Face Width Research Articles

Wear prediction and meshing characteristics for the planetary gear set considering angular misalignment and rotating carrier

This paper focuses on the effect of angular misalignment on wear prediction of the planetary gear set with the rotating carrier, which is more complex than that of the parallel-axis gear due to the position changing of the planet gears. The wear depth is calculated based on the Archard’s wear theory, the load distribution is obtained by the three dimensional loaded tooth contact analysis (3D-LTCA) method, and the effect of angular misalignment is considered by introducing the change in meshing clearance between the two engaged tooth flanks. The load distribution, the time-varying mesh stiffness (TVMS) and the backlash of gear pairs are presented under different wear levels. The results show that, with the combined effect of angular misalignment and rotating carrier, the wear depth on both sides of the face width is larger than that in the middle, and the difference is the greatest at the root. Moreover, TVMS is affected not only by the tooth wear, but also by the rotating carrier, and the cycle of TVMS and the backlash is half of that of the carrier rotation.

Gender Determination Using Nasofacial Anthropometry in the Iranian Population

Objectives: The aim of this study is to predict the gender among Iranian population by determining the facial and nasal indices.
 Methods: Four hundred healthy individuals including 200 males and 200 females (age range of 20-25 years) with normal craniofacial configuration took part in the present study. Facial and nasal variables including facial length (FL), facial width (FW), facial index (FI), nasal length (NL), nasal height (NH), nasal width (NW), and nasal index (NI) measured and SPSS 22.0 software was used for statistical analysis.
 Results: The mean age was 22.16±2.44 years. The mean FL, FW, FI, NL, NW, and NI were 11.53±0.592 cm, 12.54±1.41 cm, 92.67±6.24 cm, 6.02±0.53 cm, 2.78±0.25 cm, and 90.79±8.22 cm, respectively. There were significant differences in the mean value of all nasofacial parameters (P<0.05). The most prominent face and nose types were hyperleptoprosopic (46%) and platyrrhine (57.75%) among Iranian population. The highest sensitivity and specificity to predict the gender dimorphism were related to FL and FW (.990 and 0.97) and (.970 and 0.94), respectively.
 Conclusion: Nasofacial parameters such as FL and FW that are able to sexual dimorphism with high sensitivity and specificity can be helpful in forensic medicine. Also, the nasofacial parameters are useful in in reconstructive, and aesthetic surgeries.

A descriptive study of Cephalic and prosopic anthropometric indices in one-day-old infants in Imam Ali and Kamali Hospital, Karaj

Objective: There is no published literature about the types of head and face shapes in the Alborz newborns. Therefore, we designed this study to report these important regional data.
 Methods: Head length, head width, face length, face width, prosopic and cephalic indices were made on 150 newborns (75 males and 75 females) with a view to establish the criterion of cephalometry for this age group in Iran. A special emphasis was given to delivery mode in this study.
 Results: There was no significant difference in cephalic and prosopic indices between two groups (female: p=0.46; male: p=0.43). Brachycephalic type was dominant and dolicocephalic type was rare in male and female neonates. There was no significant difference in head shape between two groups. Hypereuriprosopic and hyperleptoprosopic were dominant and rare types, respectively, in newborns without significant difference between two groups.
 Conclusion: The head and face indices in the newborn infants born by cesarean section were not significantly different from vaginal delivery group. However, the role of nutrition and climate could not be ruled out.

Profiles of facial soft tissue changes during and after orthodontic treatment in female adults

BackgroundSome female adults who received orthodontic treatment often complain about thinner faces, protruding cheekbones and sunken temples, even at the beginning of treatment. The present research aimed to explore facial soft tissue changes during and after orthodontic treatment, as well as the related factors affecting facial soft tissue changes.MethodsThis study used 3dMD stereo photography technology to compare facial soft tissue changes among adult females who received orthodontics. A total of 52 adult females (24 teeth extraction, 28 non-teeth extraction cases) were included and potential correlations between related factors (facial morphology features, the change of occlusal height and dental arch width) were evaluated during different treatment periods.ResultsOverall, 3D-negative soft tissue changes of the buccal region moderately correlated with distances of nasion-menton and subnasale-menton (both rs = 0.30, P < 0.05) as well as the ratio of subnasale-menton/right gonion-left gonion (rs = 0.33, P < 0.05) and nasion-menton/right zygomatic point-left zygomatic point (rs = 0.30, P < 0.05). Among the 3D angle measurements, the right chelion-median upper lip point-left chelion angle was found to have a moderate correlation with 3D negative changes of the upper cheilion region (rs = 0.31, P < 0.01). Analysis of occlusal height changes revealed that an increase in the posterior facial height (nasion-gonial distance) may be a risk factor for negative changes in the zygomatic arch area. In addition, a moderate positive correlation between the maxillary first molar width changes and 3D-negative changes of the lower cheek region was found (rs = 0.41, P < 0.05).ConclusionsAfter orthodontic treatment, adult females with wide and short faces may be prone to negative soft tissue changes. Changes of posterior facial height and arch width between the first molars were also risk factors for negative changes of facial soft tissues. Extraction is not a major factor producing facial soft tissue changes.

