ISSN 1794-6190 e-ISSN 2339-3459 https://doi.org/10.15446/esrj.v26n3.100754 EARTH SCIENCES RESEARCH JOURNAL Earth Sci. Res. J. Vol. 26, No. 2 (September, 2022): 205 - 210 C O M PU T E R S C IE N C E A PP L IE D T O G E O PH Y SI C S Rigorous spherical bearing with Soldner coordinates and azimuth angles on sphere Sebahattin Bektaş Ondokuz Mayis University, Turkey sbektas@omu.edu.tr ABSTRACT Meridian systems, called Soldner coordinates (parallel coordinate) systems, have found wide application in geodesy. In particular, the meridian system constitutes a suitable base for the Gauss-Kruger projection of the ellipsoid and the sphere. Soldner coordinates can be used in Cassini-Soldner projection without any processing. As it is known, the di- rections of the edges are shown with azimuth angles in the geographic coordinate system and the bearing angles in the Soldner coordinate system. Bearing or azimuth angles are frequently used in geodetic calculations. These angles give the direction of sides in the clockwise direction from a certain initial direction. Both angle values range from 0 to 360 degrees and are usually calculated from the arctan function. But the arctan function returns an angle value between -90 and +90 degrees. Therefore, it is necessary to analyze the quarter for the angle found. For practical computations, the quadrants of the arctangents are determined by the signs of the numerator and denominator in the tangent formulas. Determining the quarter of the angles is done with if…, then…, end..., blocks on the computer. It should be noted that each comparison requires a separate processing time. This study will be given how to calculate both bearing and azimu- th angles with direct formulas without any need to examine them. In addition, a solution proposal will be given against the division by zero errors in the bearing and azimuth angles calculations. Keywords: Bearing Angles; Azimuth Angles; Geographical coordinates; Soldner coordinates; Meridian convergence; Zero division error; Rodamiento esférico riguroso con coordenadas Soldner y ángulos azimut en la esfera RESUMEN Los sistemas meridianos, llamados coordenadas de Soldner (coordenadas paralelas), han gozado de una amplia aplicación en geodesia. En particular, el sistema meridiano constituye una base adecuada para la proyección de Gauss-Kruger del elipsoide y la esfera. Las coordenadas de Soldner se pueden usar en la proyección Cassini-Soldner sin ningún procesa- miento. Como es conocido, las direcciones de los bordes se muestran como el ángulo acimut en el sistema geográfico de coordenadas y los ángulos rumbo en el sistema de coordenadas de Soldner. Los ángulos rumbo y acimut se usan frecuen- temente en los cálculos geodéticos. Estos ángulos dan la dirección de los lados en el sentido del reloj desde una dirección inicial específica. Ambos ángulos van de 0 a 360 grados y usualmente se calculan desde la función arctan. El resultado de la función arctan, sin embargo, está en el rango de -90 y +90 grados. Además, es necesario analizar el cuartil del ángulo resultado. Para computaciones prácticas, los cuadrantes de los arcotangentes se determinan por los signos del numerador y denominador en las fórmulas tangentes. La determinación de los cuartiles de los ángulos se hace en el computador con los bloques si..., entonces..., y fin... Se debe tener en cuenta que cada comparación requiere un tiempo de procesamiento por separado. Este estudio presenta cómo calcular los ángulos rumbo y