Should the game come down to conversions of tries, as we believe it will, it’ll be the kicker who picks the optimal place to convert the try.
So how should a player (without a measuring tape or protractor) choose which place to kick a conversion from? He can choose from anywhere along the red dotted line as shown here (having scored a try at the goal line where the red dotted line begins):
Firstly, it can be seen that maximising the Hughes Angle H (named after the man who first tackled (!) this problem) will help:
Maximising this angle H will increase the margin for error and increase the chances of the ball going between the posts (we’ll deal with the height and angle of the kick in a second):
With a bit of Pythagoras and some other useful mathematical techniques (details of which are available in another post to go up this weekend for those interested), it can be shown that kicking from where the circles touch the conversion line (in other words, where the conversion line is a tangent to the circle) will maximise this angle:
It can then be shown, that the optimal kicking lines are:
Now, given that the ball has to travel over the bar, we can add this constraint to the maths. When a kicker kicks the ball, the ball will follow some line which can be traced by of a cone shape of some sort and we have a new angle R, the raised Hughes angle to consider:
There is a point where E, the angle of elevation of the kick, is minimised while still permitting a successful kick: this is the point from where the flattest feasible kick will be flying horizontally as it clears the crossbar. We then, as always, have a trade off, between wanting both to minimise E and to maximise H. The precise point where the raised angle R is maximised will be somewhere in between.
The long and the short of it means that for example, kicking with an elevation of E = 35 degrees, the effective angle is 18% smaller than the Hughes angle. By comparison, kicking with an elevation E = 20 degrees, the effective angle is only 6% smaller.
As Hughes himself puts it: “The moral is, all else being equal, and as long as the goals can comfortably be reached, the player should kick at as shallow an angle as possible.” So the kicker will need to try and minimise the angle of elevation of the flight of the ball (E) while also maximising ‘Hughes Angle’.
God Help the kicker when he starts placing the ball, because if he gets this right, the rest will surely follow.
So, familiarity with the pitch, its dimensions and scale will be a factor for the kickers on Saturday, now that Ian Madigan got the nod coupled with his academic background, Leinster will outmuscle Munster on the mathematics front and win by a conversion!