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Author Topic: Calculating LBW  (Read 3234 times)

DERRINALPHIL

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Calculating LBW
« on: 30/08/2012 23:44:54 »
I have a problem for a maths person: LBW.

I wish to calculate the window that a cricket ball, delivered by a bowler must pass through to gain an LBW decision. There are two variables: where the bowler lets the ball go from and where the batsman is standing.

The stumps are nine inches wide.

As a batsman moves forward from the popping crease he will reduce the chance of being out LBW. If the bowler delivers the ball from wide of the crease he also makes it more difficult to gain an LBW decision.

Could anyone calculate the window that the ball has to go through if the bowler delivers a ball one foot, two feet, three feet and four feet from the centre stump, with the batman standing on the popping crease?

Could anyone calculate the effect of the batsman moving forward from the popping crease ? The popping crease is four foot forward of the stumps so five, six and seven foot would be a help.

A genius could combine the two.

I have just taken on a role of advising a group of umpires, most of who are post graduate students and they ask these sorts of questions.

I do not have the computer skills to draw the problem. It sounded simple but I am too old.

damocles

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Re: Calculating LBW
« Reply #1 on: 31/08/2012 04:38:52 »
There is some doubt about which window you are referring to. I am assuming that the reference is to the horizontal position where the ball contacts the batsman's body.
I question your assertion that "there are two variables". Surely there would be several other variables of relevance. Perhaps the total number of variables could be reduced to about 6 or 7 in a realistic model. Some relevant factors are: Speed of the delivery, hardness or softness of the pitch, hardness or softness of the ball, state of wear of the ball's surface -- prominence of seams, polished, smooth, or rough surface, uniform or hemispherically differentiated wear, altitude of the venue (i.e. local air pressure),
amount and axis of rotation applied to the ball on delivery (spin, cut, and swing), orientation of seam on delivery.

Firstly, let us review the rules about LBW. The decision may not be given if the ball pitches outside the leg stump. That puts a rather simple limit on the "inside" limit of your window. The decision must also be based on "whether the ball would have hit the stumps" rather than "assuming that the ball continued a straight line trajectory". However the law does state that the umpire is to base the decision on the ball's trajectory at the time of impact. That is a very ambiguous instruction. The Law states:
Quote
(b)   In assessing point (e) in 1 above, it is to be assumed that the path of the ball before interception would have continued after interception, irrespective of whether the ball might have pitched subsequently or not.

That seems to mean (and this is probably its main point of application) that if the bowler pitches up a slow delivery with a lot of spin that would bring it in from outside off stump, and the batsman moves forward and places his leading pad so that the ball strikes it before pitching, then the umpire must assume that the spin would not have changed the ball's direction at all when it hit the pitch. But it is also clear that the law does allow the umpire to consider the normal effects of gravity in assessing the vertical component of "the path of the ball". What about a ball that is swinging due to aerodynamic effects? Should the umpire be assuming that the ball would have continued to curve on its current arc after the time of impact, or to have continued with a straight line horizontal component after impact when assessing its hypothetical "would have..." trajectory? For a human umpire, this particular issue almost certainly does not arise. S/he assesses it instinctively, rather than with a conscious calculation. But for a computer calculation it is quite another matter.

There are also two more major issues of judgement in the law:
(1) Was the batsman intending to hit the ball or to block with the pad?
if the batsman has "made a genuine attempt" to contact ball with bat, then the decision may not be given if the ball pitches outside line of off stump, otherwise the decision can be given even if the ball does pitch outside the line of off stump.
(2) Has the umpire seen enough of a trajectory after pitching to extrapolate it, and how should "benefit of the doubt" be apportioned?
If the ball contacts a player's pad or body shortly after the ball has pitched, there is a huge problem. Even on a "good pitch" there is a significant difference between the ball's trajectory, both direction and velocity, before and after pitching. Exploitation of this difference is a large part of a bowler's stock in trade. The umpire has to base a decision, in part, on "the path of the ball before interception". The umpire who makes an LBW decision stands close to the line of the trajectory, and so gets a good opportunity to judge at least the horizontal component of the new trajectory after pitching. Any doubt is to be resolved in the batsman's favour.
Quote
Each umpire shall answer appeals on matters within his own jurisdiction. If an umpire is doubtful about any point that the other umpire may have been in a better position to see, he shall consult the latter on this point of fact and shall then give the decision. If, after consultation, there is still doubt remaining, the decision shall be Not out.
But an assessment of "doubt" is a very individual judgement.

