Ocular surface changes with short-term contact lens wear

Contact lenses are a common method for the correction of refractive errors of the eye. While there have been significant advancements in contact lens designs and materials over the past few decades, the lenses still represent a foreign object in the ocular environment and may lead to physiological as well as mechanical effects on the eye. When contact lenses are placed in the eye, the ocular anatomical structures behind and in front of the lenses are directly affected. This thesis presents a series of experiments that investigate the mechanical and physiological effects of the short-term use of contact lenses on anterior and posterior corneal topography, corneal thickness, the eyelids, tarsal conjunctiva and tear film surface quality. The experimental paradigm used in these studies was a repeated measures, cross-over study design where subjects wore various types of contact lenses on different days and the lenses were varied in one or more key parameters (e.g. material or design). Both, old and newer lens materials were investigated, soft and rigid lenses were used, high and low oxygen permeability materials were tested, toric and spherical lens designs were examined, high and low powers and small and large diameter lenses were used in the studies. To establish the natural variability in the ocular measurements used in the studies, each experiment also contained at least one “baseline” day where an identical measurement protocol was followed, with no contact lenses worn. In this way, changes associated with contact lens wear were considered in relation to those changes that occurred naturally during the 8 hour period of the experiment. In the first study, the regional distribution and magnitude of change in corneal thickness and topography was investigated in the anterior and posterior cornea after short-term use of soft contact lenses in 12 young adults using the Pentacam. Four different types of contact lenses (Silicone hydrogel/ Spherical/–3D, Silicone Hydrogel/Spherical/–7D, Silicone Hydrogel/Toric/–3D and HEMA/Toric/–3D) of different materials, designs and powers were worn for 8 hours each, on 4 different days. The natural diurnal changes in corneal thickness and curvature were measured on two separate days before any contact lens wear. Significant diurnal changes in corneal thickness and curvature within the duration of the study were observed and these were taken into consideration for calculating the contact lens induced corneal changes. Corneal thickness changed significantly with lens wear and the greatest corneal swelling was seen with the hydrogel (HEMA) toric lens with a noticeable regional swelling of the cornea beneath the stabilization zones, the thickest regions of the lenses. The anterior corneal surface generally showed a slight flattening with lens wear. All contact lenses resulted in central posterior corneal steepening, which correlated with the relative degree of corneal swelling. The corneal swelling induced by the silicone hydrogel contact lenses was typically less than the natural diurnal thinning of the cornea over this same period (i.e. net thinning). This highlights why it is important to consider the natural diurnal variations in corneal thickness observed from morning to afternoon to accurately interpret contact lens induced corneal swelling. In the second experiment, the relative influence of lenses of different rigidity (polymethyl methacrylate – PMMA, rigid gas permeable – RGP and silicone hydrogel – SiHy) and diameters (9.5, 10.5 and 14.0) on corneal thickness, topography, refractive power and wavefront error were investigated. Four different types of contact lenses (PMMA/9.5, RGP/9.5, RGP/10.5, SiHy/14.0), were worn by 14 young healthy adults for a period of 8 hours on 4 different days. There was a clear association between fluorescein fitting pattern characteristics (i.e. regions of minimum clearance in the fluorescein pattern) and the resulting corneal shape changes. PMMA lenses resulted in significant corneal swelling (more in the centre than periphery) along with anterior corneal steepening and posterior flattening. RGP lenses, on the other hand, caused