acimuth con formulas directas sin necesidad de examinarlas. Adicionalente, se dará una propuesta de solución frente a la división por cero errores en el cálculo de los ángulos de rumbo y acimut. Palabras clave: ángulos rumbo; ángulos azimuth; coordenadas geográficas; coordenadas Soldner; convergencia meridiana; error de división cero. Record Manuscript received: 25/02/2022 Accepted for publication: 27/10/2022 How to cite item: Bektaş, S. (2022). Rigorous spherical bearing with Soldner coordinates and azimuth angles on sphere. Earth Sciences Research Journal, 26(3), 205-210. https://doi.org/10.15446/esrj.v26n3.100754 https://doi.org/10.15446/esrj.v26n3.100754 mailto:sbektas@omu.edu.tr https://doi.org/10.15446/esrj.v26n3.100754 206 Sebahattin Bektaş 1. Introduction The reciprocal bearing angles are exactly 180 degrees different on the plane. Unlike the plane on the sphere surface, the difference between reciprocal bearing or azimuth angle is approximately 180 degrees. Therefore, while a single bearing angle is sufficient in plane calculations, it is desired to find reciprocal bearing or azimuth angles separately on surfaces such as spheres and ellipsoids. Classic the bearing calculation or the azimuth calculation formulas only work correctly if the side is in the 1st quarter. If the side is located in the other quarters, the angles of the bearing or azimuth should be examined. For example, if the angle value given by the arctan function is positive, the bearing or azimuth angle is in the 1st or 3rd quarter. If the angle value is negative, the angle is in quarter 2 or 4. In order to find the value of the angle as univocal, the angle values found from the classical formulas need to be examined. Bearing or azimuth angles are critical in geodetic calculations. It is impossible to produce correct position information with wrong bearing or azimuth angles. Calculation of angles is done with the inverse (second) geodetic problem. While solving the inverse geodetic problem in order to find reciprocal bearing or azimuth angles, first, the angle from 1 to 2 is found. As a result of the analysis, if the 1st bearing or azimuth angle is wrong, it is likely to be wrong in the opposite (from 2 to 1) bearing or azimuth. In the proposed method, reciprocal bearing or azimuth angles are obtained independently and directly. In our literature research, we came found some studies only for determining plane bearing angle (Antes, 1990; Breuer et al., 1985; Gellert et al., 1972) demonstrated direct bearing angle determination in the plane without quarter analysis. Hekimoglu (1991) also examined the condition problem and the direct formulas. We could not find any studies that give a direct bearing and azimuth angle on the spherical surface. Formulas that give a direct azimuth angle on the sphere surface without examining the quarter with geographical coordinates are shown by the author in (Bektas, 2019; Bektas, 2021). For the first time, in this study, formulas that give a direct bearing angle without examining the quarter with Soldner coordinates on the sphere surface will be discussed. On the other hand, a solution proposal will be given against the division by zero errors in the bearing and azimuth angles calculations. In this paper, firstly, bearing and azimuth angle, meridian system and Soldner coordinates on sphere will be mentioned in order to form a basis for the subject. After the relationship between azimuth, bearing angle and meridian convergence is given, direct azimuth angle determination with geographical coordinates, then direct bearing angle determination with Soldner coordinates will be discussed. 