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math

Curvature

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This post is the result of my first experiment with the capability of including TeX in WordPress blogs (that capability is the reason I switched from Blogger).  This article will eventually appear as an example in abstractmath.org with lots of links to posts in the website that are germane to reading and understanding the article.

parabolasmall

Measuring bending

The curve y=x^2 (graph above) has a fairly sharp bend near the origin but as you move away from the origin in either direction it looks more and more like a straight line.   To measure this bendiness, each curvey=f(x) in the real plane has an associated curvature function  that measures how bent the curve is at each point.  (For this to work, f must have first and second derivatives.)  By definition, the curvature ofy=f(x) at x is given by:

\kappa_f(x)=\frac{\left|f''(x)\right|}{\left(f'(x)^2+1\right)^{3/2}}

The curvature function has the following three properties:

  • The curvature at any point on a straight line is 0.
  • The curvature at any point on a circle of radius r is\frac{1}{r}.  (Proving that this is true of the formula above is a nice freshman calculus exercise.)
  • The circle that best approximates a curve at a point (x,y) is the circle that is tangent to the curve at  (x,y) and that has radius 1/k, where k is the curvature of the curve at (x,y) .   This circle is called the osculating circle at (x,y) . 

Curvature of the parabola

You can calculate that the curve of the parabola y=x^2 at x is given by

\kappa(x) = \frac{2}{\left(4 x^2+1\right)^{3/2}}

For example, the curvature at (0,1) is 2, at the point (1/2, 1/4)  it is  \scriptstyle 1/\sqrt{2}\:\approx\: 0.71, and at (1,1) it is about 0.18.  The radii of the osculating circles are  1/2,  \sqrt{2}, and 5.59 respectively.  For large numbers the curvature is nearly 0; for example, at (10, 100) the curvature is about .00025.  To the eye the parabola near (10,100)  looks like a straight line.

This graph shows the osculating circles at x = 0, 1/2 and 1:

threecircles

You can see animated osculating circles at the Wolfram Demonstration Project (click on “web preview”).  From that site you may download Mathematica Player for free, which allows you to operate the slidebars yourself. 

This graph shows the parabola and its curvature function.

   curvature21

Turning the wheel

If you think of the graph of the curve as a path and you imagine bicycling along the path, the size of the curvature corresponds to the specific angle to the right or left the front wheel must be turned to stay on the path. 

A circle has constant curvature, so to bike around a circle means keeping the front wheel at a constant angle. 

As you can see the curvature of the parabola goes up gradually as you move from a negative x– value to 0, and after that it goes down gradually.   So biking along that path from left to right means gradually turning your wheel to the left, and then at (0,0) you gradually turn it back closer to straight front. 

Notice that going faster or slower makes no difference to the angle you must turn the wheel (as long as you don’t skid).  The curvature at a point on the path depends on the path (which doesn’t move), not on the speed of your bicycle moving along the path. 

Another curve

You may have a seen a model electric train in action.   What I am about to say applies particular to cheap model trains.  They tend to have two kinds of track pieces, straight segments and segments of circles of fixed radius.  You could make a layout with these pieces that looks like this:

circleline1

When the train starts at the left, it goes along a straight track (curvature 0) until it reaches the point (0, 2), where it enters a stretch of constant curvature 1/2.   At (0, 2) the curvature jumps instantaneously from 0 to 1/2.   Of course, “instantaneous” does not exist in the physical world (at this scale — don’t start carrying on about quantum jumps, please).   Where the track starts to curve, the front wheels of the train are forced by the change in the track to suddenly jump from facing straight front to angling right by a fixed amount.  If you have the track on the floor and stand looking down at it, and the train is going pretty fast, you will notice that the front car jerks to the right as it enters the curve.

You can see this in action in this You-Tube movie at 15 seconds and 1:14 minutes.

Fancier model trains have track pieces with varying curvatures.  Look up “model train” on YouTube and you will see dozens of them.

If a highway were laid out like the graph above, and you were driving pretty fast, then at (0,2) you would have to turn your steering wheel suddenly to the right and you would probably swerve a little.  But you probably can’t find any highways like that.  In the 1960’s a Kentucky highway engineer told me that they knew better; they used French Curves with curvature that increases continuously from 0.  Nowadays highway engineers lay out highways using CAD systems that can calculate the track transition curves directly.

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