Design considerations and stress analysis of helical gear drives with unequal and offset face widths

Design considerations and stress analysis of helical gear drives with unequal and offset face widths

Assessing the capacity of dredging excavations to place dredging tailings dumps in the course of deep and man-made placer mining

Introduction. The dredge stern is often backed up by dredging tailings in the course of deep and man-made placers dredging. One of the reasons for that is the decreased capacity of dredging excavations at the dumping point caused by collapsed pit walls, which is not taken into account by known design methods. Research objective is to evaluate the degree of the developed space capacity reduction at dredging pits for dredging tailings storage, depending on the dredging conditions. Methods of research. Capacity prediction for a dredging excavation for dredging tailings storage should consider natural slope formation of open pit walls during the period of dredge advance to a distance equal to the distance of the poop decks from the face. The degree of reduction in dredging excavation lower part receiving capacity should be set based on pit walls flattening, which depends on rock composition (cohesiveness degree), loose deposits thickness, face width, and dredge working parameters. Results. The results of graphic-analytical calculation revealed that dredging excavation lower part receiving capacity is reduced from 3 to 10% due to pit walls collapse as loose deposits thickness increases from 10 to 34 m for the most favorable face width of a 380-liter dredge, which may result in a significant increase in dredge stern backing up by dredging tailings. Conclusions. At deep and man-made placers, when calculating the parameters of a dredging tailings dump, it is recommended to introduce an amendment into the existing methods, that considers the developed space lower part capacity reduction due to pit walls collapse.

친환경 천마스크 디자인을 위한 마스크 패턴 개발 연구

The purpose of this study was to reduce the use of disposable masks, which are one of the main causes of environmental pollution, and to present the basic data for the design of functional and stable eco-friendly cloth masks that take into account both environment and health. The 2D type masks used in the study were selected from the top 9 masks ranked by Naver Shopping (www.naver.com). The 9 masks collected were size L (large), which corresponds to the size of an adult. The mask shape of the L size was compared and analyzed with Size Korea (2010, 2020) using human body dimensions for women in their 20s and 50s. The mask sizes and morphological features of the 9 brands were as follows: First, the length of the vertical line was measured. This is the basic line required for the mask pattern design and is used to compute the ‘Face length’ of Size Korea (2010, 2020) and the ‘Mask vertical length’ of the mask. The results show that the vertical line length between 15.12 to 15.62 cm, was suitable for the size L (15 to 17cm). Second, ‘Face Breadth’, ‘Mask width 1 (Upper)’ and ‘Mask width 2 (Lower)’ were used as horizontal lines of the mask. Ideally, the Mask width 1 should be between 11 to 11.8 cm and Mask width 2 between 10 to 10.6 cm. Third, the dimensions required to form the curved surface of the mask are ‘Nasal dart’ and ‘Jaw dart’. The ideal sizes are 3.5 to 4.5 cm for the “Nasal dart” and 3.0 to 3.7 cm for the “Jaw dart”. A “Jaw dart” 0.5 to 1 cm smaller than the “Nasal dart,” would result in a mask with a good fit. The conclusions of this study are expected to be used as basic data for the design of masks, an important tool to protect our health and safety amid the global pandemic.

