Away from reality (the unattended Goal):

                                Little nearer to reality( the static goal keeper)

                                Still nearer ( the moving goal keeper)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Away from reality (the unattended Goal):
 
 
 

        Imagine that you are Ronaldo , or for that matter Zidane if you like it.You have the ball at your feet . Allow yourself another delusion that the goal lies unattended (it happens sometimes).You being one of the better shooters you can put your shots wherever you like them to be put;ie they are deterministic.The situation presented above is one of the easiest that you can get in a match.You aim at any point within the goal say just a whisker within the post .For a moment the crowds go silent at the apparent slip up but the next moment ,as the ball trickles in , they explode into cheers.

        Enough of wishful thinking.Lets accept that we are humans and thus fallible. So however earnestly you may wish you cannot direct the ball to its intended spot exactly.So you do the next best thing .You bring with you that statistician friend of yours who makes you take thousands of shots at a pole that you have erected.He fits some probability distribution to the collected data to some degree of confidence .

        You tell this friend of yours to come up with that sweet spot where to aim , which will maximise your chance of scoring .The statiscian says that it can be done and offers another suggestion.He says why do all the hard work running around in that field .Get yourself a Soccer playing robot which will do the  menial task for you.He will program it to take the necessary steps when encountering this special situation while for the rest, you program it to behave like yourself (lets assume that it is possible).No doubt it is a tempting proposition and you accept it.

        The match begins .By a great stroke of luck the robot emulating your behaviour finds itself in a situation like the one discussed above.So everything is set.

    In reality something has gone horribly wrong.The robot doesnot even take the shot , it just stands there ,stuck ! When the whistle signifying the end of the match blows it is still standing rooted to the spot doing its Nth iteraration to calculate that spot where it was supposed to aim.

        The above strategy just wouldnt work.Embarassed  the friend of yours offers to make for you a table of solutions for a huge set of discrette points evenly distributed over the field. But even that will not help as the robots memory is far too small and inadequate for meaningful extrapolation and interpolation from that data.
    So you take up an AI route to find some solutions. First you review what the statistician had done.
 
 

shooter has got the ball at (x , y)
the goal posts are at  (0 , a)  ( 0 , -a)
the shooter aims at a point (0 , h)
Shooters angular accuracy is given by some PDF  :      f

(i still havent learnt the art of putting mathematical formulations in html pages.)