Hodograph Turtles

In classical turtle graphics a line is drawn to connect the turtle’s position vector before and after executing each FORWARD command. A hodograph turtle shadows the classical turtle and draws a line connecting the classical turtle’s direction vector before and after executing each TURN command. Here we study examples of the hodograph turtle in action along with several extensions. We show that some shapes are easier to generate using a hodograph turtle instead of the classical turtle. More importantly, we can extract information about the program of the classical turtle from the geometry generated by a hodograph turtle. We shall see that especially for complicated turtle paths such as fractals, the path of a hodograph turtle is often much simpler and easier to understand than the path of the classical turtle. This simplicity reflects the simplicity of the underlying turtle program which is not always evident from the actual fractal path traversed by the classical turtle. We also show how to extend the power of the hodograph turtle so that the hodograph turtle can draw the same class of fractals as the classical turtle.