less corneal swelling (more in the periphery than centre) along with opposite effects on corneal curvature, anterior corneal flattening and posterior steepening. RGP lenses also resulted in a clinically and statistically significant decrease in corneal refractive power (ranging from 0.99 to 0.01 D), large enough to affect vision and require adjustment in the lens power. Wavefront analysis also showed a significant increase in higher order aberrations after PMMA lens wear, which may partly explain previous reports of "spectacle blur" following PMMA lens wear. We further explored corneal curvature, thickness and refractive changes with back surface toric and spherical RGP lenses in a group of 6 subjects with toric corneas. The lenses were worn for 8 hours and measurements were taken before and after lens wear, as in previous experiments. Both lens types caused anterior corneal flattening and a decrease in corneal refractive power but the changes were greater with the spherical lens. The spherical lens also caused a significant decrease in WTR astigmatism (WRT astigmatism defined as major axis within 30 degrees of horizontal). Both the lenses caused slight posterior corneal steepening and corneal swelling, with a greater effect in the periphery compared to the central cornea. Eyelid position, lid-wiper and tarsal conjunctival staining were also measured in Experiment 2 after short-term use of the rigid and SiHy contact lenses. Digital photos of the external eyes were captured for lid position analysis. The lid-wiper region of the marginal conjunctiva was stained using fluorescein and lissamine green dyes and digital photos were graded by an independent masked observer. A grading scale was developed in order to describe the tarsal conjunctival staining. A significant decrease in the palpebral aperture height (blepharoptosis) was found after wearing of PMMA/9.5 and RGP/10.5 lenses. All three rigid contact lenses caused a significant increase in lid-wiper and tarsal staining after 8 hours of lens wear. There was also a significant diurnal increase in tarsal staining, even without contact lens wear. These findings highlight the need for better contact lens edge design to minimise the interactions between the lid and contact lens edge during blinking and more lubricious contact lens surfaces to reduce ocular surface micro-trauma due to friction and for. Tear film surface quality (TFSQ) was measured using a high-speed videokeratoscopy technique in Experiment 2. TFSQ was worse with all the lenses compared to baseline (PMMA/9.5, RGP/9.5, RGP/10.5, and SiHy/14) in the afternoon (after 8 hours) during normal and suppressed blinking conditions. The reduction in TFSQ was similar with all the contact lenses used, irrespective of their material and diameter. An unusual pattern of change in TFSQ in suppressed blinking conditions was also found. The TFSQ with contact lens was found to decrease until a certain time after which it improved to a value even better than the bare eye. This is likely to be due to the tear film drying completely over the surface of the contact lenses. The findings of this study also show that there is still a scope for improvement in contact lens materials in terms of better wettability and hydrophilicity in order to improve TFSQ and patient comfort. These experiments showed that a variety of changes can occur in the anterior eye as a result of the short-term use of a range of commonly used contact lens types. The greatest corneal changes occurred with lenses manufactured from older HEMA and PMMA lens materials, whereas modern SiHy and rigid gas permeable materials caused more subtle changes in corneal shape and thickness. All lenses caused signs of micro-trauma to the eyelid wiper and palpebral conjunctiva, although rigid lenses appeared to cause more significant changes. Tear film surface quality was also significantly reduced with all types of contact lenses. These short-term changes in the anterior eye are potential markers for further long term changes and the relative differences between lens types that we have identified provide an indication of areas of contact lens design and manufacture that warrant further development.