Evaluation of Naso-facial proportion of the Face and Their Correlation in Bangladeshi Buddhist Rakhain Ethnic Males

Background: The face and nasal dimensions are most important cephalometric parameters describing human morphology. Nose is the most important aesthetic unit of face. There is wide variation in size and shape of nose and face due to ethnic influences. Objective: The purpose of the present study was to evaluate the naso-facial proportion of the face and their correlation in adult healthy Bangladeshi Buddhist Rakhain Ethnic males. Methodology: This cross-sectional observational study was carried out in the Department of Anatomy, Bangabandhu Sheikh Mujib Medical University (BSMMU), Dhaka from January 2011 to December 2011 for a period of one (01) year. Adult Healthy Bangladeshi Buddhist Rakhain males with the age group of 18 to 30 years were included as study population. Results: This study shows mean of nose width was 41.16 (±3.03) and mean of ¼ face width was 34.80(±1.70). This study shows nose width was 41.16 (±3.03) and ¼ face width was 34.80(±1.70). Conclusion: It was observed that nose width was significantly wider than ¼ of face width in the adult healthy Bangladeshi Buddhist Rakhain. This parameter was the importance of using patients as observers, as well as clinicians, in facial attractiveness research. Journal of Current and Advance Medical Research, January 2022;9(1):32-35

Facial Anthropometry and Analysis in Egyptian Women.

Background:Human facial analysis can be considered both an art and a science, and is used extensively to measure soft tissue proportions. Remarkable changes exist in anthropometric measures due to changes over centuries of geographical, genetic, and environmental factors, as well as waves of migration causing facial proportions to vary among the different ethnic groups. The purpose of this study was to establish facial soft tissue norms for the Egyptian female population between the ages of 18 and 50, as well as several age-related changes in facial measurements that are described fairly scarcely in the literature.Methods:A prospective study was carried out on 300 Egyptian women between June 2019 and December 2020. All were volunteers and between 18 and 50 years old. Facial measurements were obtained by both direct (caliper-based) and indirect (3D Crisalix software) anthropometric analysis. Anthropometric measurements used in the study included seventeen facial measurements derived from different anthropometric soft tissue landmarks from each subject and eight measurements for analysis.Results:Our results were able to describe the average facial and nasal measurements of the Egyptian female population, as well as the horizontal and vertical analysis of facial proportions. We also determined similarities with other Middle Eastern female measurements, with subtle differences in facial width and nasal height and width compared with Turkish and Iranian women.Conclusions:The current study is valuable because it delivers facial anthropometric measures for Egyptian female faces. This can provide a database for a multitude of uses, including operative planning, postoperative measurements, and forensic and ergonomic purposes.

Association Between Breastfeeding Type and Duration and the Molar and Facial Characteristics of Preschoolers Aged 2 To 6 Years: A Cross-Sectional Study.

To calculate the association between the type and duration of breastfeeding and the deciduous molar relationship and facial characteristics of preschoolers aged 2 to 6 years. This crosssectional study included 160 pre-schoolers aged 2 to 6 years old, enrolled in the 2019 academic year from 3 public schools in Data collection involved two phases. The first phase consisted in determining the facial type, anteroposterior and vertical profile, facial dimensions, and the molar relationship between primary second molars. The second phase included interviews with the parents or legal guardians. Children with exclusive breastfeeding and with mixed/artificial feeding (breastfeeding and bottle-feeding, or exclusive artificial bottle-feeding) were included. Chi-square test, t-test, multiple linear regression, and binary logistic regression tests were applied (p<0.05). The facial length, width, and the lower third were greater in the exclusive breastfeeding group than in the mixed/artificial feeding group (3.74mm, p=0.002; 2.06mm, p=0.047; and 2.94mm, p=0.015; respectively). In addition, children with breastfeeding for more than six months showed greater facial length, facial width and lower third (3.20mm, p=0.038; 3.68mm, p=0.006; and 3.57mm, p=0.026; respectively). The duration of exclusive breastfeeding influenced facial width, noting an increase of 3.32mm (p=0.011) if a pre-schooler had exclusive breastfeeding for more than six months. There were no significant associations between type and duration of breastfeeding and the molar terminal plane, facial type, or profile. However, facial dimensions were greater in the exclusive breastfeeding group, and when breastfeeding was longer than six months.

Head and facial dimensions of Chilean workers for design purposes and the differences with other populations.

