direction of menton deviation was consistent with that of ramus deviation and the amount of menton deviation was smaller than that of ramus deviation. In Type 3, the direction of menton deviation was inconsistent with that of ramus deviation. The maxillary occlusal plane (OP), anterior occlusal plane (AOP), and posterior occlusal plane (POP) were measured on reconstructed CBCT images. The vertical, transverse, and anteroposterior distances from maxillary teeth to reference planes and the 3D angles between the long axis of these teeth and reference planes were measured. These dental variables measured from the deviated and non-deviated sides were compared within each group, as well as among each other. Of the 81 patients with asymmetrical Class III malocclusion, 52 patients were categorized in Type 1, 12 patients in Type 2, and 17 patients in Type 3. There were significant differences between deviated and non-deviated sides in Type 1 and Type 3 (p < 0.05). In Type 1, the vertical distances of maxillary teeth on the deviated side were lower than those on the non-deviated side, and AOP, OP, and POP on the deviated side were larger than those on the non-deviated side (p < 0.05). In Type 3, the vertical distances of the maxillary teeth on the deviated side were lower (p < 0.05), and the AOP and OP on the deviated side were larger than those on the non-deviated side. In all three groups, the transverse distances of the maxillary teeth from the mid-sagittal plane on the deviated side were larger than those on the non-deviated side (p < 0.05), and the angles between the long axis of maxillary teeth and the mid-sagittal plane on the deviated side were larger, respectively (p < 0.05). The maxillary teeth on the deviated side were observed to have smaller eruption heights in Type 1 and Type 3. In Type 1, AOP, POP, and OP were greater on the deviated side, while in Type 3, only AOP and OP were greater on the deviated side. The maxillary teeth of patients in all three groups on the deviated side were buccal and buccally inclined. Larger sample observations are still needed to further verify these findings.

The identification of possible reference planes on a CBCT scan taken with a small FOV—an observational analytical study

Abstract Objectives Unlike a CBCT scan which has a large field of view (FOV), a scan with a small FOV may not involve the base of the skull. Therefore, routinely used reference planes may not be available to evaluate treatment outcome. The present study was undertaken to determine if the palatal plane and a molar occlusal plane may be used as reference planes to measure the change in the axial inclination of the maxillary central incisor and the maxillary first permanent molar towards the end of orthodontic treatment on a CBCT scan captured with a small FOV. Materials and methods A lateral cephalogram and a CBCT scan with a small FOV were taken for fifteen young adults prior to, and six months after, mini-implant supported, distal movement of the maxillary arch. CBCT multi-planar slices were sequentially viewed to obtain an image that contained the entire length of the maxillary central incisor and the maxillary first permanent molar in the tangential view. The axial inclination of the maxillary central incisor and the maxillary first permanent molar was measured from the palatal plane and the molar occlusal plane on both sides. The same parameters were evaluated on the lateral cephalogram using the palatal plane as a reference. An independent t test (p = 0.05) was conducted to compare the results. A Bland–Altman plot with linear regression was constructed to measure accuracy, method agreement and reliability. Results There was no statistically significant difference when a change in the axial inclination of the maxillary central incisor and the maxillary first permanent molar was measured using the two reference planes. The change in axial inclination of the maxillary central incisor and maxillary first permanent molar had a 95% confidence interval of -1.82 to 4.19 and -4.48 to 3.05 and a p-value of 0.425 and 0.701 on the right side, respectively, and a 95% confidence interval of -1.76 to 4.79 and -4.41 to 2.79 and p-value of 0.350 and 0.650 on the left side. Conclusions The molar occlusal plane and the palatal plane may be used as reference planes for measuring change in the axial inclination of the maxillary central incisor and the first permanent molar in a CBCT scan taken with a small FOV.

Laser-Induced Thermoelastic Response in an Isotropic Medium Having Variable Material Moduli

Thermal displacement and stress distribution are analysed in the domain of plane-strain geometry of an isotropic thermoelastic medium having variable material moduli. In this context of the analysis, laser induced heat sources are planted on the reference plane. To analyse the thermal responses due to the injected heat sources into the medium, hyperbolic type heat conduction model with three phase-lags incorporated. Analytical solutions are expressed in-terms of Laplace–Fourier integral transform domain and the physical behaviour of displacement and stresses are depicted through a discretised form of inverse-integral transformation technique. Finally, parameterized characteristic analysis are presented graphically and the salient features of thermal displacements are highlighted.