Structural behaviour and design of cold-formed steel wall systems under fire conditions

In recent times, light gauge steel framed (LSF) structures, such as cold-formed steel wall systems, are increasingly used, but without a full understanding of their fire performance. Traditionally the fire resistance rating of these load-bearing LSF wall systems is based on approximate prescriptive methods developed based on limited fire tests. Very often they are limited to standard wall configurations used by the industry. Increased fire rating is provided simply by adding more plasterboards to these walls. This is not an acceptable situation as it not only inhibits innovation and structural and cost efficiencies but also casts doubt over the fire safety of these wall systems. Hence a detailed fire research study into the performance of LSF wall systems was undertaken using full scale fire tests and extensive numerical studies. A new composite wall panel developed at QUT was also considered in this study, where the insulation was used externally between the plasterboards on both sides of the steel wall frame instead of locating it in the cavity. Three full scale fire tests of LSF wall systems built using the new composite panel system were undertaken at a higher load ratio using a gas furnace designed to deliver heat in accordance with the standard time temperature curve in AS 1530.4 (SA, 2005). Fire tests included the measurements of load-deformation characteristics of LSF walls until failure as well as associated time-temperature measurements across the thickness and along the length of all the specimens. Tests of LSF walls under axial compression load have shown the improvement to their fire performance and fire resistance rating when the new composite panel was used. Hence this research recommends the use of the new composite panel system for cold-formed LSF walls. The numerical study was undertaken using a finite element program ABAQUS. The finite element analyses were conducted under both steady state and transient state conditions using the measured hot and cold flange temperature distributions from the fire tests. The elevated temperature reduction factors for mechanical properties were based on the equations proposed by Dolamune Kankanamge and Mahendran (2011). These finite element models were first validated by comparing their results with experimental test results from this study and Kolarkar (2010). The developed finite element models were able to predict the failure times within 5 minutes. The validated model was then used in a detailed numerical study into the strength of cold-formed thin-walled steel channels used in both the conventional and the new composite panel systems to increase the understanding of their behaviour under nonuniform elevated temperature conditions and to develop fire design rules. The measured time-temperature distributions obtained from the fire tests were used. Since the fire tests showed that the plasterboards provided sufficient lateral restraint until the failure of LSF wall panels, this assumption was also used in the analyses and was further validated by comparison with experimental results. Hence in this study of LSF wall studs, only the flexural buckling about the major axis and local buckling were considered. A new fire design method was proposed using AS/NZS 4600 (SA, 2005), NAS (AISI, 2007) and Eurocode 3 Part 1.3 (ECS, 2006). The importance of considering thermal bowing, magnified thermal bowing and neutral axis shift in the fire design was also investigated. A spread sheet based design tool was developed based on the above design codes to predict the failure load ratio versus time and temperature for varying LSF wall configurations including insulations. Idealised time-temperature profiles were developed based on the measured temperature values of the studs. This was used in a detailed numerical study to fully understand the structural behaviour of LSF wall panels. Appropriate equations were proposed to find the critical temperatures for different composite panels, varying in steel thickness, steel grade and screw spacing for any load ratio. Hence useful and simple design rules were proposed based on the current cold-formed steel structures and fire design standards, and their accuracy and advantages were discussed. The results were also used to validate the fire design rules developed based on AS/NZS 4600 (SA, 2005) and Eurocode Part 1.3 (ECS, 2006). This demonstrated the significant improvements to the design method when compared to the currently used prescriptive design methods for LSF wall systems under fire conditions. In summary, this research has developed comprehensive experimental and numerical thermal and structural performance data for both the conventional and the proposed new load bearing LSF wall systems under standard fire conditions. Finite element models were developed to predict the failure times of LSF walls accurately. Idealized hot flange temperature profiles were developed for non-insulated, cavity and externally insulated load bearing wall systems. Suitable fire design rules and spread sheet based design tools were developed based on the existing standards to predict the ultimate failure load, failure times and failure temperatures of LSF wall studs. Simplified equations were proposed to find the critical temperatures for varying wall panel configurations and load ratios. The results from this research are useful to both structural and fire engineers and researchers. Most importantly, this research has significantly improved the knowledge and understanding of cold-formed LSF loadbearing walls under standard fire conditions.

An example of cognitive obstacles in advanced integration: the case of scalar line integrals

Cognitive obstacles that arise in the teaching and learning of scalar line integrals, derived from cognitive aids provided to students when first learning about integration of single variable functions are described. A discussion of how and why the obstacles cause students problems is presented and possible strategies to overcome the obstacles are outlined.

On selecting an optimal wavelet for detecting singularities in traffic and vehicular data

Serving as a powerful tool for extracting localized variations in non-stationary signals, applications of wavelet transforms (WTs) in traffic engineering have been introduced; however, lacking in some important theoretical fundamentals. In particular, there is little guidance provided on selecting an appropriate WT across potential transport applications. This research described in this paper contributes uniquely to the literature by first describing a numerical experiment to demonstrate the shortcomings of commonly-used data processing techniques in traffic engineering (i.e., averaging, moving averaging, second-order difference, oblique cumulative curve, and short-time Fourier transform). It then mathematically describes WT’s ability to detect singularities in traffic data. Next, selecting a suitable WT for a particular research topic in traffic engineering is discussed in detail by objectively and quantitatively comparing candidate wavelets’ performances using a numerical experiment. Finally, based on several case studies using both loop detector data and vehicle trajectories, it is shown that selecting a suitable wavelet largely depends on the specific research topic, and that the Mexican hat wavelet generally gives a satisfactory performance in detecting singularities in traffic and vehicular data.

A new tip area function for instrumented nanoindentation at extremely small contact depths

The accuracy of measurement of mechanical properties of a material using instrumented nanoindentation at extremely small penetration depths heavily relies on the determination of the contact area of the indenter. Our experiments have demonstrated that the conventional area function could lead to a significant error when the contact depth was below 40. nm, due to the singularity in the first derivation of the function in this region and thus, the resultant unreasonable sharp peak on the function curve. In this paper, we proposed a new area function that was used to calculate the contact area for the indentations where the contact depths varied from 10 to 40. nm. The experimental results have shown that the new area function has produced better results than the conventional function. © 2011 Elsevier B.V.

