Abstract
The great circle is of special importance as the shortest distance between two points in navigation. Navigating officers must always be able to calculate the path of the great circle and the turning points on that path. One of the drawbacks of calculations in the path of the great circle is the large volume of calculations and the uncertainty of the correct result. In this research, using the method of composite triangles, the path of the great circle is solved, and two general methods for calculating the rotation points of the great circle, which include a) calculating the rotation point with equal longitude difference and b) calculating the rotation point with distance difference equal is used. Next, using the method of equal longitude difference in calculating the points of rotation, a fast and reliable relationship has been developed to calculate the points of rotation. The premise of the developed method is that the point of rotation is always equal to the difference in longitude between the origin and destination. The developed method has the advantage that it will be able to calculate the turning points on the path of the Great Circle by calculating only 5 sentences from Norris's book in a much shorter time and with fewer calculations.

INTRODUCTION

The sailing course is an angle between the true north and the ship's head. In normal maps, the road angle is constantly changing and therefore not navigable. The great circle route as the shortest route between two points cannot be used by sailors due to the constant change of route. Mercator first designed a map in 1569 where the lines drawn for the ship's route had the same angle along the route (rhumb line route). Today, this map is known as the Mercator map and is used for navigation. The path of the great circle on the Mercator map is a curve and the line is not straight, so it is impossible to draw it on the Mercator map without knowing the intermediate points. The accepted method for navigation on the Great Circle is that points of the Great Circle are selected and a rhumb line route is drawn between the selected points.
With the advent of computers and navigation aids, the solution of navigation problems by computers was proposed, one of the most important of which is solving the great circle problem by computer, for which a suitable algorithm should be considered. Solving the great circle with the vertex point is called an indirect method; because to solve the rotation points, additional operations are needed to find the vertex point; Therefore, researchers looked for direct methods that require less computation.

MATERIALS AND METHODS

First, the mathematical basics of the rhumb line path must be explained, and then the mathematical basics of the great circle path must be discussed. Rhumb line navigation is a navigational that is used to calculate the course and speed of a ship from one latitude to another latitude. The reason why this type of navigation is known as plane navigation is that the earth is considered a flat surface, so this type of navigation in a long distance (distance more than 600 miles) does not have the necessary accuracy and it is necessary Corrections for the sphericity of the earth should be made on the problem. Spherical trigonometry formulas are widely used to solve various spherical geometry problems in various fields such as navigation, aviation, geodesy, and astronomy. However, current spherical trigonometry formulas only express relationships between the sides and angles of a single spherical triangle. Many problems may involve different types of spherical shapes, such as spherical quadrilaterals and spherical polygons, which cannot be solved directly by adopting single spherical triangle formulas; Therefore, the combined spherical triangles approach is a suitable method for solving the path of the great circle.

RESULTS

The great circle is of special importance as the shortest distance between two points in navigation. The path of the great circle on the Mercator map is a curve and the line is not straight, so it is impossible to draw it on the Mercator map without knowing the intermediate points. The accepted method for navigation on the Great Circle is that the points of the Great Circle are selected and the rhumb line route is drawn among the selected points. In the navigation on the great circle, turning points on the route should be calculated using existing methods, then a rhumb line should be drawn and calculated between the turning points of the route. An important point in ocean navigation that uses the great circle route is determining the turning points. Most of the methods can not be used in training texts and cannot be used practically by an officer on the watch while watching the bridge. To solve the problem using the vertex method, both a longer path and more calculations are needed to reach the vertex point and after that, we need to perform the calculations related to the turning point, which increases the amount of calculations, in this research by using the method of combined triangles, the path of the great circle has been solved, and two general methods have been used to calculate the turning points of the great circle, which include:

a) calculating the turning point with an equal longitude difference
b) calculating the turning point with equal distance difference.

In the following, using the method of equal longitude difference in the calculation of turning points, a fast and reliable relationship has been developed for calculating turning points. The default of the developed method is that the turning point is always located in the middle of the longitude difference between the origin and the destination. The developed method can calculate the turning point width by only calculating 5 sentences from Norris's book.

DISCUSSION AND CONCLUSION

In the new method obtained in thin research, it can be seen that the officer of the watch succeeded in calculating a turning point between the origin and destination point by calculating only 5 terms from Norris's book. This method is significant and useful from two perspectives:

a) The calculation of the latitude and longitude of the turning point is very simplified and there is no need for long calculations of the vertex point, And the coordinates of the turning point are solved directly.
b) There is no doubt about the correctness of the calculations and the obtained result is completely true and the officer of the watch does not need any additional arguments to calculate the turning point.

According to the defined relationship, the officers who are sent to ocean navigation, no longer need to calculate the midpoints of the route through the vortex method for navigation in the ocean, and they can easily use this relationship to find the turning points on the path of the great circle