This approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.

We considered the spacelike sweeping surface with rotation minimizing frames at Minkowski 3-space E31
. We presented the new geometric invariant to demonstrate geometric properties and local singularities for this surface. 'en, we derived sufficient and necessary conditions of the surface to become developable ruled surfaces. Additionally, its singularities are studied. Finally, examples are illustrated to explain the applications of the theoretical results.

In this work, we introduce a line congruence as surface in the space of lines in terms of the
E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces.
In addition, the well known equation of the Plucker’s conoid has been obtained and its kinematicgeometry
are examined in details. At last, an example of application is investigated and explained in
detail.

We investigate a spacelike sweeping surface with rotation minimizing frames at Minkowski 3–Space E31. Then, we statemany results related to the differential geometry of sweeping surfaces that are resulted from these frames. Subsequently, the problem of constructing spacelike/timelike developable surfaces from that sweeping surfaces is analysed.

In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is
a non-constant linear function of its dual arc length parameter. Thereafter, a dual dierential equation
of third order is constructed for every dual curve. Then, several well-known characterizations of dual
spherical, normal and rectifying curves are consequences of this dierential equation. Finally, we
prove a simple new characterization of dual spherical curves in terms of the Darboux vector.

We consider a timelike sweeping surface with rotation minimizing frames in Minkowski 3-
Space E31. We introduce a new geometric “invariant”, which demonstrates the geometric
properties and local singularities of the surface. Subsequently, we give the sufficient
and necessary conditions for this surface to be a developable ruled surface. Finally, the
singularities of these ruled surfaces are investigated.

In this paper, we introduce a method for determination of spacelike ruled surface from the coordinates and the first
derivatives of the base curve by making use of dual vector calculus. Consequently, we discuss the method for spacelike developable
ruled surfaces, and obtain a linear differential equation of first order. Finally, this method is demonstrated through several examples.

In the Middle East, water shortage is becoming more and more serious due to the development of agriculture and industry and the increase in population. Saudi Arabia is one of the most water-consuming countries in the Middle East, and urgent measures are needed. Therefore, we integrated data from Gravity Recovery and Climate Experiment (GRACE), and other relevant data to estimate changes in groundwater storage in Saudi Arabia. The findings are as follows: 1) Average annual precipitation (AAP) was calculated to be 76.4, 90, and 72mm for the entire period, Period I (April 2002 to March 2006) and Period II (April 2006 to July 2016), respectively. 2) The average TWS variation was estimated to be −7.94 ± 0.22, −1.39 ± 1.35, and −8.38 ± 0.34 mm/yr for the entire period, Period I and Period II, respectively. 3) The average groundwater storage was estimated to be +1.56 ± 1.35 mm/yr during Period I. 4) The higher average groundwater depletion rate was calculated to be −6.05 ± 0.34 mm/yr during Period II. 5) Both soil texture and surface streams in the study area promote lateral flow and carry surface water to the Arabian Gulf and the Red Sea. 6) During Period II, average annual recharge rates were estimated to be +9.48 ± 2.37 and +4.20 ± 0.15 km3 for Saudi Arabia and the Saq aquifer, respectively. 7) This integrated approach is an informative and cost-effective technique to assess the variability of groundwater resources in large areas more efficiently.

This study used land gravity and airborne magnetic data to investigate the depth to the magmatic chamber and map the heat flow distribution beneath the active volcanoes of Hawaii Island using the Curie point depth (CPD) and gravity modeling. Obtaining some of the ground-based geophysical measurements was problematic due to accessibility limitations; therefore, this study used available data. The CPD and magnetic data were used to map the depth to the bottom of the magnetic layer by calculating the depth to the Curie isotherm (540°C) beneath Hawaii Island. The spectral peak method was used to calculate the depths to the shallow and deep magnetic sources for the entire island, and the CPD was calculated using the centroid method. A two-dimensional density model for two Earth layers was constructed using forward modeling of the gravity data. A large plume of dense intrusive material was observed beneath the three adjacent volcanoes of Mauna Loa, Mauna Kea, and Kilauea, and two small chambers were found to be located beneath the Kohala and Hualalai volcanoes. Based on the gravity modeling results, the depth to the magma layer varied from 0.5 to 10 km, and the heat flow was higher close to the volcanic eruption zones. The current study is informative and cost effective for the world's most active volcanic areas.