When I look again at your question, I am thinking that you are wanting an analysis of a couple of factors in a very simple model. I think you would be wanting to assume that there are no aerodynamic factors at work, that there is no spin on the ball, that the ball is a uniform sphere, and that the pitch is perfect. I think I can identify at least two other variables that need to be either included or specified at standard values, even in a simplistic model of the type that you seem to be aiming for.

One fairly obvious variable is the speed with which the ball leaves the bowler's hand. If a bowler releases the ball with a slow run and arm action, its bounce will not take it above stump height no matter where it pitches. A fast bowler, on the other hand, needs to pitch the ball fairly close to the batsman's toes to achieve a trajectory where the ball will not bounce over the stumps.

A second closely related one is the elasticity of the pitch. When the ball strikes the pitch it will lose both horizontal and vertical momentum. and probably a different proportion of each. In the real world (as opposed to that of your simplistic model) this will vary over quite a large range, depending on the state of the pitch, the state of the ball, and the speed of the delivery.

However, one of the variables you have indicated does not come into play in your simplistic model. Whether or not the batsman moves forward can affect only three things: whether or not
• the ball is contacted before or after pitching.
• the question of doubt in any judgement the umpire has to make about a changed trajectory after pitching.
• a question of judgement the umpire has to make about whether a serious attempt to play the ball was being made.

It has no effect on a hypothetical trajectory, especially in a simplistic model.

So my assessment of what I think you are needing is a diagram of a horizontal window of LBW region for point of impact. If factors like spin, swing, seam, cut, pitch irregularities are not to be taken into account, then the right hand and left hand boundaries are completely and simply geometrically defined by the positions of the stumps and the point of release of the ball. The rear boundary is trivial -- it is necessarily at the stumps.
The front boundary is a bit more complicated -- it depends on the point where the pitch of the ball would have led either to a bounce over the stumps (hard pitch, fast delivery) or where a third bounce would occur before the ball reached the stumps (soft pitch, slow delivery). This in turn will depend on delivery speed and pitch/ball elasticity.

Of course, if the ball strikes the batsman before pitching, there is a different type of LBW window. This is a vertical window, extending from ground upward. The upper limit is somewhere above the height of a straight line between height of release and stump height. It is higher for a slower ball. There is a bottom limit only if the batsman has moved a long way forward, the ball/pitch interaction is very elastic, and the pitch is very fast. That particular boundary is a little problematic, but would only rarely come into play.

damocles

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Re: Calculating LBW
« Reply #2 on: 31/08/2012 06:11:40 »
The diagram is from a simplistic geometric model which assumes

• A simple straight line horizontal component of the ball's path
• A right-handed batting stance
• A bowler bowling right arm over

If the ball's first impact is with the batsman's body or padding anywhere in the green area there should be a LBW decision.
If in the orange area then the umpire must make a judgement whether a serious attempt was made to play the ball.

The geometric calculations that go with this model are very simple. However the front boundary does depend on the speed of the delivery and the elasticity of the ball/pitch interaction (i.e. the amount of bounce, discounting its spin component)

Edited by Mod to increase diagram size
« Last Edit: 03/09/2012 11:39:44 by imatfaal »

DERRINALPHIL

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Re: Calculating LBW
« Reply #3 on: 31/08/2012 08:35:26 »

There are some physiological issues of perception, for example visual summation that I can go on about. These are not relevent to my request. The issue here is that I want to demonstrate ,to my umpires, that the horizontal window that the ball, being delivered by a right hand bowler, over the wicket, to a right hand batsman, narrows as the bowler delivers the ball from futher out from the stumps.

