Sectrix Curves on the Sphere

In [3] a geometrical construction was defined from which a generalization of sectrix of Ceva comes. Let e be a line given by the origin O and angle α between axis x+ and e, as the angle of polar coordinates of e (Figure 3). Let the point A0 coincide with O. Let the point A1 be given on e such that the distance between the points O and A1 is 1. Let the point A2 be on axis x such that the distance of A1 and A2 is equal also to 1 and A2 6= O if it is possible. Then let the new point A3 be on the line e again such that A2A3 = 1 and A3 6=A1 if it is possible. Recursively, we can define the point Ai (i≥ 2) on the line e or on axis x if i is odd or even, respectively, where Ai−1Ai = 1 and Ai 6= Ai−2 if it is possible. For all α the point Ai exists. Figure 3 shows the first six points. If α is small enough then Ai is between points O and Ai+2. Let angle OAi+1Ai be αi, then αi = iα can be proved easily. If A1 is on the axis x we obtain a similar geometric construction (Figure 4). These constructions gave a new proof for some trigonometric connections [5].