Written for Canadian Community News by Mike Sterling
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Driving between Port Elgin and Southampton on highway 21 there is a long curving transition called Spark's Corner. It has been the scene of a great many traffic accidents in the past. There is a similar curve just north of Tiverton. That one is kind of like a bent elbow.
Spark's corner has been greatly improved over a 60 year period. If you pay attention going from Port Elgin to Southampton, you hardly have to adjust the wheel as the curve seems to have the right shape and a decent little bank. This has to do with a good traffic engineer someplace crouched over a drafting board.
Traffic engineers prefer straight lines because they are safer. The world is not a rectangular grid.
These engineers don't like ragged twisting and turning curves, but they have to deal with the real estate that they are given.
If they are left alone and given a chance they will use some simple geometry to help transition from one curve or straight to another.
A Log Spiral showing every angle equal to 80 degrees. Of course any angle can be used and just a portion of the curve is needed for most transitions.
I suspect that at Spark's corner they are using a segment of the magical curve called a logarithmic spiral first investigated by Jacob Bernoulli. But, why the log spiral?
The reason is the log spiral is the only curve other than a circle which is equal angular with a ray coming from the centre. See Figure 1 where the rays form constant 80 degrees with the curve. Any angle can be used. The important thing is that it is ALWAYS constant.
The traffic engineers love this curve because the driver can just hold on to the steering wheel while following a long transition curve. There is no adjustment required on a perfect log spiral This is important especially in winter because 'fiddling' with the steering wheel can cause accidents as little transitions throw the driver off a bit and skids occur
The Euler Spiral used for transitions from the straight to the circular
Another magic and beautiful curve is called the Euler Spiral named after one of the greatest mathematician and scientist ever and my personal choice as at or near the topt. It is also called the Spiral of Cornu
Why is it important? You will see from Figure 2 the beauty of the curve, but also something just as important is hidden there.
Starting in the centre of the figure you have a straight line that gradually morphs into a circle. So traffic engineers use sections of this curve to enter and exit areas that are formed by circular arcs.
From a technical standpoint the centres of curvature of the log spiral form another log spiral, which is nothing short of magical.
The curvature of the Euler Spiral undergoes a linear transition which is easy for a driver to handle as it is gradual and winds tighter and tighter to become tangent to the target circle. So the driver gripping the wheel just turns it gradually at the same rate either tighter or looser coming into or out of the arc of a circle or for that matter a straight line that is tangent to the Euler spiral..
These two curves should be known together as Traffic Spirals. So as you are driving, please look for them. There won't be too many because most roads are slaves to the topography and the amount of real estate available.
Also, remember that there are some good engineers that may shake their head in worry about some curves that they cannot change.
The over arching idea here is that there is mathematics and good engineering at work everywhere. Did you know about it? Does it surprise you? Could this be exposed to young students using such things as Lego blocks and little race cars?
Notice too that art picks up on these shapes. Artists should know a little about the mathematics, it would improve some of their work.
I noticed the Euler Spiral used as a support on a chair the other day. Kind of neat and easy to form because of the linear curvature.
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Saturday, April 12, 2014