Curviness and Squigglyness -
New Paradigms in Automotive/Road Theory

Copyright © 1995-2004
Marc J. Zeitlin

Abstract:

On a recent business trip⁽¹⁾ for the Hewlett Packard Company's Mechanical Design Technology Council, I found myself and three colleagues driving down a relatively curvy road in Germany, near Stuttgart and Shloss Lichtenstein. As engineers⁽²⁾ tend to do, we began discussing the physical characteristics and parametric nature of the "squigglyness" of roads (having just climbed a steep switchback road up a small mountain).  Out of this discussion came the "Squigglyness, Curviness, and Motoring" hypothesis, known as SCAM.  I will attempt to apprise the reader of the development of the SCAM hypothesis, the methodology used in SCAM, as well as the parameters used to perform the critical calculations.

First, a narrative description of the nature of SCAM.  As anyone who has ever driven knows, roads have straight sections (where the radius of curvature is infinite) and curved sections (where the radius ofcurvature is not infinite [known as finite]).  Colloquially, people refer to roads which have a lot of curved sections as "curvy", and roads which do not as "straight".  People refer to roads which have left and right alternating curved sections as "squiggly".   Until now, we⁽³⁾ had not known of any methodology (or method [or just way]) of comparing the relative curviness or squigglyness of two roads.  Previous attempts at this have generally involved copious amounts of scientific hand-waving (with the consequent danger of eye injuries to those who don't wear eyeglasses) as well as yelling and threats.  We felt the problem cried out for some unambiguous mathematical equations, however arbitrary they might end up.

We felt intuitively that the radius of curvature of a curve, the distance between subsequent curves and the curve arc determined the "curviness"⁽⁴⁾. As each curve adds to the curviness of the road, the total curviness would involve a summation of the curviness contributions of each curve in the road.  However, we felt⁽⁵⁾ that a long road with particular curves didn't have as much curviness as a short road with the same curves, so we decided to normalize the summation by using the total length of the road.  In a flash of insight while taking the train⁽⁶⁾ to the airport at 8:30 A.M. the next morning, I realized that the number of lanes in the curve would also affect the apparent curviness.  I have unilaterally incorporated this into the SCAM.

The "squigglyness" of the road obviously depends on the curviness of the road (a straight road cannot have squigglyness, while a curvy road may or may not have squigglyness).  We determined that squigglyness also depended on the changing of the curvature from "left" curves to "right" curves (or "right" curves and "left" curves, depending on the automobile's direction of travel).   We called these changes "inflection points"⁽⁷⁾, and divided by the total length of the road to get an "inflection density".  Multiplying the curviness by the inflection density gives the squigglyness (measured in squiggs, which have units of meters^-4).  We consider perfectly circular roads and roads which have curves with zero radius curvature (corners) degenerate from both a curviness and squigglyness standpoint, and will not address them in this treatise⁽⁸⁾.   We consider roads with curves in only one direction (conical mountain ascents or decents) to have non-degenerate curvature, but degenerate squigglyness.

Here, then, I present the list of parameters used in this SCAM.

The equations of SCAM, as derived:

Curviness:

                        n
                      ----|       am
                      \       ------------
                      /         Lm Rm Xm
                      ----|
                       m=1
            Cv =  ---------------------------  meters-3
                                 D

Squigglyness:

                        Z
            Sq = Cv * -----  meters-4
                        D

The reader can use these relatively simple equation to determine the actual values of the curviness and squigglyness for any road.  Let's do that for a few simple examples of 10 kilometer long roads:

Example #1 (Virtual Interstate):

                   -12  -3                     -15  -4
        Cv= 30 x 10    m            Sq= 15 x 10    m

                        Cv
                       ---- = 2000 meters
                        Sq

Example #2 (Virtual Country Road):

        n  = 100
        D  = 10,000
        Z  = 60
        am = 0.5 rad
        Lm = 300 meters
        Rm = 100 meters
        Xm = 1

                     -9  -3                      -9  -4
        Cv = 167 x 10   m             Sq = 1 x 10   m

                        Cv
                       ---- = 167 meters
                        Sq

Example #3 (Virtual Dirt Mountain Road):

        n  = 400
        D  = 10,000
        Z  = 300
        am = 1.57 rad
        Lm = 100 meters
        Rm = 30 meters
        Xm = 1

                      -6  -3                       -9  -4
        Cv = 31.4 x 10   m            Sq = 942 x 10   m

                        Cv
                       ---- = 33 meters
                        Sq

We can see that Mountain Roads have much higher "curviness" values (by a factor of a million!)  and higher "squigglyness" values (by a factor of a hundred million [within an order of magnitude or so]) than Interstates.

We can also see that as the road gets "curvier and squigglyer" the S_q seems to asymptotically approach C_v.  Does this represent an innate characteristic of roads?  Who can say.

Clearly, we will need to obtain a new grant to perform wheel-on verification of these calculations.  The SCAM team has begun investigation of continuing grant proposal submissions for this express purpose⁽⁹⁾ at this time.

Conclusions:

We believe we have created a new paradigm in road parameterization techniques.   This SCAM will allow us to objectively compare road curviness and squigglyness, assign a number to each road and post this (as well as the speed and weight limits) for use by the observant driver.

Footnotes:

1. read: boondoggle #1
   
2. read: nerds
   
3. read: the four of us in the car
   
4. Other factors include the "Bank Angle" of the curve, and the   "Radius Change" in the curve.  We have arbitrarily relegated these factors to the scrap heap of engineering known as "second order effects" so that we could ignore them.  Engineers do this when they have only a slight clue as to the importance of the factor, and so make the assumption that they cannot have any importance.   We did not want to break with tradition.
   
5. no intuition this time
   
6. the S2 out of Hauptbahnhof
   
7. (a.k.a. "Nausea Activators").  One may determine the existance of an inflection point by drawing a straight line from the center of one curve to the center of the next. If the line crosses the road, you've found an inflection point.  If the line stays on one side of the road, you do not have an inflection point.
   
8. they also tend to have very little utility and a great deal of danger, respectively
   
9. read: boondoggle #2