Changes in Soft Tissue Variable of Lips Following Retraction of Anterioir Teeth- A Cephalometric Study.

The purpose of this study was to assess the short-term perioral soft tissue variations of the lips before and after treatment cases in 15 patients with bi-maxillary protrusion using treated lateral cephalograms who had already achieved active growth. Fifteen pre-treatment and post-treatment lateral cephalometric radiographs of 18-25-year-old individuals with bimaxillary protrusion treated with all four 1st premolar extractions were accessed from the records. From the reference planes and landmarks, 13 horizontal, 10 vertical, and 2 angular measurements were noted. Statistical comparisons between pre-treatment and post-treatment measurements were measured by a paired t-test to assess the importance of the mean variations at the predetermined significance level. Pearson's correlation coefficient (R) was utilized to assess the strength and significance of the linear relationship between the mean differences for paired (dependent and independent) variables. Pearson's correlation exhibited a noteworthy positive association between the horizontal changes in upper lip position and the horizontal changes of the upper incisor tip point (H-tU1) (R = 0.748), the upper incisor cervical point (H-cU1) (R = 0.707), the lower incisor tip point (H-tL1) (R = 0.839), and the lower incisor cervical point (H-cL1) (R = 0.767). This indicated that upper lip changes are the aftermath of the retraction of the upper and lower incisors in class I bi-maxillary protrusion malocclusion. Thick upper lips showed more retraction of the upper lip in correlation with retraction of the incisors as compared with thin lips. The lower incisor cervical point displayed the strongest association with lower lip retraction.

Mapping the Posterior Ledge and Optic Foramen in Orbital Floor Blowout Fractures.

Background The posterior ledge (PL) is a vital structure that supports the implant posteriorly during orbital floor reconstruction. This study describes a technique for mapping the PL in relation to the infraorbital margin (IM) in patients with orbital floor blowout fractures. This study establishes the location of the optic foramen in relation to the PL. Methods Facial computed tomography (FCT) scans of 67 consecutive patients with isolated orbital floor blowout fractures were analyzed using Osirix. Planes of reference for orbital fractures, a standardized technique for performing measurements on FCT, was used. Viewed coronally, the orbit was divided into seven equal sagittal slices (L1 laterally to L7 medially) with reference to the midorbital plane. The distances of PL from IM and location of optic foramen were determined. Results The greatest distance to PL is found at L5 (median: 30.1 mm, range: 13.5-37.1 mm). The median and ranges for each slice are as follows: L1 (median: 0.0 mm, range: 0.0-19.9 mm), L2 (median: 0.0 mm, range: 0.0-21.5 mm), L3 (median: 15.8 mm, range: 0.0-31.7 mm), L4 (median: 26.1 mm, range: 0.0-34.0 mm), L5 (median: 30.1 mm, range: 13.5-37.1 mm), L6 (median: 29.0 mm, range: 0.0-36.3 mm), L7 (median: 20.8 mm, range: 0.0-39.2 mm). The median distance of the optic foramen from IM is 43.7 mm (range: 37.0- 49.1) at L7. Conclusion Distance to PL from IM increases medially until the L5 before decreasing. A reference map of the PL in relation to the IM and optic foramen is generated. The optic foramen is located in close proximity to the PL at the medial orbital floor. This aids in preoperative planning and intraoperative dissection.

Paper 65: Novel CT-Derived 3D Modeling Assessing Dysplastic Hips Versus Controls Demonstrates Decreased 2D Lateral Center Edge Angle, but not Tönnis Angle, Predicts Decreased 3D Anterolateral Coverage

Paper 65: Novel CT-Derived 3D Modeling Assessing Dysplastic Hips Versus Controls Demonstrates Decreased 2D Lateral Center Edge Angle, but not Tönnis Angle, Predicts Decreased 3D Anterolateral Coverage

A Compact Broadband Common-Mode Suppression Filter That Integrates Series-Mushroom into Defected Corrugated Reference Plane Structures.