An appropriate match between a product and its end-users requires anthropometric data, which show variations among different countries. Proper Personal Protection Equipment (PPE) fit is key for safety and comfort. Chile had no head and facial anthropometric data available in order to design face/head PPE. To describe face/head anthropometric characteristics of Chilean workers. Additionally, this study compared those dimensions against other populations (United States (US), South Korea and China). An anthropometric survey involving 21 measures was conducted between September 2013 to May 2016 using stratified sampling. The measurements were based on ISO/TS 16976-2 and ISO 15535 to ensure the highest standards possible, and a total of 474 workers, aged from 18 to 66 years old, participated in the survey. The biggest differences were in Neck circumference, Weight, Nose breadth, Nose protrusion, Bitragion chin arc, Face length, Subnasale-sellion length, Face width, Bigonial breadth and Bitragion subnasal arc. Head length of Chileans were longer than Chinese and South Korean ones, but shorter than US Head length. Chilean Head breadth is smaller than Chinese and South Korean ones. Chileans Face length was the largest observed. Face width of Chileans was smaller than US and Chinese ones. Gender specific differences were also observed in the inter-country comparisons. Chilean males had larger anthropometric dimensions than females. Chilean Head and face dimensions differed significantly when compared against US, South Korean and Chines Head and Face dimensions. Chileans have longer and narrower faces with wider mouths, with a head size in between the US and Asian heads. Face and head PPE for Chileans should use dimensions in the current paper in order to ensure proper fit.

Changes in facial width according to the ostectomy level of the proximal bone segment in intraoral vertical ramus osteotomy for mandibular prognathism

BackgroundThis study aimed to investigate the changes in facial width according to the ostectomy level of the proximal segment after orthognathic surgery using intraoral vertical ramus osteotomy (IVRO) in patients with mandibular prognathism.MethodsThe participants included 32 individuals who were diagnosed with class III malocclusion prior to surgery. All participants underwent orthognathic surgery using either version of IVRO. The surgery patients were categorized into two groups depending on the type of proximal bone-segment ostectomy technique used: patients whose osteotomy height was at the level of the mandibular tooth occlusal surface (the mandibular tooth surface–level group) and patients whose osteotomy height was at the level of the mandibular inferior border (the mandibular inferior border–level group). The distances between the mandibular width and soft tissue width at the height of the sigmoid notch, mandibular foramen, and alveolar bone and at the anterior-posterior location of the mandibular condyle, mandibular foramen, and coronoid process were compared between the groups. All data were compared to identify differences between preoperative and postoperative measurements.ResultsThe postoperative change in facial soft tissue width at the intersection of the coronal plane with the coronoid process and the horizontal plane at the height of the mandibular alveolar bone in the group with osteotomy at the level of the mandibular occlusal surface differed significantly from that in the group with osteotomy at the level of the mandibular inferior border, with respective increases (mean ± SD) of 1.3 ± 3.5% and 4.7 ± 5.6%, compared to preoperative measurements (p = 0.050).ConclusionsProximal segment ostectomy at the level of the mandibular occlusal surface must be considered with regard to postoperative facial soft tissue width in vertical ramus osteotomy. Additionally, it is necessary to study the visual effect of the width of the mandible appearing small because of the posterior position of the mandible, even when the mandibular facial width is maintained.

3D Scanner’s Accuracy In Different Races Of Head And Face Measurement For Ergonomic Design

Introduction: 3D scanning of the head and face has been growing to a vital technique in the field of anthropometrics such as the creation of a sizing chart for medical wearables and motorcycle helmet. This research project targets to identify the optimize setting so that we can obtain better accuracy of head and face scanning for various races. Methods: The review process involved few factors such as brightness, skin colour, glossiness, scanning angle, head and face shape and scanning duration. From the 3 chosen factors, scanner brightness, skin colour, and process layout, we have conducted DOE experiment to identify the significant factors affecting the 3D scanner accuracy. Results: The ANOVA result of experiment imply that Factor A-Scanner Brightness is the most significant factor (p&lt;0.0001) in affecting the 3D scanning result of head circumference and face width. Factor B-Skin Colour is only significant factor affecting face width dimension. Factor C-Process Layout is not significant. Increasing the scanner brightness will further worsen the accuracy of 3D scanner due to increased light exposure to 3D scanner camera. Factor B-Skin Colour only appears to be significant on the face width dimension because some of the landmark stickers are pasted on the grey wig cap. Conclusion: In coherent with the objective discussed, the study of significant factors from scanner brightness to the skin colour has been identified for the 3D scanner accuracy which creates a milestone to contribute to the diagnostic of skin cancer, forensic studies and medical wearable product design.

