Cliff's question

Last Saturday, we were standing on the balcony in pleasant August sunshine with a beer in hand, watching Petersfield's pre-season warm-up match against a Chichester XV, refereed by the excellent Mike Gill.  In the third quarter, Petersfield scored a try close to the near touchline.  The kicker brought the ball back to the 10 metre line to gain the extra two points for the conversion. Cliff asked why they brought the kick so far back, and I agreed - it looked too far.  Indeed the kick missed.

 

 Obviously, the reason that the kicker brings the ball back from the try line is to make the apparent target as big as possible.  The parallax effect means that the goal posts look wider apart as you move back, but at the same time the distance for the kick grows, making the kicker's task more difficult. What's the ideal distance to bring the ball back?  Is there a simple way to work this out in the heat of the match?  Probably every professional kicking coach knows this, and maybe there are doctoral theses on the subject, but here's my take on the question. 

The geometry is simple - the width of the target depends on the distance back from the try line and the distance of the mark made by the referee from the mid-point of the goal posts.  The width of the target can be shown as the angle viewed by the kicker from the point where the ball is placed.  So assuming that the pitch has been marked up properly and is rectangular in shape, the following table shows this angle for various positions (distances in metres, angles in degrees):

Offset from centre line		0	5	10	15	20	25	30	35
Distance from try line	10	31.3	25.5	16.2	10.0	6.5	4.5	3.2	2.4
	15	21.1	19.1	14.8	10.8	7.8	5.7	4.3	3.3
	20	15.9	15.0	12.8	10.3	8.0	6.3	5.0	4.0
	25	12.8	12.3	11.0	9.4	7.8	6.4	5.3	4.3
	30	10.7	10.4	9.6	8.6	7.4	6.3	5.4	4.5
	35	9.1	9.0	8.5	7.7	6.9	6.1	5.3	4.6
	40	8.0	7.9	7.5	7.0	6.4	5.8	5.1	4.5
	45	7.1	7.0	6.8	6.4	6.0	5.4	4.9	4.4
	50	6.4	6.3	6.2	5.9	5.5	5.1	4.7	4.3

The widest angle for each 5 metre distance (wide and back) is highlighted in the table.  With the crossbar at a height of 3 metres, I don't think many people will choose to kick from closer than 10.

So on this basis, it looks as though the widest angle is always given by taking the ball back as far out as the mark is from the centre of the posts.  On a full-sized pitch, this means that a line through the crossing point of the 15 metre dashed lines (20 metres from the centre of the pitch) and the 22 metre line is very close to the best apparent target.  And conversions from the touchline should never be further back than about 5 metres short of the 10 metre line.

The other important factor in deciding whether to kick, and where from, is the distance.  A kick taken too far back will need a huge knock.  Here's the distance table with the same squares highlighted as in the table above:

Offset from centre line		0	5	10	15	20	25	30	35
Distance from try line	10	10.0	11.2	14.1	18.0	22.4	26.9	31.6	36.4
	15	15.0	15.8	18.0	21.2	25.0	29.2	33.5	38.1
	20	20.0	20.6	22.4	25.0	28.3	32.0	36.1	40.3
	25	25.0	25.5	26.9	29.2	32.0	35.4	39.1	43.0
	30	30.0	30.4	31.6	33.5	36.1	39.1	42.4	46.1
	35	35.0	35.4	36.4	38.1	40.3	43.0	46.1	49.5
	40	40.0	40.3	41.2	42.7	44.7	47.2	50.0	53.2
	45	45.0	45.3	46.1	47.4	49.2	51.5	54.1	57.0
	50	50.0	50.2	51.0	52.2	53.9	55.9	58.3	61.0

So you can see that a 10 metre error in the positioning of a touchline conversion will change a 49.5 metre kick to 57 metres, as well as making the angle worse.  On the other hand, taking the same kick from just outside the 22 metre line will worsen the angle by only 0.3 degrees (next to nothing) and make the kick 6.5 metres shorter - could be important for some kickers. 

All of these calculations assume that a kicker can hoof the ball 50-60 metres - easier at altitude - and they ignore weather conditions and pitch slopes that will lead to a more complex decision-making process.  But it does look as though many kickers are making their task more difficult.

Feedback welcome.