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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
To find the distance between two points on the surface of a spherical earth, let point P1 have a (latitude, longitude) of (ϕ1,θ1) $ (\phi_{1} , \theta_{1} ) $ and point P2 have a (latitude, longitude) of (ϕ2,θ2) $ (\phi_{2} , \theta_{2} ) $ . Two different computational methods are as follows:Let A be the North pole and let B and C be the points P1 and P2. Then the spherical law of cosines gives the desired distance, a, on a sphere of unit radius:
Introduction to Modeling Daylight
Published in Daryl R. Myers, Solar Radiation, 2017
The solar azimuth and zenith angle of the sun are calculated, and the gradation function ϕ(0) is computed. Next, for each patch of sky, say in steps from zenith angle 0° to 90° and steps of azimuth from 0° to 360°, at the desired angular resolution, the angular separation gradation function ϕ(z) and the indicatrix functions f(χ) and f(Zs) are calculated for each desired location in the sky. The indicatrix functions depend on the angular distance χ between the sky patch of interest and the sun. This is computed using the spherical law of cosines: cos(χ)= cos(Zs) cos(Zp)+ sin(Zs) sin(Zp) cos(|Azp−Azs|)
Geodesy Fundamentals
Published in Julio Sanchez, Maria P. Canton, William Perrizo, Space Image Processing, 2018
Julio Sanchez, Maria P. Canton
The problem of calculating the great circle distance between two points has been around for centuries. In spherical trigonometry, the problem consists of solving an oblique spherical triangle, such as the one in Figure 6.14. The conventional methods are based on using the law of sines for solving the sides and the law of cosines for both the sides and the angles. The objection to using the law of cosines is that the inverse cosine is ill-suited for small distances since the difference between the cosine of small angles can be a very small value. For instance, when calculating to seven significant figures, we cannot distinguish cosines of angles smaller than one minute of arc.
Enhenced cell adhesion on collagen I treated parylene-C microplates
Published in Journal of Biomaterials Science, Polymer Edition, 2021
Lijun Zhao, Weiwei Lan, Xiao Dong, Han Xu, Lili Wang, Yan Wei, Jinchuan Hou, Di Huang, Weiyi Chen
3 μm-thick parylene-C microplates were produced. In detail, the parylene was deposited by chemical vapor deposition with a parylene deposition machine (LABCOTER PDS2010, USA) on a glass substrate spin-coated with 0.3% gelatin (Sigma-Aldrich, USA) at 2000 rpm. The parylene-C film was then etched away with O2 plasma (10 mL/min, 25 W) (RIE-10NR, Japan) at the defined regions with photoresist mask that was patterned using standard photolithographic technique. Before removing the photoresist, we coated the glass substrate with MPC (2-methacryloyloxyethyl phosphorylcholine) polymer to inhibit cell adhesion. Specifically, All the groups were then immersed in MPC solution for 3 min. After the MPC coating is dry, the photoresist was washed with acetone to expose the microplate area. Then the glass slides were dipped into the Col-I solution (0.1 mg/mL) and stored at 4 °C overnight. We allowed only one cell to grow on each microplate by adjusting the concentration of the C2C12 cell suspension. While cells were spreading on a microplate, the cell traction force (CTF) on its surface would be generated [40–42]. The adhesion effect between cells and microplates was further observed by folding microplates caused by cell contraction. The edges of individual microplates were pushed with a glass tip manipulated by a micromanipulator. Because of the existence of sacrificial layer, the microplate and substrate would be detached due to manual triggering [38]. The folding angle, θ (between the folded microplate and the glass substrate), was measured by microscope image and the law of cosines.
A scalable mobile context-aware recommender system for a smart city administration
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
For our approach, we are using a multi-criteria rating, where each institution is rated according to four criteria: distance, stay points, citizen rating and opening time each one is calculated as follows: Distance: calculating the distance between the actual position of the citizen and the location of the institution according to the spherical law of cosines [24] which is defined by the following formula: Where ϕ1 and λ1 are respectively the latitude and longitude of the citizen position, and ϕ2 and λ2 are the latitude and longitude of the institution location, as for R represents the radius of the earth which is equivalent to R = 6371 km.
Detecting visually salient scene areas and deriving their relative spatial relations from continuous street-view panoramas
Published in International Journal of Digital Earth, 2020
Fangli Guan, Zhixiang Fang, Tao Yu, Mingxiang Feng, Fan Yang
In Figure 15(a), Pa represents the Baidu street-view panorama, Pb and Pc represent the GoPro panoramas, and Pd represents the homonymy points for the three panoramas. The relative azimuth angles between Pd and Pc, Pd, and Pb were measured using an electronic compass. This enabled the relative distance of Pc and Pd to be calculated using the law of sines. Similarly, the relative distance of Pa and Pc were derived using the same method. The ∠PaPcPd angle was then calculated, and the relative distance of Pa and Pd was derived using the law of cosines. Finally, the relative azimuth angle was derived for Pa and Pd. In Figure 15(b), Pa represents the Baidu street-view panorama, and Pb and Pc represent the GoPro panoramas. Moreover, Pd represents the homonymy point for the three panoramas, and Pe is the projected point of Pd on the ground. The relative elevation relation of Pd was derived based on the distance between Pa and Pe, and the height of Pd and Pe.