Problem Description

Design a roadbed which will allow "square wheels" to roll smoothly.

Background & Techniques

While browsing for information about the catenary shape of the Gateway Arch recently, I ran across a paragraph in the Wikipedia article on Catenary curves which mentioned that regular polygons rolling over a series of catenary shaped bumps could keep their centers at a constant height.   Not at all practical of course, but it motivated me to see if I could design such a roadbed. 

Designing the road turned out to be a good mental exercise.  It was clear early on that there were two constraints on each "bump".  First, as the polygon rolled on a level surface, its center would vary in height from the perpendicular distance to an edge (at its low point) to the distance to a corner (at its high point).  So the height of  each bump had to be the difference of those two heights.  Also if the edge is to roll without slipping, the length of each bump must equal the length of each edge. 

Those two parameters, curve height, H, and curve length, S, should define our catenary.  The problem is, the standard definition of a catenary does not include either of these parameters as independent variables.   Look it up and you'll find and equation for catenaries  like Y=C cosh(X/C).  No H or S in sight as  independent variables!   Y, which could be the curve height if we have the correct origin, is determined by X, the horizontal distance, and C, some sort of width controlling parameter.    Cosh(X/C), the hyperbolic cosine is  (e^x/c + e^-x/c)/2 by definition.   Note that negative X values give the same result a positive values, so the curve is symmetric about X=0 and we only need to work with 1/2 of the curve.  Also, height increases as X increases and is at its minimum  when X=0.  For X=0, Y=C ( (e⁰ + e^-0)/2 =C.  In order to have the expression measure height directly, we need to shift the curve down by C units so that it crosses the Y axis at 0.  If we subtract C from the expression, we can  replace Y with H (for height). The modified equation then is H=C cosh(X/C) - C.  

We have an expression for H, how about S, the arc length?  There is an expression for that also.  In the interval 0 to X, the arc length  S  = C sinh(X/C).  Sinh, hyperbolic sine, is defined as (e^x - e^-x)/2.   Let's denote  the specific X value where height and arc length both match out target values as A.  

So given H and S, we'd like to solve the two equations  C cosh(A/C )- C - H = 0 and C sinh(A/C) - S = 0 for A and C.  Note we are solving the upright version of the curve, but the inverted catenary will have exactly the same C and A values.   Also, we are working with only half of the "bump" so our target arc length will be half the length of a polygon side.

With the problem finally defined, now we can work on a solution.  The "Find catenary parameters" page has three solution techniques, developed and listed in the order they were developed:

I display Hans' derivation on the results page in the program so I won't repeat it here.   How he came up with the derivation is a mystery to me, but it all looks valid, and the results agree with (are better than) the results from the preceding methods.   .  

 Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

Programmer's Notes

Since the math discussion was quite long, I'll keep these notes brief.

Running/Exploring the Program 

Suggestions for Further Explorations