2. Direct Azimuth Angles with Geographical Coordinates Figure 1. Problem on Sphere Classically, the reciprocal azimuth angles (A12, A21) from two points whose geographic coordinates P1(φ1, λ1), P2(φ2,λ2) are given obtained from the following formulas (Fig.1). (1) (2) It is important to remember that these classic formulas only work correctly if the side is in the 1st quarter. If the side is located in the other quarters, the angles of the bearing angle should be examined. The necessary additions should be made for correct angles according to Table-1 seen below. Table 1. Fixed value to add for azimuth angles Quadrant ∆φ=φ2−φ1 ∆λ=λ2− λ1 fixed value to add for A12 fixed value to add for A21 1.Quadrant + + - - 2.Quadrant + - +180o +180o 3.Quadrant - - +180o -180o 4.Quadrant - + +360o - In this proposed method, formulas are given for obtaining the azimuth angle directly without any examination. For direct determination of azimuth by geographic coordinates, we provide the below formulas. In this proposed method, formulas are given to obtain the azimuth angle directly without any examination. The proposed method can calculate direct azimuth angles on the sphere and ellipsoid surface (Bektas, 2019). I=sinΔ λ II=tan 2cos 1 − sin 1cosΔ III=tan 1cos 2 − sin 2cosΔ A 12 = 2.arctan( −√ 2 + 2 ) + 180 A 21 = 2.arctan(− +√ 2 + 2 ) + 180 (3)I=sinΔ λ II=tan 2cos 1 − sin 1cosΔ III=tan 1cos 2 − sin 2cosΔ A 12 = 2.arctan( −√ 2 + 2 ) + 180 A 21 = 2.arctan(− +√ 2 + 2 ) + 180 (4) I=sinΔ λ II=tan 2cos 1 − sin 1cosΔ III=tan 1cos 2 − sin 2cosΔ A 12 = 2.arctan( −√ 2 + 2 ) + 180 A 21 = 2.arctan(− +√ 2 + 2 ) + 180 (5) I=sinΔ λ II=tan 2cos 1 − sin 1cosΔ III=tan 1cos 2 − sin 2cosΔ A 12 = 2.arctan( −√ 2 + 2 ) + 180 A 21 = 2.arctan(− +√ 2 + 2 ) + 180 (6) I=sinΔ λ II=tan 2cos 1 − sin 1cosΔ III=tan 1cos 2 − sin 2cosΔ A 12 = 2.arctan( −√ 2 + 2 ) + 180 A 21 = 2.arctan(− +√ 2 + 2 ) + 180 (7) The given formulas in Equations 6 and 7 do not work if the denominator equals zero. The program will be interrupted because division by zero is done. This problem also arises in the use of classical formulas. In order not to interrupt the program, very small numbers are added to the numerator and denominator at a rate that will not affect the result. If μ1=1E – 15,... μ2 =1E – 27,... are selected, ten digit sensitive bearing angle values are obtained (Hekimoğlu, 1991). The division by zero error will be avoided if the following formulas (Eq. 8 and Eq. 9) are used. A12 = 2.arctan ( +µ1 µ2+ −√ 2 + 2 ) + 180 A21 = 2.arctan ( +µ1 µ2− +√ 2 + 2 ) + 180 (8) A12 = 2.arctan ( +µ1 µ2+ −√ 2 + 2 ) + 180 A21 = 2.arctan ( +µ1 µ2− +√ 2 + 2 ) + 180 (9) 207 Rigorous spherical bearing with Soldner coordinates and azimuth angles on sphere 3. Direct Bearing Angles with Soldner Coordinates in Meridian System on Sphere This meridian system, as the name suggests, is based on the meridian. The meridian that defines the system is called the prime meridian, and a meridian near the study area is taken as the prime meridian (Heck, 2003).oSoldner coordinates system, have found wide application in geodesy. In particular, the Soldner coordinates system constitutes a suitable base for projection of the ellipsoid and the sphere. Soldner coordinates can be used in Cassini-Soldner projection without any processing. In this system, the x-axis is the selected prime meridian (Figure 