Equation for graph of constant helming Constant helming is (or was) a tactic used by ships in wartime to frustrate torpedo attacks by submarines.  It is a modified form of zigzagging.  It differs from zigzagging in that every time the helmsman completes a new zig, instead of bringing the rudder amidships, he maintains a slight angle on the rudder.  If an ordinary zigzag course results in a sine curve , then constant helming results in a modified sine curve that does not have a point of inflection at x=π.  Instead of inflecting at x=π, the curve continues to bend until the next zig, when it begins to curve in the opposite direction.
So if zigzagging can be described by y=sin(x), then what equation describes constant helming?   I presume the constant angle maintained on the helm after reaching maxima and minima at x=π/2, 3π/2, etc., until the next zig, is a parameter in this equation.
 A: If the helm is held constant between direction changes, then (I believe) the ship would trace out a circular arc. When the helmsman moves the rudder, the ship's position and direction initially remain the same, but the ship starts tracing out a circle pointing in the opposite direction. That means that the course overall will be a $C^1$ and piecewise $C^\infty$ curve where each piece is a circular arc.
You can construct the curve as follows: choose a circle which intersects the $x$-axis. Where it meets the $x$-axis, reflect the initial circle in the tangent line to the circle. This represents the helmsman swapping the rudder to the other side. Repeat this process to construct your curve. The centres of the circles will stay a constant distance from the $x$-axis, and the spacing will stay constant.
Note that the points where the curve meets the $x$-axis are essentially "discontinuous points of inflection". The second derivative jumps from positive to negative when the curve crosses the $x$-axis. You can't have the point of inflection at a local maximum distance from the $x$-axis, i.e. at "$x=\frac{\pi}{2}$", because the curve has to be parallel to the $x$-axis at the local maximum. The second derivative must be negative on both sides of the local maximum (unless the curve isn't differentiable there, which would correspond to the ship stopping and changing direction before it starts moving again).
