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Accuracy of Cameras on Bridges etc.

 
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M8TJT
The Other Tired Old Man
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PostPosted: Thu May 14, 2009 8:11 pm    Post subject: Accuracy of Cameras on Bridges etc. Reply with quote

Having seen a couple of questions along the lines of ‘How much difference in speed measurement does the height of the camera make when it is sited on an over bridge make.’ I decided to crack on with my trusty calculator and do some sums. Here are my findings. (And the short answer is: ‘Not Much, and it’s in your favour’)

Consider a car approaching an observer on the surface of a straight, flat road with the observer looking straight down the road at the approaching car. Its velocity towards that point (radial velocity) is the same as the speed of the car and the difference between the direction that the car is travelling and the direction that the observer is looking (angle of approach ) is zero (I.e. Directly towards the observer). Strictly speaking the difference is 180 degrees, but this makes no difference to the calculations.

Now consider a car travelling along the top bar of a letter T with the observer at the bottom of the upright. Whilst it is travelling towards the top of the upright, its range to the observer is closing, and when it is travelling away from the top of the upright, its range to the observer is lengthening. I.e. whilst approaching the upright, it has a positive radial velocity (towards the observer) and vice versa. When the car is exactly at the top of the upright, it is neither approaching nor receding from the observer, so its radial velocity (in the direction of the observer) is zero, and the angle between the direction of the car and the observer looking at the top of the upright (angle of approach) is 90 degrees.

Given the above, there must be a relationship between the angle of approach and the radial velocity towards the observer. (It’s the latter that is measured by a speed camera.) There is; it’s the cosine of the angle of approach.
Take the situation of an observer on a motorway over bridge (A Plod with a laser gun) at (say) 33ft directly above the carriageway and a car approaching the over bridge. Let’s draw a right angled triangle to sort out the angles involved. Draw a longish horizontal line representing the car’s distance to the over bridge with a short vertical line at the end of it representing the height of the Plod above the road. Label the junction of the two lines ‘C’, the other end of the long line ‘A’ and the top of the vertical line ‘B’. We now have a triangle ‘ABC’. Imagine that the car is passing point ‘A’ at exactly 70 mph, the Plod is at point ‘B’ and point ‘C’ is on the carriageway immediately below the plod.

The angular difference (angle of approach) between the direction of the car (A-C) and the direction that the plod is looking (B-A) is defined as the angle ‘CAB’ or angle ‘A’ i.e. the angle between the road surface and the straight line between the car and the plod. It’s the cosine of this angle that determines the velocities.
Ok so let’s now shove in some figures. We have said that the height ‘BC’ is 33 feet. If we now assume that Mr. Plod waits until the car gets to about 50 yards from the bridge (point C) we now have a right triangle with base 150 feet and height of 33 feet. Calculate the angle A from this as arcTan(BC/AC) = arcTan(33/150) = 12.4 degrees. Cos 12.4 = 0.9766. So the radial velocity of the car towards the plod is 70x.9766=68.36 mph. This is also a pretty typical setup for a Gatso, except that the triangle is horizontal. You will notice that the velocity difference is in your favour, i.e. the gun is under reading your actual speed.

Lets put a different distance into the equation. Mr. Plod hits you at 100 yards. arcTan(33/300)=6.28 degrees. Measured radial velocity = 70 x Cos(6.28) = 69.6mph
One more? Mr. Plod hits you at 100 yards but is now 66 feet above the road (20m) arcTan(66/300) = 12.4 degrees. 70xCos(12.4) = 68.36 mph.