This paper proposes a common-mode noise suppression filter scheme for use in the servers and computer systems of high-speed buses such as SATA Express, HDMI 2.0, USB 3.2, and PCI Express 5.0. The filter uses a novel series-mushroom-defected corrugated reference plane (SMDCRP) structure. The measured results are similar to the full-wave simulation results. In the frequency domain, the measured insertion loss of the SMDCRP structure filter in differential mode (DM) can be kept below -4.838 dB from DC to 32 GHz and can maintain signal integrity characteristics. The common-mode (CM) suppression performance can suppress more than -10 dB from 8.81 GHz to 32.65 GHz. Fractional bandwidth can be increased to 115%, and CM noise can be ameliorated by 55.2%. In the time domain, using eye diagram verification, the filter shows complete differential signal transmission capability and supports a transmission rate of 32 Gb/s for high-speed buses. The SMDCRP structure filter reduces the electromagnetic interference (EMI) problem and meets the quality requirements for the controllers and sensors used in the server and computer systems of high-speed buses.

Accuracy and clinical validity of automated cephalometric analysis using convolutional neural networks.

This study aimed to assess the error range of cephalometric measurements based on the landmarks detected using cascaded CNNs and determine how horizontal and vertical positional errors of individual landmarks affect lateral cephalometric measurements. In total, 120 lateral cephalograms were obtained consecutively from patients (mean age, 32.5 ± 11.6) who visited the Asan Medical Center, Seoul, Korea, for orthodontic treatment between 2019 and 2021. An automated lateral cephalometric analysis model previously developed from a nationwide multi-centre database was used to digitize the lateral cephalograms. The horizontal and vertical landmark position error attributable to the AI model was defined as the distance between the landmark identified by the human and that identified by the AI model on the x- and y-axes. The differences between the cephalometric measurements based on the landmarks identified by the AI model vs those identified by the human examiner were assessed. The association between the lateral cephalometric measurements and the positioning errors in the landmarks comprising the cephalometric measurement was assessed. The mean difference in the angular and linear measurements based on AI vs human landmark localization was .99 ± 1.05°, and .80 ± .82 mm, respectively. Significant differences between the measurements derived from AI-based and human localization were observed for all cephalometric variables except SNA, pog-Nperp, facial angle, SN-GoGn, FMA, Bjork sum, U1-SN, U1-FH, IMPA, L1-NB (angular) and interincisal angle. The errors in landmark positions, especially those that define reference planes, may significantly affect cephalometric measurements. The possibility of errors generated by automated lateral cephalometric analysis systems should be considered when using such systems for orthodontic diagnoses.

Collision avoidance analysis of human-robot physical interaction based on null-space impedance control of a dynamic reference arm plane.

When the terminal upper limb rehabilitation robot is used for motion-assisted training, collisions between the manipulator links and the human upper limb may occur due to the null-space self-motion of the redundant manipulator. A null-space impedance control method based on a dynamic reference arm plane is proposed to realize collision avoidance during human-robot physical interaction motion for the collision problem between the manipulator links and the human upper limb. Firstly, a dynamic model and a Cartesian impedance controller of the manipulator are established. Then, the null-space impedance controller of the redundant manipulator is established based on the dynamic reference plane, which manages the null-space self-motion of the redundant manipulator to prevent collision between the manipulator links and the human upper limb. Finally, it is experimentally verified that the method proposed in this paper can effectively manage the null-space self-motion of the redundant manipulator, and thus achieve collision avoidance during the human-robot physical interaction motion. This research has significant potential in improving the safety and feasibility of motion-assisted training with rehabilitation robots.

Relationship between the amount of mesial movement of the mandibular first molar and angular changes of the mandibular third molar: A retrospective comparative study in growing patients with and without first premolar extraction