P104. ASSOCIATIONS OF MASCULINITY-FEMININITY, FACIAL RATIOS, AND ATTRACTIVENESS: INSIGHTS FROM AN ANALYSIS OF 827 INDIVIDUALS

PURPOSE: Males and females have known differences in anatomic facial features coding for sex, which are important for planning gender affirming facial surgery. This study hypothesized certain anatomic facial ratios are associated with perceptions of masculinity-femininity and sex. METHODS: This study used Chicago Face Database, a domain of facial measurements and metrics by 1,087 raters. Sex and ratings of masculinity-femininity were compared to facial anatomic indices, which characterized facial feature relativity (higher ratio scores indicated greater facial feature representation), including “nasal width ratio”= (interalar width)/(face width)*100, nasal length, eye height, eye width, forehead, lip, and chin. RESULTS: In 828 individuals, femininity was associated with eye height/width, lip, and nasal length indices, while masculinity was associated with chin, forehead, and nasal width indices (p<0.01). Among males alone (n=406), femininity was associated with eye height, lip, and nasal length indices; among females alone (n=421), masculinity was associated with nasal width and chin indices (p<0.01). Multivariate logistic regression models revealed nasal width and chin indices interact to predict male sex (OR=1.012, p=0.010), while nasal length and lip indices interact to predict female sex (OR=0.987, p=0.001). CONCLUSION: Our data suggest nasal length and lip indices interact to predict female sex while nasal width and chin indices interact to predict male sex. An understanding of these facial proportions may benefit those who perform facial feminization/masculinization surgery.

Comparison of Face Width and Frame Design between Korean and Chinese Spectacle Wearers

Comparison of Face Width and Frame Design between Korean and Chinese Spectacle Wearers

First study on phenotypic and morphological characteristics of Malaysian Kedah-Kelantan cattle (Bos indicus) and method of estimating their body weight.

Background and Aim:Indigenous Kedah-Kelantan (KK) cattle are well adapted with distinguished reproductive capabilities; they account for more than 70% of the domestic beef production in Malaysia. The published literature on the phenotypic and morphometric characteristics of KK cattle are sparse and require further improvement. Therefore, this study was aimed to determine the phenotypic and morphometric characteristics of Malaysian KK cattle and method of estimating live body weight (BW).Materials and Methods:Morphometric and phenotypic measurements were taken from 184 KK cattle (102 males and 82 females) sourced from three regions. Each animal’s color pattern was recorded for their coat, muzzle, face, eyelashes, horns, tail switch, hoof, and legs through visual observation. Length measurements were taken of the body, face, ear, horn, tail, and rump. Several morphological features such as length, width, and girth were measured using a measuring tape, while wither height and hip height were assessed with a measuring scale.Results:Brown is the predominant coat color in KK cattle (>82%). The overall means of head length, face width (FW), ear length, horn length, wither height, heart girth (HG), body length (BL), and rump length were 42.5±4.5, 17.3±2.9, 19.8±3.1, 9.9±4.4, 104.3±7.1, 127.4±13.2, 98.3±12.3, and 32.4±4.1 cm, respectively. Different morphometric parameters of length, width, and circumference were significantly ( p<0.01) larger in males than females, except for tail length and TG. Correlation coefficient and multiple regression analysis clearly revealed that BL is the best parameter for estimating live BW in KK cattle.Conclusion:Phenotypic and morphometric measurements in this study showed that Malaysian KK cattle generally possess a brown coat pattern with smaller body size, while BL revealed to be the best parameter to predict BW. The data generated from this study would be useful as baseline data for the identification and selection of KK cattle based on their phenotypical- and morphological-features for further improvement of this breed.

Facial width to height ratio and perceived aggression: The disjunction effect of horizontal and vertical components

Facial width-to-height ratio (fWHR) is associated with aggressive behavior and perceived aggression. This study examined the independent effects of the horizontal and vertical components of fWHR on general aggression perception and specific aggression in romantic relationship (intimate partner violence, IPV) perception. By increasing width, the horizontal component of fWHR significantly increases the IPV perception. By reducing height, the vertical component of fWHR significantly decreases the IPV perception (Study 1). Upper and lower facial heights were reduced simultaneously in the vertical manipulation, indicating mixed effects. In Study 2, we separated the upper and lower face effects. Reducing upper face height could facilitate aggression perception. However, reducing the lower face could suppress aggression perception. The results indicated the disjunction effect of horizontal and vertical components on aggression perception. Our findings highlighted the effect of the lower face on aggression perception, which was ignored in previous studies.

