On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the di...On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the direction of the unknown point. The requirement must be fulfilled: the distance from each point to an unknown point must be equal to the sum of the length of the segment outgoing from this point, and some increment, the same for all three segments. In the article, the conditions for the solvability of a geometric problem are established by the methods of spherical trigonometry and vector algebra. It is proved that when they are fulfilled, the problem is always solvable. The number of solutions is two, except in rare cases where there is only one solution. A solution method is presented. One of the practical applications is the problem of determining the time and location of a cloud-to-ground lightning discharge, which is directly reduced to this problem.展开更多
文摘On the unit sphere, the geometric problem of calculating the position of a point relative to three given points is considered. We know the length of three spherical segments that go out from the given points in the direction of the unknown point. The requirement must be fulfilled: the distance from each point to an unknown point must be equal to the sum of the length of the segment outgoing from this point, and some increment, the same for all three segments. In the article, the conditions for the solvability of a geometric problem are established by the methods of spherical trigonometry and vector algebra. It is proved that when they are fulfilled, the problem is always solvable. The number of solutions is two, except in rare cases where there is only one solution. A solution method is presented. One of the practical applications is the problem of determining the time and location of a cloud-to-ground lightning discharge, which is directly reduced to this problem.