2). The coordinates of a point P1 on the sphere in the meridian system are defined as follows. The spherical distance (arcs of a great circle) from the P1 point to the prime meridian (P1F1) is the y value of the P1 point (y1 = P1F1). The distance of the perpendicular point F1 to the starting point Po on the prime meridian also gives the x coordinate (x1 = PoF1). The starting point of Po can also be taken on the equator. Figure 2. Meridian System and Soldner Coordinates (Bektas, 2021) These sides used in the calculations on the sphere s, y1, x1, y2, x2 in Figure 2 must be arcs of a great circle. Point O in Figure 2 becomes the pole of the prime meridian circle. So it is always OF1= OF2 = OPo = π/2. In the Soldner coordinate system, the reference direction of the bearing angles of the sides is the parallel y=constant circle drawn to the prime meridian at that point (Figure-2). The prime meridian determines the Soldner system. Each meridian system creates a separate coordinate system. The ordinates of the points (y) to the left of the prime meridian are negative. Likewise, the abscissas of points (x) below the starting point take negative values. The azimuth angle is used for the direction values of the sides in main problem solutions with geographical coordinates. The azimuth angle (A12, A21) is the angle from the north wing of the meridian at a point to the clockwise side (Fig. 2). The bearing angle is used for the direction values of the sides in main problem solutions with Soldner coordinates meridian system. The bearing angle (α12, α21) is from the north wing of the parallel to the prime meridian at a point to the clockwise side (Fig. 2). Figure 3. Azimuth angle, bearing angle, meridian convergence (Bektas, 2021) There is a meridian convergence ( γ ) between bearing and azimuth angle (Fig. 3). Meridian convergence can be easily calculated from the geographic or Soldner coordinates of the point. If the bearing angle (α) is given, it is converted to the azimuth angle by using the meridian convergence at that point A = α + γ (10) For example, γ1 meridian convergence on the P1 point tan γ1 = sin φ1 tan( λ1 -λprime) from geographic coordinates (11) tan 1 = 1 1 𝛾𝛾 from Soldner coordinates (12) Reciprocal angles between two points whose coordinates are given in geodesy are obtained by solving the inverse (second) geodetic problem. Classically, the reciprocal bearing angles (α12, α21) from two points whose Soldner coordinates P1(y1, x1), P2(y2, x2) are given obtained from the following formulas (Fig. 4). Solving P1P2O spherical triangle Figure 4. Inverse geodetic problem in the meridian system on the sphere (Bektas, 2021) dy = y2 - y1 (13) α12 = arctan ( ) ( 1 ) ( )− ( 2 ) ( 1 ) α21 = arctan ( ) ( 1 ) ( 2 )− ( 2 ) ( ) (14) α12 = arctan ( ) ( 1 ) ( )− ( 2 ) ( 1 ) α21 = arctan ( ) ( 1 ) ( 2 )− ( 2 ) ( ) (15) The bearing angle values found from the classical formula need to be examined. According to the quarter where the angle is located, the additions in the Table 2 below should be made. Table 2. Fixed value to add for bearing angles Quadrant ∆y=y2− y1 fixed value to add for α12 fixed value to add for α21 1 and 2 + 90o 270o 3 and 4 - 270o +90o 208 Sebahattin Bektaş The following values of I, II and III are calculated to find the direct bearing angle without examining the sphere surface. dx = x2 - x1 (16) = ( 1 ) ( 2 ) − ( 1 ) ( ) = ( ) = ( 2 ) ( 1 ) − ( 2 ) ( ) α12 = 2.arctan ( −√ 2 + 2 ) + 180 α21 = 2.arctan(− −√ 2 + 2 ) + 180 (17) = ( 1 ) ( 2 ) − ( 1 ) ( ) = ( ) = ( 2 ) ( 1 ) − ( 2 ) ( ) α12 = 2.arctan ( −√ 2 + 2 ) + 180 α21 = 2.arctan(− −√ 2 + 2 ) + 180 (18) = ( 1 ) ( 2 ) − ( 1 ) ( ) = ( ) = ( 2 ) ( 1 ) − ( 2 ) ( ) α12 = 2.arctan ( −√ 2 + 2 ) + 180 α21 = 2.arctan(− −√ 2 + 2 ) + 180 (19) = ( 1 ) ( 2 ) − ( 1 ) ( ) = ( ) = ( 2 ) ( 1 ) − ( 2 ) ( ) α12 = 2.arctan ( −√ 2 + 2 ) + 180 α21 = 2.arctan(− −√ 2 + 2 ) + 180 (20) = ( 1 ) ( 2 ) − ( 1 ) ( ) = ( ) = ( 2 ) ( 1 ) − ( 2 ) ( ) α12 = 2.arctan ( −√ 2 + 2 ) + 180 α21 = 2.arctan(− −√ 2 + 2 ) + 180 (21) The given formulas in Equation 20 and Equation 21 do not work if the denominator equals zero. The program will be interrupted because division by zero is done. This problem also arises in the use of classical formulas. In order not to interrupt the program, very small numbers are added to the numerator and denominator at a rate that will not affect the result. If μ1=1E – 15,... μ2 =1E – 27 are selected, ten-digit sensitive bearing angle values are obtained. The division by zero error will be avoided if the following formulas Equation 22 and Equation 23 are used. α12 = 2.arctan ( +µ1 µ2+ −√ 2 + 2 ) + 180 α21 = 2.arctan( +µ1 µ2− −√ 2 + 2 ) + 180 (22) α12 = 2.arctan ( +µ1 µ2+ −√ 2 + 2 ) + 180 α21 = 2.arctan( +µ1 µ2− −√ 2 + 2 ) + 180 (23) 4. Numerical Example In order to show the validity of the given formulas, numerical applications were made separately in the east and west of Greenwich in both the northern and southern hemispheres. Inverse problem was solved between twenty-point pairs in total. The radius of the earth is taken as R=6370000m, the prime meridian is taken as λprime=30 o to the east of Greenwich and λprime= -30 o to the west of Greenwich. In numerical applications (Y, X) Soldner coordinates and (γ) meridian convergences of twenty-point pairs are calculated whose (φ, λ) geographic coordinates are given. Then, the bearing (α) and the azimuth angles (A) calculations were made separately with direct formulas. In order to control between the bearing and azimuth angles, the azimuth angle was found by adding the meridian convergence to the bearing angle (Aij=αij +γi) between each pair of points. Numerical application results can be seen in Table 3. Table 3. The bearing and azimuth angles calculations from direct formula φ1= 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2= 32.0 λ2= 31.0 Y2= 94282.5003 X2= 3558115.1919 γ2= 0.5299579646 A12= 11.9669826036 A21=192.2245420693 α12= 11.7169778439 α21=191.6945841047 A12=α12+γ1= 11.9669826036 φ1= 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2= 29.0 λ2= 32.0 Y2= 194466.7324 X2= 3225792.8907 γ2= 0.9699205898 A12=127.0797659802 A21=307.8184614062 α12=126.8297612205 α21=306.8485408164 A12=α12+γ1=127.0797659802 φ1= 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2= 28.0 λ2= 29.0 Y2= -98162.7839 X2= 3113371.4602 γ2= -0.4695087290 A12=213.6299774554 A21= 32.9026204737 α12=213.3799726956 α21= 33.3721292026 A12=α12+γ1=213.6299774554 φ1= 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2= 32.0 λ2= 29.0 Y2= -94282.5003 X2= 3558115.1919 γ2= -0.5299579646 A12=327.6475721969 A21=146.8748649794 α12=327.3975674372 α21=147.4048229439 A12=α12+γ1=327.6475721969 φ1= 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2= 30.0 λ2= 31.0 Y2= 96281.2941 X2= 3335744.3496 γ2= 0.5000380801 A12= 89.8749994050 A21=270.1250005950 α12= 89.6249946453 α21=269.6249625149 A12=α12+γ1= 89.8749994050 φ1= 0.0 λ1= 30.5 Y1= 55588.7367 X1= 0.0000 γ1= 0.0000000000 φ2= 0.0 λ2= 32.0 Y2= 222354.9467 X2= 0.0000 γ2= 0.0000000000 