## Greatest Area given a 3D Boundary

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 $z=0$ and then applies this formula, $A =\frac{1}{2}*|\sum_{i=1}^{n}x_i*y_{i+1} - x_{i+1}*y_i |$ where $A$ stands for the area of a polygon with vertex coordinates in order ${(x_1,y_1),(x_2,y_2),...,(x_n,y_n)}\text{ and }x_{n+1} = x_1, \text{ }y_{n+1}=y_1$. .  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 $\oint_C xdy$ for the area.  I have a few projects that need doing first and I might never get back to this. 