Time dependency of molecular rate estimates and systematic overestimation of recent divergence times

Studies of molecular evolutionary rates have yielded a wide range of rate estimates for various genes and taxa. Recent studies based on population-level and pedigree data have produced remarkably high estimates of mutation rate, which strongly contrast with substitution rates inferred in phylogenetic (species-level) studies. Using Bayesian analysis with a relaxed-clock model, we estimated rates for three groups of mitochondrial data: avian protein-coding genes, primate protein-coding genes, and primate d-loop sequences. In all three cases, we found a measurable transition between the high, short-term (<1–2 Myr) mutation rate and the low, long-term substitution rate. The relationship between the age of the calibration and the rate of change can be described by a vertically translated exponential decay curve, which may be used for correcting molecular date estimates. The phylogenetic substitution rates in mitochondria are approximately 0.5% per million years for avian protein-coding sequences and 1.5% per million years for primate protein-coding and d-loop sequences. Further analyses showed that purifying selection offers the most convincing explanation for the observed relationship between the estimated rate and the depth of the calibration. We rule out the possibility that it is a spurious result arising from sequence errors, and find it unlikely that the apparent decline in rates over time is caused by mutational saturation. Using a rate curve estimated from the d-loop data, several dates for last common ancestors were calculated: modern humans and Neandertals (354 ka; 222–705 ka), Neandertals (108 ka; 70–156 ka), and modern humans (76 ka; 47–110 ka). If the rate curve for a particular taxonomic group can be accurately estimated, it can be a useful tool for correcting divergence date estimates by taking the rate decay into account. Our results show that it is invalid to extrapolate molecular rates of change across different evolutionary timescales, which has important consequences for studies of populations, domestication, conservation genetics, and human evolution.

Investigating the change in three dimensional deformity for idiopathic scoliosis using axially loaded MRI

Background: Adolescent idiopathic scoliosis is a complex three-dimensional deformity, involving a lateral deformity in the coronal plane and axial rotation of the vertebrae in the transverse plane. Gravitational loading plays an important biomechanical role in governing the coronal deformity, however, less is known about how they influence the axial deformity. This study investigates the change in three-dimensional deformity of a series of scoliosis patients due to compressive axial loading. Methods: Magnetic resonance imaging scans were obtained and coronal deformity (measured using the coronal Cobb angle) and axial rotations measured for a group of 18 scoliosis patients (Mean major Cobb angle was 43.4 o). Each patient was scanned in an unloaded and loaded condition while compressive loads equivalent to 50% body mass were applied using a custom developed compressive device. Findings: The mean increase in major Cobb angle due to compressive loading was 7.4 o (SD 3.5 o). The most axially rotated vertebra was observed at the apex of the structural curve and the largest average intravertebral rotations were observed toward the limits of the coronal deformity. A level-wise comparison showed no significant difference between the average loaded and unloaded vertebral axial rotations (intra-observer error = 2.56 o) or intravertebral rotations at each spinal level. Interpretation: This study suggests that the biomechanical effects of axial loading primarily influence the coronal deformity, with no significant change in vertebral axial rotation or intravertebral rotation observed between the unloaded and loaded condition. However, the magnitude of changes in vertebral rotation with compressive loading may have been too small to detect given the resolution of the current technique.

Unit commitment in power systems with plug-in hybrid electric vehicles

With the continued development of renewable energy generation technologies and increasing pressure to combat the global effects of greenhouse warming, plug-in hybrid electric vehicles (PHEVs) have received worldwide attention, finding applications in North America and Europe. When a large number of PHEVs are introduced into a power system, there will be extensive impacts on power system planning and operation, as well as on electricity market development. It is therefore necessary to properly control PHEV charging and discharging behaviors. Given this background, a new unit commitment model and its solution method that takes into account the optimal PHEV charging and discharging controls is presented in this paper. A 10-unit and 24-hour unit commitment (UC) problem is employed to demonstrate the feasibility and efficiency of the developed method, and the impacts of the wide applications of PHEVs on the operating costs and the emission of the power system are studied. Case studies are also carried out to investigate the impacts of different PHEV penetration levels and different PHEV charging modes on the results of the UC problem. A 100-unit system is employed for further analysis on the impacts of PHEVs on the UC problem in a larger system application. Simulation results demonstrate that the employment of optimized PHEV charging and discharging modes is very helpful for smoothing the load curve profile and enhancing the ability of the power system to accommodate more PHEVs. Furthermore, an optimal Vehicle to Grid (V2G) discharging control provides economic and efficient backups and spinning reserves for the secure and economic operation of the power system