The main objective of this study was to assess the relationship of the amount of mesial movement of mandibular first molar (MFM) and angular changes of the mandibular third molar (MTM) in orthodontic patients. The secondary objective of this study was to compare the obtained values in extraction and non-extraction orthodontic patients. In this cross-sectional retrospective study were enrolled all the eligible patients (12-16 years) with and without first premolar extraction that met the inclusion criteria. The angle between the longitudinal axis of MTM and horizontal reference plane (HRP) (to calculate the angular change of MTM), and the distance between the cementoenamel junction of the mesial surface of MFM and the bisector of the anterior nasal spine and nasal septum (to determine the amount of mesial movement of MFM) were measured on pre- and post-treatment panoramic radiographs. The correlation between the two variables was analysed and compared between extraction and non-extraction patients using STATA v. 14.2 software. A total of 100 fixed orthodontic patients with and without first premolar extraction (n=50) whose treatment had been accomplished were included in the study. In the non-extraction group, the mean amount of mesial movement of MFM was 1.45mm and the mean angular change of MTM was 4.28 degrees; this correlation was significant (P<0.05). These values were 2.98mm and 7.17 degrees in the first premolar extraction group, respectively, with a significant correlation (P<0.05). However, the difference in this respect was not significant between the two groups (P>0.05). According to the regression model, 1mm mesial movement of MFM would averagely cause 2.2-degree angular change in MTM, adjusting for the effect of extraction/non-extraction treatment plan. The mesial movement of MFM was significantly correlated with the angular changes of MTM in extraction and non-extraction orthodontic patients, with no significant difference between them.

A new technique to transfer the upper maxillary arch position using a facebow, a transfer table, and a reference block with a CAD application.

This paper presents a method for the digital transfer of the upper maxillary arch position using a facebow, a transfer table, and a reference block with a CAD application without requiring physical casts mounted with articulating gypsum. This technique facilitates the prosthetic digital workflow when the impression is made via intraoral scanning, to obtain the placement of the maxillary arch in the anatomical reference planes and in relation to the axes of rotation of the mandibular movements.

CBCT Based Assessment of First Molar Inclination in Adult Subjects

OBJECTIVE: The ideal buccolingual (BL) inclination of posterior dentition allows maximum intercuspation and avoids occlusal interferences at rest and during jaw movements. Therefore the objective of present study was to evaluate the BL inclination of upper and lower first molars in adult patients. METHODOLOGY: A cross-sectional study was conducted on 52 subjects visited dental clinics. The evaluation of BL inclination of maxillary and mandibular first molars was done using CBCT images on Galaxis/Galileos implant viewer software. After sagittal orientation, the angle was measured on the tranverse plane from long axis of tooth to the vertical line drawn perpendicular to horizontal reference plane. RESULTS: Mandibular first molars were found to be lingually inclined with mean value of 14.07°±4.49°. Similarly, mean buccalinclination of maxillary first molars was found to be 9.42°±6.70°. CONCLUSIONS: It can be concluded from this study that maxillary molars are buccally incline over the basal bone whereas, mandibular molars are lingually inclined. KEYWORDS: Molar, maxilla, mandible, torque

Advances in algorithms for three-dimensional craniomaxillofacial features construction based on point clouds

In light of the increasing digitalization of dentistry, the automatic determination of three-dimensional (3D) craniomaxillofacial features has become a development trend. 3D craniomaxillofacial landmarks and symmetry reference plane determination algorithm based on point clouds has attracted a lot of attention, for point clouds are the basis for virtual surgery design and facial asymmetry analysis, which play a key role in craniomaxillofacial surgery and orthodontic treatment design. Based on the studies of our team and national and international literatures, this article presented the deep geometry learning algorithm to determine landmarks and symmetry reference plane based on 3D craniomaxillofacial point clouds. In order to provide reference for future clinical application, we describe the development and latest research in this field, and analyze and discuss the advantages and limitations of various methods.

INDIAMAN-20 (INstant DIAgnosis of 20 Major ANomalies) protocol: application of IOTA diagnostic strategy to fetal anomalies.

