The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems.The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system. Symmetry 2019, 11, 191 2 of 17 of the grid. In Figure 1a, the first coordinate value is ascending right-upwardly, the second values are ascending into the right-downward direction, and the third one is ascending into the left-upward direction [7,8]. We should mention that this system could be seen as the extension of the oblique coordinate system using two independent integer values [9] by concerning the third value to obtain zero-sum for every triplet. The digital distance based on the neighborhood relation is computed in Reference [9]. Since the vectors describing the grid are not orthogonal, some geometric descriptions based on Cartesian coordinates are not very clear. However, to simplify the expressions of the constrained three-dimensional coordinate system is recommended. We should also mention that 0-sum triplets allowing real numbers were used in Reference [8] to describe rotations (that may not map the hexagonal grid to itself). In this way, a useful digitization operator is found. Her's system was mentioned and used in References [10,11] for various imaging-related disciplines.Symmetry 2019, 11, 191 3 of 17

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Many environmental parameters affect the performance of solar photovoltaics (PV), such as dust and temperature. In this paper, three PV technologies have been investigated and experimentally analyzed (mono, poly, and flexible monocrystalline) in terms of the impact of dust and thermal energy on PV behavior. Furthermore, a modular neural network is designed to test the effects of dust and temperature on the PV power production of six PV modules installed at Sohar city, Oman. These experiments employed three pairs of PV modules (one cleaned daily and one kept dusty for 30 days). The performance of the PV power production was evaluated and examined for the three PV modules (monocrystalline, polycrystalline, and flexible), which achieved 30.24%, 28.94%, and 36.21%, respectively. Moreover, the dust reduces the solar irradiance approaching the PV module and reduces the temperature, on the other hand. The neural network and practical models’ performance were compared using different indicators, including MSE, NMSE, MAE, Min Abs Error, and r. The Mean Absolute Error (MAE) is used for evaluating the accuracy of the ANN machine learning model. The results show that the accuracy of the predicting power of the six PV modules was considerable, at 97.5%, 97.4%, 97.6%, 96.7%, 96.5%, and 95.5%, respectively. The dust negatively reduces the PV modules’ power production performance by about 1% in PV modules four and six. Furthermore, the results were evident that the negative effect of the dust on the PV module production based on the values of RMSE, which measures the square root of the average of the square’s errors. The average errors in predicting the power production of the six PV modules are 0.36406, 0.38912, 0.34964, 0.49769, 0.46486, and 0.68238.

The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

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