Is there a way to construct a tractrix curve geometrically without resorting to drawing individual X and Y points based on the parametric equation?
Theoretically. Check this page for an explanation of what a tractrix is–it results from a geometric construction.
Thanks, but that doesn’t really help. I found a number of sites on the internet explaining the origin of the curve, and its real-world significance. What I am looking for is a way to lay it out on a surface for cutting, without having to define the geometry in terms of a number of X and Y points and interpolating between them for the best smooth curve.
I guess the only thing I can do is CAD draw it using x and y points based on the equation, with splines between the points, and then transfer that to the surface. I just thought that there might have been a pencil and compass shortcut to recreating this.
The definition of a tractrix is based on a construction, so you can do it. Take a compass that will hold its distance and place its point on the origin, and the pencil at (0, 1). Start dragging the point of the compass along the x-axis, and the pencil will trace out a tractrix.
It would be better if you stuck a rubber band around the compass towards the bottom, I think.
The problem with that approach is that, in trying to hold the compass upright, I may inadvertently apply a torque to it about the z axis, which would pull the trace off the curve. I suppose I could devise an apparatus that used the string idea to plot this, but for that much effort, it would be just as easy to plot a bunch of points and connect them.