Dear 100 Hour Board,
Hi! I have a mathy problem that I can't figure out how to solve. I would gladly do the legwork, I'm just not sure where to start. A pointer or two would be greatly appreciated.
Suppose I have a differentiable surface in three dimensions, defined by the function z = f(x, y). Given two points (x0, y0) and (x1, y1), I am seeking a function y = g(x) such that the integral of f(x, g(x)) with respect to x from x = x0..x1 is minimized.
I had the crazy idea of superimposing a grid (an infinite graph where each node is equidistant from each of its neighbors; squares or triangles or something) on the manifold and using a graph search algorithm such as A* to find the shortest path... and then taking the limit as the distance between each node approaches zero? But I would prefer an entirely symbolic method, if at all possible... or at least a numerical one.
Any help you can give me would be appreciated. :D I haven't taken Multivariable Calculus yet (this semester, though!), so if tools exist to solve this sort of problem, I'd appreciate even just a link to an explanation of them. I can use Khan Academy for the rest.
Thanks for your question! You've caused quite a stir amongst my mathy friends. I'll admit, the wording of your question is a little unclear to me. The way I interpret your question is to imagine a continuous 3D surface (no kinks), choose any two points along the surface, and then determine a function that expresses the minimal traversed arc length along the surface that connects those two points. While this might be an incorrect interpretation of your question, this is what I've put time into answering. If someone gave you this question to solve (a professor, a math club, whatever), then I'm sure I am interpreting your question incorrectly. The situation I described above is extremely advanced math, and the answer is no, not yet.
For starters, you'll learn about arc length integrals in Math 314, probably in the first few weeks. I consulted my old math book when I first started thinking about it. The textbooks teach you formulas for finding arc length along three dimensional traces (paramaterized curves), not 3D surfaces You can look those formulas up here. I visited the math lab on campus to talk to one of the TAs, who have historically been pretty nerdy when it comes to these things. I asked a math TA about an arc length formula for two points on a surface, and they said they didn't know. So I headed upstairs and tried to bother a pale grad student who brushed me off with a nasally "too busy to help" line. Next, I found Dr. Cardon in his office, one of my past math professors. Initially, he said he only had a short minute to entertain board questions. Thirty minutes later, we were going crazy on the white board with hypotheticals on how to derive such a function. I'll walk you through our thought process.
Let's imagine a simple scenario, where our surface is a 3D sphere, or a globe. If your two points are simply the north and south pole, the minimal path is simply any tie line that you see below (half of the circumference). Pretty easy, right? Airplanes use similar math when determining the flight path between two destinations.
Let's consider a more complex surface, the torus! Imagine the first point (red dot) on the outer rim of the donut, and the second point (blue dot) on the opposite side, on the interior. Deriving an expression that expresses the shortest possible line drawn between those two points is a lot harder now. At first I thought the problem could be solved by intersecting a plane between those two points, similar to the sphere surface. The intersection between a plane and a surface would yield a line, but not necessarily the minimum arc length (it would depend on how the plane was rotated). With this surface, the minimum arc length expression could still be determined with other methods in mathematics, only because the surface is so uniform and behaves so predictably.
Consider a more complex torus. With this twisted surface, the same algorithmic system that dictates the minimal traversed arc length between two points on the regular torus now breaks down.
Consider the double torus. Imagine the first point being on the top center, and the second point directly underneath (not visible below). From inspection, we can predict that the minimal path between those points would trace a line through either hole, wrapping around to the bottom. How would a single function be able to negotiate which hole the path goes through, if both are equal? Dr. Cardon presented this as a fundamental problem of seeking such a function for a 3D surface. This is a tricky part of mathematics at work here. Imagine a 2D squiggly function with several local minimums and a global minimum. An algorithm that scans the squiggly line and searches for the lowest point will get stuck in a local minimum, not knowing that there are deeper points yet ahead. This is the same scenario at work on a complex 3D surface, such as the double torus below. Any expression that can define the arc length, or the traversed line as a paramaterized function in space, would be incapable of determining if it was the absolute minimal distance. You could potentially arrive at a function that expresses an arc length between two points, but there's no guarantee that it is the shortest traversed arc length. As you probably guessed, the larger issue here is dealing with the complexity of the surface.
To further demonstrate the difficulty of finding such a universal expression, consider the triphombul repeating torudial sinking cylinder. I actually just made up that name. But really, putting two points on this surface and seeking the minimal traversed arc length between them becomes dramatically more difficult, depending on where the points are. Holding one point constant on the surface, the slightest move of the second point could require the line the go through another hole, or around it, to achieve minimal traversed distance.
Dr. Cardon and I spent some more time talking about hypothetical approaches for seeking a universal method that could determine the minimal arc length on all surfaces. I am convinced that one exists, and have been thinking day and night about it. Dr. Cardon recommended that I post your question on Stack Exchange, a math forum where professionals can help each other answer questions. You can view my question here. The responses reference Geodesics, which I'm still researching. One of the math professors at BYU researches similar problems of minimal surfaces on soap bubbles. His name escapes me. He could be a good person to talk to. I told Dr. Cardon that I would continue to work on the problem and let him know if I had any breakthroughs, to which he asked me to come to him right away if I did, because he believes this is a publishable topic, and there are likely others working on this (or similar) problem in universities around the world.
I've talked through the problem with several of my more brilliant friends, and we've spent lots of time coming up with possible algorithms that can solve this. It seems that our ability to define a function that works is always less than our ability to imagine a surface in which the algorithm breaks down. Needless to say, it's been hard to stop thinking about this, because I believe it can be solved.
For those that have read this far, and have no idea what any of this means, think of it this way: imagine you are at the Clyde Building and you want to walk to the Tanner Building by taking as few steps as possible. Which way do you go? As you scan the terrain ahead, you make decisions about which buildings to go around, right or left, knowing what comes next. How is it that you know that your path is the absolute shortest? Deriving a mathematical expression to find that shortest path involves programming/teaching/including your decision making process into a single formula. As you can imagine, there's a lot of complexity that goes into it.
I'm sorry if this wasn't anything close to what you were hoping for, and also for the delay in response. Because I'm now hooked, I will undoubtedly continue to work on this problem, so feel free to email me if you have any questions, comments, concerns, moans, groans, complaints, or childhood memories.
Dear Phaedrus and Arandur,
I am not the mathiest person in the world, but I made you another picture.