To compile a list of instant diagnoses of major fetal anomalies, and to present their sonographic descriptors and test them in a retrospective series of fetuses with congenital anomalies managed at our center. The first step was to identify major anomalies that meet the following criteria: (1) can be diagnosed directly and unequivocally on the basis of a set of descriptors evident on a single sonographic view; (2) are among those more commonly diagnosed in utero; (3) represent major lethal or life-threatening malformations or anomalies for which no or only palliative surgery can be performed. For each of these anomalies, a sonographic descriptor was produced. The second step was to test the descriptors of the instant diagnoses in all cases with these anomalies evaluated since January 2000 in our Fetal Medicine Unit, retrieved from our electronic database. For each congenital anomaly-sonographic descriptor pair we then evaluated the presence vs absence of an unequivocal relationship between anomaly and descriptor, assessing: whether a particular descriptor in a set reference plane identified unequivocally the anomaly, i.e. the same descriptor could not be found in any other malformation; and whether a particular anomaly presented the corresponding descriptor in all cases. Finally, we evaluated whether the sonographic descriptors could be applied in all trimesters or only selectively, considering the natural history of the 20 instant diagnoses. Of the 20 instant diagnoses, five involved the central/peripheral nervous system (exencephaly/anencephaly, alobar/semilobar holoprosencephaly, cephalocele, open spinal dysraphism, complete agenesis of corpus callosum), five involved the heart (hypoplastic left heart syndrome, complete atrioventricular septal defect, severe Ebstein's anomaly, critical aortic stenosis, transposition of the great arteries), five involved the abdominal wall (left-sided congenital diaphragmatic hernia, exomphalos, gastroschisis, body-stalk anomaly) or gastrointestinal tract (duodenal atresia), and five involved the remaining systems (three urinary (autosomal recessive polycystic kidney disease, multicystic dysplastic kidney, lower urinary tract obstruction), two skeletal (transverse limb defect, radial ray defect)). All of the descriptors were found to identify unequivocally the related malformations, whereas four of the malformations were not associated with their reference descriptors in every case. For example, congenital diaphragmatic hernia (left-sided) presented without the stomach in the thorax in 42/93 (45.2%) cases. Due to the natural history of the condition, not all signs were recognizable in the first trimester, some becoming evident only from the second trimester onwards. We have defined sonographic descriptors that enable an unequivocal instant diagnosis for a list of 20 major congenital anomalies: the 'INDIAMAN-20' (INstant DIAgnosis of Major ANomalies) protocol. We believe that using this approach may facilitate focused training of less experienced operators and that it may be incorporated into artificial intelligence and deep learning protocols, further boosting the instant prenatal recognition of these lethal or life-threatening malformations. © 2022 International Society of Ultrasound in Obstetrics and Gynecology.

ADENIUM — A demonstrator for a next-generation beam telescope at DESY

High-resolution beam telescopes for charged particle tracking are one of the most important and equally demanding infrastructure items at test beam facilities. The main purpose of beam telescopes is to provide precise reference track information of beam particles to measure the performance of a device under test (DUT). In this report the development of the ADENIUM beam telescope (ALPIDE sensor based DESY Next test beam Instrument) as a demonstrator and prototype for a next-generation beam telescope is presented. The ADENIUM beam telescope features up to six pixelated reference planes framed by plastic scintillators for triggering. ADENIUM is capable of replacing the currently used EUDET-type beam telescopes without impacting existing DUT implementations due to the integration of the telescope DAQ into EUDAQ2.In this report the concept and design of the ADENIUM telescope as well as its performance arediscussed. The telescope's pointing resolution is determined in different configurations. For anoptimal setup at a momentum of 5.6 GeV with an ALPIDE as DUT, a resolution better than3 μm has been extracted. No rate limitations have been observed at the DESY II test beam.

Clinical analysis of successful insertion of orthodontic mini-implants in infrazygomatic crest

BackgroundThe insertion positions of mini-implant in infrazygomatic crest has been reported, but due to the anatomical variation, the precise location of this site is not clear yet. This study used cone-beam computed tomography (CBCT) to analyze the position and angle of mini-implants successfully inserted in the infrazygomatic crest, with the goal of providing reference data for clinical practice.MethodsCBCT was used to image 40 mini-implants and their surrounding tissues in adult orthodontic patients who successfully underwent mini-implant insertion in the infrazygomatic crest. The insertion positions and angles of mini-implants were measured, and the thicknesses of buccal and palatal bone adjacent to the mini-implants were also recorded. Then, we proposed the position and implantation angle for infrazygomatic crest insertion. According to the position and angle, the cortical bone thickness and distance to the root of another 54 randomly selected infrazygomatic crests were recorded to verify its feasibility.ResultsIn the coordinate system, the implantation position of the 40 successful mini-implants was (-0.4 ± 2, 8.2 ± 2.5) and the implantation angle between the long axis of the mini-implant and horizontal reference plane was 56.4° ± 7.7°. The bone thicknesses on buccal and palatal sides of infrazygomatic crest adjacent to mini-implants were 4.1 ± 2.5 mm and 7.2 ± 3.2 mm, respectively, and the cortical bone thickness was 2.4 ± 0.6 mm. Among 54 infrazygomatic crests, 75.9% of them met the safety and stability requirements. When the implantation height was increased by 1, 2, and 3 mm, the proportions of implants that met requirements for success were 81.5%, 90.7%, and 94.4%, respectively. But, the proportions of eligible implants were limited at implantation angle increases of 5° and 10°.ConclusionsUsing the long axis of the maxillary first permanent molar (U6) as the vertical reference line, mini-implants could be safely inserted in the infrazygomatic crest at a distal distance of 0.4 mm and height of 8.2 mm from the central cementum-enamel junction of U6, with an implantation angle of 56.4°. The success rate increased when the implant height increased, but the proportion of eligible implantation was limited with the increase of implantation angle.