Size and Perception of Facial Features with Selfie Photographs, and Their Implication in Rhinoplasty and Facial Plastic Surgery.

Patients increasingly use photographs taken with a front-facing smartphone camera-"selfies"-to discuss their goals with a plastic surgeon. The purpose of this study was to quantify changes in size and perception of facial features when taking a selfie compared to the gold standard of clinical photography. Thirty volunteers took three series of photographs. A 12-inch and 18-inch series were taken with a front-facing smartphone camera, and the 5-foot clinical photography series was taken with a digital single-lens reflex camera. Afterward, subjects filled out the FACE-Q inventory, once when viewing their 12-inch selfies and once when viewing their clinical photographs. Measurements were taken of the nose, lip, chin, and facial width. Nasal length was, on average, 6.4 percent longer in 12-inch selfies compared to clinical photography, and 4.3 percent longer in 18-inch selfies compared to clinical photography. The alar base width did not change significantly in either set of selfies compared to clinical photography. The alar base to facial width ratio represents the size of the nose in relation to the face. This ratio decreased 10.8 percent when comparing 12-inch selfies to clinical photography (p < 0.0001) and decreased 7.8 percent when comparing 18-inch selfies to clinical photography (p < 0.0001). This study quantifies the change in facial feature size/perception seen in previous camera-to-subject distance studies. With the increasing popularity of front-facing smartphone photographs, these data allow for a more precise conversation between the surgeon and the patient. In addition, the authors' findings provide data for manufacturers to improve the societal impact of smartphone cameras. Diagnostic, III.

Facial indices and morphology: A study among Bono and Ewe adult population in the Bono region, Ghana

Facial parameters are of utmost importance in identification and racial morphological classification. Facial indices serve as identification tools when used alongside hand print patterns for forensic and biometric purposes such as health insurance, voters’ identity card, driver's license and passports among others. There is paucity of information on facial types and their distribution patterns with respect to ethnicity, age and sex in Ghana. Therefore, the aim of this study was to generate baseline data on facial dimensions among the Bonos and Ewes living in the Bono Region of Ghana. A total of 291 healthy individuals (152 Bonos and 139 Ewes), aged 18–60 years were recruited for the study. The results of the present study showed that male facial parameters were numerically higher than those of the female participants. The predominant facialtype was hyperleptoprosopic (98.96%) and the least was leptoprosopic (1.03%). There were more Bono participants (51.89%) that were hyperleptoprosopic compared to the Ewes (47.07%). Also, among the hyperleptoprosopic participants, there were more males (55.32%) than females (43.29%). Very few males (0.003%) and females (0.006%) were leptoprosopic.The Bono participants recorded significantly greater morphological facial height and facial breadth than the Ewes. The results of the present study affirm population variation in facial anthropometry and reinforce the need to establish specific facial standards for each tribe of the Ghanaian population. This study has provided data for facial morphology for biometric and forensic purposes as well as facial reconstruction in Ghana.

Using variable modulus to modify a rack cutter and generate a gear pair

In this paper, two normal imaginary helical rack cutters were first established. One of these cutters is a skewed-rack cutter with an asymmetrical straight edge. The other is a rack cutter with an asymmetric parabolic profile. Second, the gear’s tooth surface of the asymmetric parabolic rack cutter is modified to be barrel-shaped based on a variable modulus. The tooth thickness of the gear is gradually reduced along the face width of the tooth from the middle of the tooth surface. Then the coordinate relationship between the gears’ blanks and the imaginary helical rack cutters was established. Through the differential geometry, crowned and uncrowned helical gear pairs were generated. Because of human factors, when the gear pair is installed, it is easy to cause the gear pair edge contact. It is necessary to add artificial assembly error settings through the tooth contact analysis to investigate the kinematic errors and contact conditions of the crowned and uncrowned helical gear pair. The mathematical models and analysis methods proposed for the crowned imaginary rack cutter using variable modulus should be useful for the design and production of double crowned helical gears with asymmetric parabolic teeth.