A study on the design of half pipe

In this paper, we will determine the shape of a snowboard course (currently known as a “half pipe”) to maximize the production of “vertical air” by a skilled snowboarder. "Vertical air" is the maximum vertical distance above the edge of the half pipe. For designing the ideal shape of the half-pipe, we want to use the method calculus of variations to determine the function of the shape of half-pipe instead of using a special curve, such as circle, ellipse, parabola and so on. Hence, our design can meet the request of the question more objectively than the design using special curve.