In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.

In this paper, we give the parametric equation of the Bishop frame for a timelike sweeping surface with a unit speed timelike curve in Minkowski 3-space. We introduce a new geometric invariant to explain the geometric properties and local singularities of this timelike surface. We derive the sufficient and necessary conditions for this timelike surface to be a timelike developable ruled surface. Afterwards, we take advantage of singularity theory to give the classification of singularities of this timelike developable surface. Furthermore, we give some representative examples to show the applications of the theoretical results.

Objective: To evaluate the prevalence of deleterious oral habits and its effect on occlusion in school going children from 7 to 13 years old, in Aseer region of Saudi Arabia. Material and Methods: A cross-sectional study was conducted by clinical examination for the total sample of 750, who were attending private/ government schools in Aseer region school. Students were selected by stratified cluster random sampling. Clinical findings of each oral habits, including assessment of malocclusion, were examined involving the following parameters, i.e., crowding, spacing, open bite, crossbite and deep bite. Logistic regression analysis was conducted. Results: Boys had 26.2 % of crowded teeth, while 16.3% of girls were found to have crowded teeth; the difference was found to be significant (p=0.002). Spacing was present in 26.4% of boys and 16.7% girls; a significant difference was found between the two (p=0.002). Sleep bruxism was observed more among male subjects than females with a statistically significant difference in detrimental habits (p<0.001). Those subjects having tongue thrusting are 1.264 times at more risk of having crowded teeth than those with no tongue thrusting. Conclusion: There was a high prevalence of malocclusion associated with harmful oral habits in children. This highlighted the need to implement programs of oral care and health education for preventive orthodontic treatment at an early age.

Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained.

This paper investigated the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group. Then, the sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained. In particular, we mostly focused on the study of the resulting developable surface is a cylinder, cone, or tangent surface. Meanwhile, to support the results in the paper, some illustrative examples are presented.