Modeling random effect and excess zeros in road traffic accident prediction

Poisson distribution has often been used for count like accident data. Negative Binomial (NB) distribution has been adopted in the count data to take care of the over-dispersion problem. However, Poisson and NB distributions are incapable of taking into account some unobserved heterogeneities due to spatial and temporal effects of accident data. To overcome this problem, Random Effect models have been developed. Again another challenge with existing traffic accident prediction models is the distribution of excess zero accident observations in some accident data. Although Zero-Inflated Poisson (ZIP) model is capable of handling the dual-state system in accident data with excess zero observations, it does not accommodate the within-location correlation and between-location correlation heterogeneities which are the basic motivations for the need of the Random Effect models. This paper proposes an effective way of fitting ZIP model with location specific random effects and for model calibration and assessment the Bayesian analysis is recommended.

Performance characteristics of compound curved sandwich shell structures

Sandwich shells have recently emerged as aesthetically pleasing, efficient and economical structural systems, with a number of applications. They combine the advantages of sandwich layer technology together with those of shell action. With different materials and thicknesses used in the sandwich layers, their performance characteristics largely remain un-quantified and there are no guidelines at present for their design. This research paper provides verification, through finite element modeling and testing, for the application of this technology to dome styled dwellings with research currently being conducted into the further application to roofing and floor structures.

Clinical performance of a new silicone hydrogel contact lens over 6 months of extended wear

Purpose: Silicone hydrogel contact lenses (CLs) are becoming increasingly popular for daily wear (DW), extended wear (EW) and continuous wear (CW), due to their higher oxygen transmissibility compared to hydrogel CLs. The aim of this study was to investigate the clinical and subjective performance of asmofilcon A (Menicon Co., Ltd), a new surface treated silicone hydrogel CL, during 6-night EW over 6 months (M). Methods: A prospective, randomised, single-masked, monadic study was conducted. N=60 experienced DW soft CL wearers were randomly assigned to wear either asmofilcon A (test: Dk=129, water content (WC)=40%, Nanogloss surface treatment) or senofilcon A (control: Dk=103, WC=38%, PVP internal wetting agent, Vistakon, Johnson & Johnson Vision Care) CLs bilaterally for 6 M on an EW basis. A PHMB-preserved solution (Menicon Co., Ltd) was dispensed for CL care. Evaluations were conducted at CL delivery and after 1 week (W), 4 W, 3 M and 6 M of EW. At each visit, a range of objective and subjective clinical performance measures were assessed. Results: N=50 subjects (83%) successfully completed the study, with the majority of discontinuations due to loss to follow-up (n=3) or moving away/travel (n=5). N=2 subjects experienced adverse events; n=1 unilateral red eye with asmofilcon A and n=1 asymptomatic infiltrate with senofilcon A. There were no significant differences in high or low contrast distance visual acuity (HCDVA or LCDVA) between asmofilcon A and senofilcon A; however, LCDVA decreased significantly over time with both CL types (p<0.05). The two CL types did not vary significantly with respect to any of the objective and subjective measures assessed (p>0.05); CL fitting characteristics and CL surface measurements were very similar and mean bulbar and limbal redness measures were always less than grade 1.0. Superior palpebral conjunctival injection showed a statistically, but not clinically, significant increase over time with both CL types (p<0.05). Corneal staining did not vary significantly between asmofilcon A and senofilcon A (p>0.05), with low median gradings of less than 0.5 observed for all areas assessed. There were no solution-related staining reactions observed with either CL type. The asmofilcon A and senofilcon A CLs were both rated highly with respect to overall comfort, with medians of 14 or 15 hours of comfortable lens wearing time per day reported at each of the study visits (p>0.05). Conclusions: Over 6 months of EW, the asmofilcon A and senofilcon A CLs performed in a similar manner with respect to visual acuity, ocular health and CL performance measures. Some changes over time were observed with both CL types, including reduced LCDVA and increased superior palpebral injection, which warrant further investigation in longer-term EW studies. Asmofilcon A appeared to be equivalent in performance to senofilcon A.

