Quasi-stationary sailplane trajectory problems are relatively simple, practical nonlinear optimization problems which for educational reasons will be of interest to a much larger audience than the sailplane pilot community alone. One interesting problem is the turning point problem which concerns the determination of the optimal velocities with which the sailplane pilot should fly when he wants to optimally round a turning point on a return or on a triangle flight in the presence of wind. This problem lends itself very well for solution by the convex-combinations approach discussed by the author in a number of recent papers dealing with other sailplane trajectory problems. This convex-combinations approach is a purely geometric approach that is based on an interesting relation between the average velocity over a broken trajectory and a certain convex combination of the vectors representing the velocities over the different legs of the trajectory. In the paper first the basic ideas governing quasi-stationary sailplane trajectory optimization problems as well as the convex-combinations approach are reviewed. Then the solution is derived of the turning point problem by means of this convex-combinations approach. With the help of a special graph, the turning point graph, introduced in the paper, this solution may be easily implemented in flight. This turning point graph may be constructed by graphical means by any sailplane pilot without too much trouble.
Keywords: Aerospace trajectories, nonlinear optimization