A12= 90.0000000000 A21=270.0000000000 α12= 90.0000000000 α21=270.0000000000 A12=α12+γ1= 90.0000000000 209 Rigorous spherical bearing with Soldner coordinates and azimuth angles on sphere φ1 = 30.0 λ1= 30.5 Y1= 48141.1054 X1= 3335429.2308 γ1= 0.2500047597 φ2 = 29.0 λ2= 30.5 Y2= 48618.8595 X2= 3224249.5773 γ2= 0.2424095173 A12=180.0000000000 A21=360.0000000000 α12=179.7499952403 α21=359.7575904827 A12=α12+γ1=180.0000000000 φ1 = 30.0 λ1= 0.0 Y1= -2852692.3684 X1= 3745576.5846 γ1=-16.1021137520 φ2 = 29.0 λ2= 0.0 Y2= -2883094.7939 X2= 3626800.4993 γ2=-15.6371970759 A12=180.0000000000 A21=360.0000000000 α12=196.1021137520 α21= 15.6371970759 A12=α12+γ1=180.0000000000 φ1= 30.0 λ1= -30.5 Y1= -48141.1054 X1= 3335429.2308 γ1= -0.2500047597 φ2= 32.0 λ2= -31.0 Y2= -94282.5003 X2= 3558115.1919 γ2= -0.5299579646 A12=348.0330173964 A21=167.7754579307 α12=348.2830221561 α21=168.3054158953 A12=α12+γ1=348.0330173964 φ1= 30.0 λ1= -30.5 Y1= -48141.1054 X1= 3335429.2308 γ1= -0.2500047597 φ2= 29.0 λ2= -32.0 Y2= -194466.7324 X2= 3225792.8907 γ2= -0.9699205898 A12=232.9202340198 A21= 52.1815385938 α12=233.1702387795 α21= 53.1514591836 A12=α12+γ1=232.9202340198 φ1= 30.0 λ1= -30.5 Y1= -48141.1054 X1= 3335429.2308 γ1= -0.2500047597 φ2= 28.0 λ2= -29.0 Y2= 98162.7839 X2= 3113371.4602 γ2= 0.4695087290 A12=146.3700225446 A21=327.0973795263 α12=146.6200273044 α21=326.6278707973 A12=α12+γ1=146.3700225446 φ1= 30.0 λ1= -30.5 Y1= -48141.1054 X1= 3335429.2308 γ1= -0.2500047597 φ2= 32.0 λ2= -29.0 Y2= 94282.5003 X2= 3558115.1919 γ2= 0.5299579646 A12= 32.3524278031 A21=213.1251350206 α12= 32.6024325628 α21=212.5951770561 A12=α12+γ1= 32.3524278031 φ1= -30.0 λ1= 30.5 Y1= 48141.1054 X1= -3335429.2308 γ1= -0.2500047597 φ2= -32.0 λ2= 31.0 Y2= 94282.5003 X2= -3558115.1919 γ2= -0.5299579646 A12=168.0330173964 A21=347.7754579307 α12=168.2830221561 α21=348.3054158953 A12=α12+ γ1=168.0330173964 φ1= -30.0 λ1= 30.5 Y1= 48141.1054 X1= -3335429.2308 γ1= -0.2500047597 φ2= -29.0 λ2= 32.0 Y2= 194466.7324 X2= -3225792.8907 γ2= -0.9699205898 A12= 52.9202340198 A21=232.1815385938 α12= 53.1702387795 α21=233.1514591836 A12=α12+ γ1= 52.9202340198 φ1= -30.0 λ1= 30.5 Y1= 48141.1054 X1= -3335429.2308 γ1= -0.2500047597 φ2= -28.0 λ2= 29.0 Y2= -98162.7839 X2= -3113371.4602 gam2= 0.4695087290 A12=326.3700225446 A21=147.0973795263 α12=326.6200273044 α21=146.6278707973 A12=α12+ γ1=326.3700225446 φ1= -30.0 λ1= 30.5 Y1= 48141.1054 X1= -3335429.2308 γ1= -0.2500047597 φ2= -32.0 λ2= 29.0 Y2= -94282.5003 X2= -3558115.1919 γ2= 0.5299579646 A12=212.3524278031 A21= 33.1251350206 α12=212.6024325628 α21= 32.5951770561 A12=α12+ γ1=212.3524278031 φ1= -30.0 λ1= -30.5 Y1= -48141.1054 X1= -3335429.2308 γ1= 0.2500047597 φ2= -32.0 λ2= -31.0 Y2= -94282.5003 X2= -3558115.1919 γ2= 0.5299579646 A12=191.9669826036 A21= 12.2245420693 α12=191.7169778439 α21= 11.6945841047 A12=α12+ γ1=191.9669826036 φ1= -30.0 λ1= -30.5 Y1= -48141.1054 X1= -3335429.2308 γ1= 0.2500047597 φ2= -29.0 λ2= -32.0 Y2= -194466.7324 X2= -3225792.8907 γ2= 0.9699205898 A12=307.0797659802 A21=127.8184614062 α12=306.8297612205 α21=126.8485408164 A12=α12+ γ1=307.0797659802 φ1= -30.0 λ1= -30.5 Y1= -48141.1054 X1= -3335429.2308 γ1= 0.2500047597 φ2= -28.0 λ2= -29.0 Y2= 98162.7839 X2= -3113371.4602 γ2= -0.4695087290 A12= 33.6299774554 A21=212.9026204737 α12= 33.3799726956 α21=213.3721292026 A12=α12+ γ1= 33.6299774554 φ1= -30.0 λ1= -30.5 Y1= -48141.1054 X1= -3335429.2308 γ1= 0.2500047597 φ2= -32.0 λ2= -29.0 Y2= 94282.5003 X2= -3558115.1919 γ2= -0.5299579646 A12=147.6475721969 