One of my favorite summer mountain bike rides is to go “around the watershed” at the base of Mt. Ashland – 25 miles with 2000 feet of elevation gain. Thus the following question: What area did I encircle? In mathematical terms what area is enclosed by a closed path in three dimensions? I am sure there is some calculus of variations solution that interpolates a minimal surface given a 3D boundary but I decided on a simpler idea – one where I could apply my favorite formula. Look at this diagram. The figure enclosed by the bicycle trail is an ellipse which projects to a circle on the horizontal plane. Here is a slightly more complicated trail. My definition of the area enclosed is the largest area which the 3D path makes when projected on any plane.
This seems well-defined if the path is not too complicated. I built a Mathematica module to explore the idea as in this diagram. (The green line shows how a point is projected.)Sure enough if I varied the direction numbers of the plane I could vary the area until it achieved a maximum something like this. The module uses analytic geometry to project the path onto a plane and then rotates the plane to be coincident with the horizontal plane with and then applies this formula, where stands for the area of a polygon with vertex coordinates in order . . I struggled a bit with the projection – it didn’t look perpendicular – until I discovered that the 3D graphics box had different scales for different axes. Oh well. I now know three ways to project a point onto a plane.
My next mission if I choose to accept it is to try for an analytic solution for maximizing a projected area given a path, probably using for the area. I have a few projects that need doing first and I might never get back to this.