Validation of a new hand-held electronic data capture method for continuous monitoring of subjective appetite sensations

Background When large scale trials are investigating the effects of interventions on appetite, it is paramount to efficiently monitor large amounts of human data. The original hand-held Electronic Appetite Ratings System (EARS) was designed to facilitate the administering and data management of visual analogue scales (VAS) of subjective appetite sensations. The purpose of this study was to validate a novel hand-held method (EARS II (HP® iPAQ)) against the standard Pen and Paper (P&P) method and the previously validated EARS. Methods Twelve participants (5 male, 7 female, aged 18-40) were involved in a fully repeated measures design. Participants were randomly assigned in a crossover design, to either high fat (>48% fat) or low fat (<28% fat) meal days, one week apart and completed ratings using the three data capture methods ordered according to Latin Square. The first set of appetite sensations was completed in a fasted state, immediately before a fixed breakfast. Thereafter, appetite sensations were completed every thirty minutes for 4h. An ad libitum lunch was provided immediately before completing a final set of appetite sensations. Results Repeated measures ANOVAs were conducted for ratings of hunger, fullness and desire to eat. There were no significant differences between P&P compared with either EARS or EARS II (p > 0.05). Correlation coefficients between P&P and EARS II, controlling for age and gender, were performed on Area Under the Curve ratings. R2 for Hunger (0.89), Fullness (0.96) and Desire to Eat (0.95) were statistically significant (p < 0.05). Conclusions EARS II was sensitive to the impact of a meal and recovery of appetite during the postprandial period and is therefore an effective device for monitoring appetite sensations. This study provides evidence and support for further validation of the novel EARS II method for monitoring appetite sensations during large scale studies. The added versatility means that future uses of the system provides the potential to monitor a range of other behavioural and physiological measures often important in clinical and free living trials.

A randomised controlled trial to prevent hospital readmissions and loss of functional ability in high risk older adults: a study protocol

Background Older people have higher rates of hospital admission than the general population and higher rates of readmission due to complications and falls. During hospitalisation, older people experience significant functional decline which impairs their future independence and quality of life. Acute hospital services comprise the largest section of health expenditure in Australia and prevention or delay of disease is known to produce more effective use of services. Current models of discharge planning and follow-up care, however, do not address the need to prevent deconditioning or functional decline. This paper describes the protocol of a randomised controlled trial which aims to evaluate innovative transitional care strategies to reduce unplanned readmissions and improve functional status, independence, and psycho-social well-being of community-based older people at risk of readmission. Methods/Design The study is a randomised controlled trial. Within 72 hours of hospital admission, a sample of older adults fitting the inclusion/exclusion criteria (aged 65 years and over, admitted with a medical diagnosis, able to walk independently for 3 meters, and at least one risk factor for readmission) are randomised into one of four groups: 1) the usual care control group, 2) the exercise and in-home/telephone follow-up intervention group, 3) the exercise only intervention group, or 4) the in-home/telephone follow-up only intervention group. The usual care control group receive usual discharge planning provided by the health service. In addition to usual care, the exercise and in-home/telephone follow-up intervention group receive an intervention consisting of a tailored exercise program, in-home visit and 24 week telephone follow-up by a gerontic nurse. The exercise only and in-home/telephone follow-up only intervention groups, in addition to usual care receive only the exercise or gerontic nurse components of the intervention respectively. Data collection is undertaken at baseline within 72 hours of hospital admission, 4 weeks following hospital discharge, 12 weeks following hospital discharge, and 24 weeks following hospital discharge. Outcome assessors are blinded to group allocation. Primary outcomes are emergency hospital readmissions and health service use, functional status, psychosocial well-being and cost effectiveness. Discussion The acute hospital sector comprises the largest component of health care system expenditure in developed countries, and older adults are the most frequent consumers. There are few trials to demonstrate effective models of transitional care to prevent emergency readmissions, loss of functional ability and independence in this population following an acute hospital admission. This study aims to address that gap and provide information for future health service planning which meets client needs and lowers the use of acute care services.

An exploration of affine group laws for elliptic curves

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.