Now let’s have a look at approaching a Mr. Plod on his bridge up a one in 25 hill (fairly steep) again at 70 mph being ‘shot’ at 50 yards as before. The horizontal part of the triangle that you drew above is no longer horizontal, but droops below the horizontal by the angle defined by the 1/25 hill = arcTan(1/25) = 2.3 degrees. So the new angle between the road and the vertical to Mr. Plod is now 92.3 degrees not 90 as it was before. We still need to calculate the angle at A to determine the radial velocity. This gives us an obtuse triangle to which we have to apply the Cosine law to find side AB and then the Sine law to find the angle A
Cosine law states c squared = a squared + b squared – 2ab Cos C. Where ‘a’ is the side opposite angle ‘A’ etc
So calculate c = Sqrt(33 squared + 150 squared – 2x33x150 x Cos92.3) = 154.9 ft
Now use the Sin law to calculate angle A. SinA=Sin(C)/c x a = Sin(92.3)/154.9 x 33 = 12.29 degrees

Now apply the same rules as before; radial velocity = velocity x Cos approach angle = 70 x Cos(12.29) = 68.4mph, which is 0.02 mph different from ‘on the flat’.
So the hill makes no appreciable difference, even at 'worst case'..

Similar calculations can be done for GPS speed ‘errors’ up and down hill assuming that the GPS velocity readout is accurate* for horizontal travel**. So going up a hill of 1 in 5 (an angle of 11.3 degrees) if the GPS is reading is 30mph, divide this by Cos(11.3) to get road speed up the hill of 30.59mph (2%higher), or at 70mph (What car have you got?) 71.4mph.

Or even a 1 in 25 (2.29 deg) hill @ GPS 70 = 70.056mph
So again, the hill makes very little difference unless it's a very steep one.

Shocked Rolling Eyes Rolling Eyes Surprised

*From the Sirf Starlll spec: Accuracy: Speed <0.01 m/s, Heading <0.01° 0.01m/s is about 0.02mph. Even if you degrade this by a factor of 20, it gives 0.4mph which pretty much agrees with Garmin’s suggested accuracy below and is just over 1% at 30mph and better at higher speeds.

**Velocity measured by a GPS is inherently 3 dimensional, but consumer GPS receivers only report 2D (horizontal) speed on their readout. Garmin's specifications quote 0.1mph accuracy but due to signal degradation (Snip), perhaps 0.5mph accuracy in typical automobile applications would be what you can count on. (http://gpsinformation.net/main/gpsspeed.htm)
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DennisN
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PostPosted: Thu May 14, 2009 11:04 pm    Post subject: Reply with quote

Yippeeeee! Now we've got the perfect answer next time we get got by a camera - we chuck cosines and sines at them.

Wasn't it something about Adjacent squared, plus Opposite squared = Hippopotamus squared?
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Border_Collie
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PostPosted: Fri May 15, 2009 1:10 am    Post subject: Reply with quote

Several months ago I wrote:-

Quote:
I read a while back that when cameras are placed on bridges, readings could not be taken within, I believe, 200 yards, because of inaccuracies caused by the downward angle used on the camera. (Do you think I can find it again by Googling? No.).


I still can't find it.
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Oldboy
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PostPosted: Fri May 15, 2009 6:41 am    Post subject: Reply with quote

Lost_Property wrote:
I still can't find it.
Looking for This?
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M8TJT
The Other Tired Old Man
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Location: Bexhill, South Sussex, UK

PostPosted: Fri May 15, 2009 8:19 am    Post subject: Reply with quote

DennisN wrote:
Wasn't it something about Adjacent squared, plus Opposite squared = Hippopotamus squared?

Yes. It's the basis of the cosine rule, with -2ab aqnd Cos(A) thrown in.
Good old Mr. Pythagoras. Wink
@LP
Now you know how big the errors are, but they are in favour of the driver. Very Happy
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DennisN
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PostPosted: Fri May 15, 2009 4:46 pm    Post subject: Reply with quote

M8TJT wrote:
It's the basis of the cosine rule, with -2ab aqnd Cos(A) thrown in.

Surely that should be "aqnd" squared? (I think? - Although when I passed GCE O level, my maths mistress accused me of cheating! I hadn't, but neither she nor I had any idea how I did it. Apart from a deep seated natural inability, I also missed calculus because of a rugby back injury). Surprised
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Border_Collie
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PostPosted: Fri May 15, 2009 5:38 pm    Post subject: Reply with quote

Yes, that's the one Oldboy.
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M8TJT
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PostPosted: Fri May 15, 2009 7:40 pm    Post subject: Reply with quote

Lost_Property wrote:
Yes, that's the one Oldboy.

The old ones are always the best. Luckily the laws of physics don't change much these days Very Happy
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