For the Mathematically Minded
Graeme and Tim recently spent some time re-working some data obtained from G. W. Hemming's, "Billiards Mathematically treated," (1899). They claim that graphs drawn from this data confirm what most players come to realise almost by instinct.
Cueball Deflection
Graph 1 confirms the fact that a plain, unforced, half-ball contact gives a maximum deflection of 34°. It also confirms that the rate of change of deflection is much smaller in this region than for a thick or a thin contact* For example: if the contact is changed from half-ball into the region of three-quarter, or quarter ball, then the deflection changes by only about 6°. However, a change to a full-ball or a very thin contact, causes a 28° difference of deflection. Mathematically a half-ball in-off is 4 times as easy as a full or a thin in-off.
The above diagram shows, on the left, the maximum angle possible by a plain, non-forcing, half-ball stroke. If the object ball is struck fuller or thinner than a true half-ball, an in-off may still be scored. On the right is seen the angle of the half-ball pot which is 30°. This does not vary and goes some way towards explaining why pots are often more difficult than in-offs.
Conclusions
Mathematically it is from two to three times more difficult to make a half-ball pot than it is to make a half-ball in-off. Thin pots are one-and-a-half times more difficult than thin in-offs. These effects are mainly due to the fact that the object ball can be deflected up to 90° whilst the cue ball can be deflected only 34° without using screw, stun, or force. It would be easy to conclude that in-offs easier to make than pots. From a scoring point of view this may be generally so, but from the aspect of break making it is not necessarily so. To make a series of in-offs (George Gray style) the object ball must be kept under control by both contact and pace.
Please Note
There is nothing new in all this. It was all quite well known in 1899. The graphs do not take into account the initial half-ball recoil of the cue ball when it is deflected from the object ball; but that is another story that many solely snooker players are quite unaware of. And by the way, how we all long for the perfect cue action.
Thank you Graeme and Tim.
* The observation that cue ball contacts in the half ball area cause only slight differences in the deflection prompted me to make a small experiment. I got a friend of mine to play a series of long losers and I noted where the red ball contacted the side cushion. Twelve long in-offs were played -and scored - but resulted in a variation of just over 24 inches as shown in the diagram. The object ball never struck the same point twice though three of the strokes caused the red ball to strike the side cushion within the space of an inch or so. This shows quite clearly that there must have been a slight variation in contact on every stroke but that the deflection was not changed sufficiently to cause the stroke to be missed. This practical demonstration convincingly bears out your theory.