If a bowler delivers the ball from the centre stump he has a nine inch window to pass the ball through. If he was allowed to deliver the ball from ten yards wide of the stumps he could not hit the pads of a batsman with a ball that would both hit the stumps AND hit the pad "in line with the stumps."

graham.d

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Re: Calculating LBW
« Reply #4 on: 31/08/2012 10:15:37 »
Damocles, that was an impressive response and quite correct. Derrinalphil, what you ask for is a simple diagram showing the simple geometry of a straight line delivery that demonstrates it is harder to get an LBW when bowling wide of the stumps on the offside (it's even harder on the legside of course). You have to assume that the batsmen's front pad is at a certain distance from the stumps then it would be simple to do this by simply doing a scale drawing.

I do wonder whether this is simplifying to too greater degree though. Most people who have played cricket and umpired (at most levels of the sport) intuitively understand this without the need of such an explanation. They may not know the maths and geometry behind it but can make judgements based on experience. Actually the narrowing of the "window" because a bowler is delivering from close to the return crease is probably a much lesser effect that that that would result from a swing bowler capable of straightening a ball from wide on the crease. This is purely down to the umpire's judgement and experience - and as we can see from Hawkeye, even the best umpires sometimes get it wrong.

damocles

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Re: Calculating LBW
« Reply #5 on: 31/08/2012 10:21:50 »
The hawkeye program and the heuristic assumptions behind it would be very interesting to examine and analyse. Does it belong to MCC, or some media network, or some consultancy? And is its operation regarded as a valuable commercial secret? (It should be!)

damocles

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Re: Calculating LBW
« Reply #6 on: 31/08/2012 10:49:35 »
Ok, then, in that case the problem is really simple.

Firstly, if the ball is delivered from anywhere in line with the stumps the window is more than 9 inches wide -- it is 9 inches plus the diameter of the ball, because the ball will hit the stumps if it is within half a ball width of the outside stump, and it will also be adjudged to have pitched in line if it pitches within half a ball width or impacts within half a ball width. That works out at just less than 12 inches, and almost exactly 300 mm.

The problem is then fairly straightforward and simple to calculate, because you can use similar triangles. You can probably simplify my metric measures into Imperial ones.

Case 1 : Ball released from anywhere in line of stumps, probably about 19 m from the wicket (I gather that the no-ball rule means that the point of delivery is usually some distance short of the wicket-to-wicket distance). The width of the target area is 30 cm (22.86 + 7.2). If the batsman is impacted

1 m in front of wickets, the window is 300 * 18/19 = 284 mm
1.5 m ===> 276 mm
2 m ===> 268 mm
2.5 m ===> 260 mm

Case 2: If the ball is delivered from wide of the line of wickets and 19 m length then

a) at 0.3 m width, the most angled ball path will intersect the line of stumps at 9.5 m from wicket
impact 1 m in front ==> 268 mm window
1.5 m ==> 253 mm
2 m ==> 237 mm
2.5 m ==> 221 mm

b) at 0.6 m width, most angled intersection at 6.33 m
impact 1 m in front ==> 253 mm window
1.5 m ==> 229 mm
2 m ==> 205 mm
2.5 m ==> 182 mm

c) at 1.0 m width, most angled intersection at 4.38 m
impact 1 m in front ==> 231 mm window
1.5 m ==> 197 mm
2 m ==> 163 mm
2.5 m ==> 129 mm

graham.d

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Re: Calculating LBW
« Reply #7 on: 31/08/2012 11:59:36 »
Hawkeye seems to be a private development used by Sky for cricket and the BBC for Tennis at Wimbledon for example:

http://www.hawkeyeinnovations.co.uk/?page_id=1008

The software is very good I think. I imagine it uses a high order curve fit to match a trajectory and then uses this to predict the subsequent path. Through the air this would be very accurate but it's not so reliable in trying to predict the path of a ball after pitching, especially if pitched up so that the software has not got many points after the ball pitches. Predicting the effect of bounce on an imperfect ground surface will be subject to error. I don't think it tries to analyse the spin of the ball or the seam position for example.

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Re: Calculating LBW
« Reply #7 on: 31/08/2012 11:59:36 »