Rapid texture depth detection method considering pavement deformation calibration

To realize the non-destructive automatic measurement of pavement texture and determine a suitable mean texture depth (MTD) calculation method, a 3D scan method was used to scan the depth information of the pavement surface. The effect of pavement wheel track deformation on 3D pavement texture was considered, and different data calibration methods and reference plane definitions on MTD calculations were compared. Simulating samples with varying abrasion levels for fine-grained and slightly textured pavements through accelerated wear experiments were formed and scanned. This method can measure fine-grade or highly abrasive pavements that cannot be measured using the mean profile depth (MPD) method. Experiments show that the calculation based on the deformation calibration is more conservative. The superiority of the deformation hypothesis and the feasibility of deformation separation were further analyzed and verified on the measured field pavement. The deformation calibration makes the calculated MTDs less volatile.

A novel 2+1 three-dimensional measurement for temporal phase unwrapping

A novel 2+1 three-dimensional (3D) measurement for temporal phase unwrapping is proposed. While two high-frequency complimentary sinusoidal fringes and one low-frequency sinusoidal fringe are projected onto the measured object respectively, the corresponding deformed patterns can be captured. Then the high-frequency wrapped phase can be extracted from the two high-frequency complimentary deformed patterns by high-accuracy computer-generated moiré profilometry (HCGMP). Due to its shared direct current (DC) component with the high-frequency deformed pattern, the low-frequency wrapped phase can still be extracted from the only one low-frequency deformed pattern by HCGMP. Because the extracted wrapped phase excludes the phase contribution of the reference plane, a sufficiently high low-frequency fringe projection can also make the low-frequency wrapped phase become its continuous phase without phase unwrapping. And then, the high-frequency wrapped phase is unwrapped, guided by the low-frequency unwrapped phase. Finally, the 3D profile of the measured object can be reconstructed successfully. Compared to the conventional 2+1 algorithm, the proposed algorithm can accomplish temporal phase unwrapping and improve the 3D measuring accuracy. It has a good application prospect in real-time 3D measurement. The experiment results show the feasibility and effectiveness of the proposed algorithm.

Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature

The theory of polyhedra and the geometric methods associated with it are interesting not only in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author’s work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other. Examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for “basic” sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.

Frankfort Horizontal Deviation Comparison in Natural Head Position on Discrepancy of Skeletal Pattern (Cephalometry Study)

Cephalometry remain imperative in diagnosis and treatment planning. Commonly used craniofacial reference plane such as Frankfort Horizontal (FH), have shortcomings, including their variation individual orientation when Natural Head Position (NHP), so FH cannot be considered a true horizontal. The purpose of this study was to determine deviant FH, to highlight effect of deviant FH on the determination cephalometric values in vertical skeletal discrepancy cases dan to construct a true horizontal line (HOR) as alternate reference plane. Lateral cephalometric roentgenograms from 104 randomly selected patients, then determined FH inclination. Control group was created from FH inclination come near to true horizontal and study group was created from FH inclination away from true horizontal. Independent t test was done for both the groups to compare the values obtained and determine the statistic difference. Statistical analysis in control group shows that there is no significant difference between FH and HOR, which prove that HOR could be used as an alternative to FH. The study group showed a significant statistic difference, resulting in erroneous analysis if the deviant DH was used.