A21=326.8748649794 α12=147.3975674372 α21=327.4048229439 A12=α12+ γ1=147.6475721969 210 Sebahattin Bektaş 5. Results and discussion In this study, formulas are given for how to obtain the bearing and the azimuth angle directly without any examination. The numerical examples also show no loss of precision in the use of direct formulas. It has been observed that the classical formulas (Equations 1, 2, 14, and 15) and the formulas we propose (Equations 6, 7, 20, and 21) give the same results in the calculations made with ten digits after the decimal point. It is suggested to use Equations 8, 9, 22, and 23) to avoid division by zero error. The advantage of the method is that no examination is required. In computer calculations, if…, then..., end…, blocks are not used when direct formulas are used. The if..., then…, end…, blocks reduce the execution speed in computer calculations. The classical formulas and the direct formulas we proposed were compared in terms of processing time. All were coded in MATLAB. The main characteristics of the hardware were Windows 10 Pro, Intel(R) Core(TM) i5- 2450M CPU @ 2.50GHz, 6 GB of RAM. The same bearing and the same azimuth angle were calculated 500 million times and the processing times are shown in Table 4 below. Table 4. Processing Time Bearing Angle Azimuth Angle Classic formulas 4.642 sec Eq. 14-Eq. 15 5.492 sec Eq. 1-Eq. 2 Proposed formulas 4.445 sec Eq. 20-Eq. 21 4.413 sec Eq. 6-Eq. 7 It has been observed that the processing time is slightly shorter in the direct formulas we recommend. This is because classical formulas need quadrant comparison. 6. Conclusion In this proposed method, formulas are given for how to obtain the bearing and azimuth angle directly without any examination. The need for comparison in classical formulas increases both the processing time and the possibility of error. Direct formulas are always more useful. While saving time on the one hand, the probability of making mistakes will also be seriously reduced on the other hand. We hope that the accuracy and non-examination of the direct formulas we proposed will increase the usability of our formulas. The formulas on the sphere surface, which are used to calculate the direct bearing and azimuth, can also be adapted to the ellipsoid surface. For future studies, researchers are advised to find more simple direct formulas. References Antes, E. (1990). Noch ein Beitrag zur Berechung des Richtungwinkels. Verm. wesen, 115. Bektaş, S. (2019). Direct bearing angles determination on globe. MOJ Civil Engi- neering, 5(4), 78–80. Bektaş, S. (2021). Jeodezi-1, Atlas Akademi Press. ISBN 978-605-7839-87-9. Breuer, P., Hirle, M., & Joeckel, R. (1985). Basic- programmierbare Taschen- rechner und Handcomputer. Mitt.des Deutscher Verein für Vermes- sungswesen-Bayern, 37, Jahr. Sonderheft 1/1985, s.218-219 Gellert, W., Küstner, H., Hellwich, M., & Kastner, H. (1972). Kleine Enzyklopadie: Mathematik. Verlag Harri Deustch, Frankfurt/ M. und Zürich. Heck, B. (2003).  Rechenverfahren und Auswertemodelle der Landesvermessung: klassische und moderne Methoden. 3. neu bearbaitete und erweiterte Au- flage. Heidelberg: Herbert Wichmann, 473 pp. Hekimoğlu, Ş. (1991). Açıklık Açısını Bölge Sorgulaması Yapmadan Hesaplayan Yeni Formüller ve Kondüsyon Sorunu. HKMO Dergisi, Sayı, 70, Trabzon Theissen, R. (1990). Berechnung des Richtungswinkels t ohne Quadrantenabfrage. Z.fg. Verm. wesen 115,193-195 _Hlk97057942 _Hlk97058323 _Hlk97058734 _Hlk97059392 _Hlk97059724